Almost Periodic Solutions of Evolution Differential Equations with Impulsive Action

Abstract

In an abstract Banach space we study conditions for the existence of piecewise continuous, almost periodic solutions for semilinear impulsive differential equations with fixed and nonfixed moments of impulsive action.

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7.1 Introduction

We consider the problem of the existence of piecewise continuous, almost periodic solutions for the linear impulsive differential equation

$$ \displaystyle\begin{array}{rcl} & & \frac{du} {dt} + (A + A_{1}(t)u = f(t,u),\quad t\not =\tau _{j}(u),{}\end{array} $$

(7.1)

$$ \displaystyle\begin{array}{rcl} & & u(\tau _{j}(u) + 0) - u(\tau _{j}(u)) = B_{j}u + g_{j}(u),\quad j \in Z,{}\end{array} $$

(7.2)

where u: R → X,  X is a Banach space, A is a sectorial operator in X, A ₁(t) is some operator-valued function, {B _j } is a sequence of some closed operators, and {τ _j (u)} is an unbounded and strictly increasing sequence of real numbers for all u from some domain of space X. 

We use the concept of piecewise continuous, almost periodic functions proposed in [7]. Points of discontinuities of these functions coincide with points of impulsive actions {τ _j }. We mention the remarkable paper [18], where a number of important statements about the almost periodic pulse system were proved. Then these results were included in the well-known monograph [19]. Today there are many articles related to the study of almost periodic impulsive systems (see, for example, [1, 3]). In the papers [8, 23, 27, 28] almost periodic solutions for abstract impulsive differential equations in the Banach space are investigated.

In this chapter we consider the semilinear abstract impulsive differential equation in a Banach space with sectorial operator in the linear part of the equation and some closed operators in linear parts of impulsive action. Using fractional powers of operator A and corresponding interpolation spaces allows us to consider strong or classical solutions. Note that such equations with periodic right-hand sides were first studied in [17]. In equations with nonfixed moments of impulsive action, points of discontinuity depend on solutions; that is, every solution has its own points of discontinuity. Moreover, a solution can intersect the surface of impulsive action several times or even an infinite number of times. This is the so-called pulsation or beating phenomenon. We will assume that solutions of ( 7.1) and ( 7.2) don’t have beating at the surfaces t = τ _j (u); in other words, solutions intersect each surface no more than once. For impulsive systems in the finite-dimensional case, there are several sufficient conditions that allow us to exclude the phenomenon of pulsation (see, [19], [22]). Unfortunately, in a Banach space this conditions cannot easily be verified. In every concrete case one needs a separate investigation.

We assume that the corresponding linear homogeneous equation (if f ≡ 0, g _j  ≡ 0) has an exponential dichotomy. The definition of exponential dichotomy for an impulsive evolution equation corresponds to the definition of exponential dichotomy for continuous evolution equations in an infinite-dimensional Banach space [5, 9, 16]. We require that only solutions of a linear system from an unstable manifold be unambiguously extended to the negative semiaxis.

Robustness is an impotent property of the exponential dichotomy [5, 10, 16]. We mention the papers [4, 14, 25, 26], where the robustness of the exponential dichotomy for impulsive systems by small perturbations of right-hand sides is proved. In this chapter we prove robustness of the exponential dichotomy also by the small perturbation of points of impulsive action. We use a change of time in the system. Then approximation of the impulsive system by difference systems (see [9]) can be used. If a linear homogeneous equation is exponentially stable, we prove stability of the almost periodic solution of nonlinear equations ( 7.1) and ( 7.2). Following [17], we use the generalized Gronwall inequality, taking into account singularities in integrals and impulsive influences.

This chapter is organized as follows. In Sect. 7.2 we present some preliminary definitions and results. In Sect. 7.3, we study an exponential dichotomy of impulsive linear equations. Section 7.4 is devoted to studying the existence and stability of almost periodic solutions in linear inhomogeneous equations with impulsive action and semilinear impulsive equations with fixed moments of impulsive action. In Sect. 7.5 we consider impulsive evolution equations with nonfixed moments of impulsive action. In Sect. 7.6 we discuss the case of unbounded operators B _j in linear parts of linear parts of impulsive action.

7.2 Preliminaries

Let $(X,\|.\|)$ be an abstract Banach space and R and Z be the sets of real and integer numbers, respectively.

We consider the space $\mathcal{P}\mathcal{C}(J,X),\ J \subset R,$ of all piecewise continuous functions x: J → X such that

• i) the set {τ _j  ∈ J: τ _j+1 > τ _j , j ∈ Z} of discontinuities of x has no finite limit points;

• ii) x(t) is left-continuous x(τ _j + 0) = x(τ _j ) and there exists $\lim _{t\rightarrow \tau _{j}-0}x(t) = x(\tau _{j} - 0) < \infty.$

We will use the norm $\|x\|_{PC} =\sup _{t\in J}\|x(t)\|$, in the space $\mathcal{P}\mathcal{C}(J,X)$.

Definition 1.

The integer p is called an $\varepsilon$-almost period of a sequence {x _k } if $\|x_{k+p} - x_{k}\| <\varepsilon$ for any k ∈ Z. The sequence {x _k } is almost periodic if for any $\varepsilon > 0$ there exists a relatively dense set of its $\varepsilon$-almost periods.

Definition 2.

The strictly increasing sequence {τ _k } of real numbers has uniformly almost periodic sequences of differences if for any $\varepsilon > 0$ there exists a relatively dense set of $\varepsilon$-almost periods common for all sequences $\{\tau _{k}^{j}\},$ where $\tau _{k}^{j} =\tau _{k+j} -\tau _{k},j \in Z.$

By Samoilenko and Trofimchuk [21], the sequence {τ _k } has uniformly almost periodic sequences of differences if and only if τ _k  = ak + c _k , where {c _k } is an almost periodic sequence and a is a positive real number.

By Lemma 22 ([19], p. 192), for a sequence {τ _j } with uniformly almost periodic sequences of differences there exists the limit

$$ \displaystyle\begin{array}{rcl} \lim _{T\rightarrow \infty }\frac{i(t,t + T)} {T} = p& &{}\end{array} $$

(7.3)

uniformly with respect to t ∈ R, where i(s, t) is the number of the points τ _k lying in the interval (s, t). Then for each q > 0 there exists a positive integer N such that on each interval of length q there are no more than N elements of the sequence {τ _j }; that is, i(s, t) ≤ N(t − s) + N. 

Also, for sequence {τ _j } with uniformly almost periodic sequences of differences there exists $\varTheta > 0$ such that $\tau _{j+1} -\tau _{j} \leq \varTheta,j \in Z.$

Definition 3.

The function $\varphi \in \mathcal{P}\mathcal{C}(R,X)$ is said to be W-almost periodic if

• i) the strictly increasing sequence {τ _k } of discontinuities of $\varphi (t)$ has uniformly almost periodic sequences of differences;

• ii) for any $\varepsilon > 0$ there exists a positive number $\delta =\delta (\varepsilon )$ such that if the points t′ and t″ belong to the same interval of continuity and | t′ − t″ |  < δ, then $\|\varphi (t') -\varphi (t'')\| <\varepsilon;$

• iii) for any $\varepsilon > 0$ there exists a relatively dense set Γ of $\varepsilon$-almost periods such that if τ ∈ Γ, then $\|\varphi (t+\tau ) -\varphi (t)\| <\varepsilon$ for all t ∈ R that satisfy the condition $\vert t - t_{k}\vert \geq \varepsilon,k \in Z.$

We consider the impulsive equations ( 7.1) and ( 7.2) with the following assumptions:

(H1) :

A is a sectorial operator acting in X and $\inf \{Re\mu:\ \mu \in \sigma (A)\} \geq \delta > 0,$ where $\sigma (A)$ is the spectrum of A. Consequently, the fractional powers of A are well defined, and one can consider the spaces X ^α = D(A ^α) for α ≥ 0 endowed with the norms $\|x\|_{\alpha } =\| A^{\alpha }x\|.$

(H2) :

The function A ₁(t): R → L(X ^α, X) is Bohr almost periodic and Hölder continuous, α ≥ 0,  L(X ^α, X) is the space of linear bounded operators X ^α → X. 

(H3) :

We shall use the notation $U_{\varrho }^{\alpha } =\{ x \in X^{\alpha }:\ \| x\|_{\alpha } \leq \varrho \}.$ Assume that the sequence {τ _j (u)} of functions $\tau _{j}: U_{\varrho }^{\alpha } \rightarrow R$ has uniformly almost periodic sequences of differences uniformly with respect to $u \in U_{\varrho }^{\alpha }$ and there exists $\theta > 0$ such that $\inf _{u}\tau _{j+1}(u) -\sup _{u}\tau _{j}(u) \geq \theta > 0,$ for all $u \in U_{\varrho }^{\alpha }$ and j ∈ Z. 

Also, there exists $\varTheta > 0$ such that $\sup _{u}\tau _{j+1}(u) -\inf _{u}\tau _{j}(u) \leq \varTheta$ for all j ∈ Z and $u \in U_{\varrho }^{\alpha }.$

(H4) :

The sequence {B _j } of bounded operators is almost periodic and there exists b > 0 such that $\|B_{j}u\|_{\alpha } \leq b\|u\|_{\alpha }$ for j ∈ Z, α ≥ 0, and u ∈ X ^α. 

(H5) :

The function $f(t,u):\ R \times U_{\rho }^{\alpha } \rightarrow X$ is continuous in u and is Hölder continuous and W-almost periodic in t uniformly with respect to $x \in U_{\rho }^{\alpha }$ with some ρ > 0. 

(H6) :

The sequence {g _j (u)} of continuous functions $U_{\rho }^{\alpha } \rightarrow X^{\alpha }$ is almost periodic uniformly with respect to $x \in U_{\rho }^{\alpha }.$

Remark 1.

We assume that operators B _j are bounded and satisfy assumption (H4). Many of our results are valid if the B _j are unbounded closed operators X ^α+γ → X ^α for α ≥ 0 and some γ ≥ 0. We discuss this case in the last section.

We use the following generalization of Lemma 7 from [7] (also, see [6] and [19]):

Lemma 1.

Assume that a sequence of real numbers {τ _j } has uniformly almost periodic sequences of differences, the sequence {B _j } is almost periodic, and the function f(t): R → X is W-almost periodic. Then for any $\varepsilon > 0$ there exist a such $l = l(\varepsilon ) > 0$ that for any interval J of length l there are such r ∈ J and an integer q that the following relations hold:

$$ \displaystyle{\|f(t + r) - f(t)\| <\varepsilon,\ t \in R,\vert t -\tau _{j}\vert >\varepsilon,j \in Z,} $$

$$ \displaystyle{\|B_{k+q} - B_{k}\| <\varepsilon,\ \|\tau _{k}^{q} - r\| <\nu,k \in Z.} $$

If A is a sectorial operator, then (−A) is an infinitesimal generator of the analytical semigroup e ^−At. For every x ∈ X ^α we get e ^−At A ^α x = A ^α e ^−At x. Further, we shall use the inequalities (see [9])

$$ \displaystyle\begin{array}{rcl} & & \|A^{\alpha }e^{-At}\| \leq C_{\alpha }t^{-\alpha }e^{-\delta t},\ t > 0,\ \alpha > 0, {}\\ & & \|(e^{-At} - I)u\| \leq \frac{1} {\alpha } C_{1-\alpha }t^{\alpha }\|A^{\alpha }u\|,\ t > 0,\alpha \in (0,1],u \in X^{\alpha }, {}\\ \end{array} $$

where C _α  ∈ R is nonnegative and bounded as α → +0. 

Definition 4.

The function x(t): [t ₀, t ₁] → X ^α is said to be a solution of the initial-value problem u(t ₀) = u ₀ ∈ X ^α for Eqs. ( 7.1) and ( 7.2) on [t ₀, t ₁] if

• (i) it is continuous in [t ₀, τ _k ], (τ _k , τ _k+1], …, (t _k+s , t ₁] with the discontinuities of the first kind at the moments t = τ _j (u) of intersections with impulsive surfaces;

• (ii) x(t) is continuously differentiable in each of the intervals (t ₀, τ _k ),  (τ _k , τ _k+1), …, (t _k+s , t ₁) and satisfies Eqs. ( 7.1) and ( 7.2) if t ∈ (t ₀, t ₁), t ≠ τ _j , and t = τ _j , respectively;

• (iii) the initial-value condition u(t ₀) = u ₀ is fulfilled.

We assume that solutions u(t) of ( 7.1) and ( 7.2) are left-hand-side continuous; hence u(τ _j ) = u(τ _j − 0) at all points of impulsive action.

Also, we assume that in the domain $U_{\rho }^{\alpha }$ solutions of ( 7.1) and ( 7.2) don’t have beating at the surfaces t = τ _j (u); in other words, solutions intersect each surface only once.

7.3 Exponential Dichotomy

Together with Eqs. ( 7.1) and ( 7.2) we consider the corresponding linear homogeneous equation

$$ \displaystyle\begin{array}{rcl} & & \frac{du} {dt} + (A + A_{1}(t))u = 0,\quad t\not =\tau _{j},{}\end{array} $$

(7.4)

$$ \displaystyle\begin{array}{rcl} & & \varDelta u\vert _{t=\tau _{j}} = u(\tau _{j} + 0) - u(\tau _{j}) = B_{j}u(\tau _{j}),\quad j \in Z,{}\end{array} $$

(7.5)

where τ _j  = τ _j (0). Denote by V (t, s) the evolution operator of the linear equation without impulses ( 7.4). It satisfies V (τ, τ) = I,  V (t, s)V (s, τ) = V (t, τ),  t ≥ s ≥ τ. 

By Theorem 7.1.3 [9, p.190], V (t, τ) is strongly continuous with values in L(X ^β) for any 0 ≤ β < 1 and

$$ \displaystyle\begin{array}{rcl} \|V (t,\tau )x\|_{\beta } \leq L_{Q}(t-\tau )^{(\gamma -\beta )_{-} }\|x\|_{\gamma },& &{}\end{array} $$

(7.6)

where $(\gamma -\beta )_{-} =\min (\gamma -\beta,0),\ t-\tau \leq Q,\ L_{Q} = L_{Q}(Q).$ Moreover,

$$ \displaystyle\begin{array}{rcl} \|V (t,\tau )x - x\|_{\beta } \leq L_{\beta,\nu }(t-\tau )^{\nu }\|x\|_{\beta +\nu },\quad \nu > 0,\beta +\nu \leq 1.& &{}\end{array} $$

(7.7)

Using the proof of Lemma 7.1.1 from [9], p. 188, one can verify the following generalized Gronwall inequality:

Lemma 2.

a ₁ ≥ 0,a ₂ ≥ 0, and y(t) is a nonnegative function locally integrable on 0 ≤ t < Q with

$$ \displaystyle{y(t) \leq a_{1} + a_{2}t^{-\alpha } + b\int _{ 0}^{t}(t - s)^{-\beta }u(s)ds} $$

on this interval; then there is a constant $\tilde{C} =\tilde{ C}(\beta,b,Q) < \infty$ such that

$$ \displaystyle{y(t) \leq \left (a_{1} + \frac{a_{2}} {(1-\alpha )t^{\alpha }}\right )\tilde{C}(\beta,b,Q).} $$

We will use the following perturbation lemma.

Lemma 3.

Let us consider the perturbation equation

$$ \displaystyle\begin{array}{rcl} \frac{du} {dt} + (\gamma A + A_{2}(t))u = 0,& &{}\end{array} $$

(7.8)

where γ = Const > 0, A ₂ (t): R → L(X ^α ,X).

For Q > 0, there exists $\varepsilon _{0} > 0$ such that for all $\varepsilon \leq \varepsilon _{0}$ and $\vert \gamma - 1\vert \leq \varepsilon,\ \sup _{t}\|A_{1}(t) - A_{2}(t)\|_{L(x^{\alpha },X)} \leq \varepsilon$ the evolution operators V (t,s) of ( 7.4 ) and V ₁ (t,s) of ( 7.8 ) satisfy

$$ \displaystyle\begin{array}{rcl} \|V (t,s) - V _{1}(t,s)\|_{\alpha } \leq R_{1}(\varepsilon ),\ t - s \leq Q,& &{}\end{array} $$

(7.9)

where $R_{1}(\varepsilon )$ depends on Q,α, and $R_{1}(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0.$

Proof.

For definiteness let γ > 1. Solutions x(t) and y(t) of Eqs. ( 7.4) and ( 7.8) satisfy the following integral equations:

$$ \displaystyle\begin{array}{rcl} x(t) = e^{-A(t-t_{0})}x_{ 0} +\int _{ t_{0}}^{t}e^{-A(t-s)}A_{ 1}(s)x(s)ds& & {}\\ \end{array} $$

and

$$ \displaystyle\begin{array}{rcl} y(t) = e^{-A\gamma (t-t_{0})}x_{ 0} +\int _{ t_{0}}^{t}e^{-A\gamma (t-s)}A_{ 2}(s)y(s)ds.& & {}\\ \end{array} $$

Then

$$ \displaystyle\begin{array}{rcl} & & \|x(t) - y(t)\|_{\alpha } \leq \| (I - e^{-A(\gamma -1)(t-t_{0})})A^{\alpha }e^{-A(t-t_{0})}x_{ 0}\| + {}\\ & & +\int _{t_{0}}^{t}\|(I - e^{-A(\gamma -1)(t-s)})A^{\alpha }e^{-A(t-s)}A_{ 1}(s)x(s)\|ds + {}\\ & & +\int _{t_{0}}^{t}\|A^{\alpha }e^{-A\gamma (t-s)}(A_{ 1}(s) - A_{2}(s))x(s)\|ds + {}\\ & & +\int _{t_{0}}^{t}\|A^{\alpha }e^{-A\gamma (t-s)}A_{ 2}(s)(x(s) - y(s))\|ds \leq {}\\ & &\leq a_{1}(\varepsilon )\|x_{0}\|_{\alpha } + a_{2}\int _{t_{0}}^{t}(t - s)^{-\alpha }\|x(s) - y(s)\|_{\alpha }ds, {}\\ \end{array} $$

where $a_{2} = C_{\alpha }\sup _{s}\|A_{1}(s)\|_{L(X^{\alpha },X)}$ and $a_{1}(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0.$ By Lemma 2, there exists a positive constant K ₁ depending on α and Q such that

$$ \displaystyle{\|x(t) - y(t)\|_{\alpha } \leq K_{1}a_{1}(\varepsilon )\|x_{0}\|_{\alpha } = R_{2}(\varepsilon )\|x_{0}\|_{\alpha }.} $$

Lemma 4.

Let us consider Eq. ( 7.4 ) and

$$ \displaystyle\begin{array}{rcl} \frac{dv} {dt} + (A + A_{2}(t))v = 0,& &{}\end{array} $$

(7.10)

such that A ₂ : R → L(X ^α ,X) is a bounded and Hlder continuous function.

Then for Q > 0, there exists $\varepsilon _{0} > 0$ such that for all $\varepsilon \leq \varepsilon _{0}$ and

$$ \displaystyle{\sup _{t}\|A_{1}(t) - A_{2}(t)\|_{L(X^{\alpha },X)} \leq \varepsilon } $$

the evolution operators V (t,s) of ( 7.4 ) and V ₁ (t,s) of ( 7.10 ) satisfy

$$ \displaystyle\begin{array}{rcl} \|(V (t,s) - V _{1}(t,s))u\|_{\alpha } \leq R_{3}(\varepsilon )\vert t - t_{0}\vert ^{1-2\alpha +\delta }\|u\|_{\delta },\ t - s \leq Q,& &{}\end{array} $$

(7.11)

where $R_{3}(\varepsilon ) = R_{3}(\varepsilon,Q,\alpha )$ and $R_{3}(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0.$

Proof.

Denote by u(t) and v(t) solutions of ( 7.4) and ( 7.10) with initial value u(t ₀) = u(t ₀) = u ₀. They satisfy the inequalities

$$ \displaystyle\begin{array}{rcl} & & \|u(t) - v(t)\|_{\alpha } \leq \int _{t_{0}}^{t}\|A^{\alpha }e^{-A(t-s)}(A_{ 1}(s) - A_{2}(s))u(s)\|ds + \\ & & +\int _{t_{0}}^{t}\|A^{\alpha }e^{-A(t-s)}A_{ 2}(s)(u(s) - v(s))\|ds \leq \\ & &\leq C_{\alpha }L_{Q}\varepsilon \|u_{0}\|_{\delta }\int _{t_{0}}^{t} \frac{ds} {(t - s)^{\alpha }(s - t_{0})^{\alpha -\delta }} + C_{\alpha }\|A_{1}\|_{L}\int _{t_{0}}^{t}\frac{\|u(s) - v(s)\|_{\alpha }ds} {(t - s)^{\alpha }} \leq \\ & &\leq \varepsilon \| u_{0}\|_{\delta }R_{4} + C_{\alpha }\|A_{1}\|_{L}\int _{t_{0}}^{t}\frac{\|u(s) - v(s)\|_{\alpha }ds} {(t - s)^{\alpha }}. {}\end{array} $$

(7.12)

Applying Lemma 2 to ( 7.12), we obtain ( 7.11).

We define the evolution operator for Eqs. ( 7.4) and ( 7.5) as

$$ \displaystyle\begin{array}{rcl} U(t,s) = V (t,s)\ \mathrm{if}\ \tau _{k} < s \leq t \leq \tau _{k+1}& & {}\\ \end{array} $$

and

$$ \displaystyle\begin{array}{rcl} U(t,s) = V (t,\tau _{k})(I + B_{k})V (\tau _{k},\tau _{k-1})\ldots (I + B_{m})V (\tau _{m},s)& &{}\end{array} $$

(7.13)

if τ _m−1 < s < τ _m  < τ _m+1 < … < τ _k  ≤ t ≤ τ _k+1. 

It it easy to verify that for fixed t > s the operator U(t, s) is bounded in the space X ^α. 

Definition 5.

We say that the system ( 7.4)–( 7.5) has an exponential dichotomy on R with exponent β > 0 and bound M ≥ 1 (with respect to X ^α) if there exist projections P(t), t ∈ R, such that

• (i) U(t, s)P(s) = P(t)U(t, s),  t ≥ s;

• (ii) U(t, s) |  _Im(P(s)) for t ≥ s is an isomorphism on Im(P(s)), and then U(s, t) is defined as an inverse map from Im(P(t)) to Im(P(s));

• (iii) $\|U(t,s)(1 - P(s))u\|_{\alpha } \leq Me^{-\beta (t-s)}\|u\|_{\alpha },\ t \geq s,\ u \in X^{\alpha }$;

• (iv) $\|U(t,s)P(s)\|_{\alpha } \leq Me^{\beta (t-s)}\|u\|_{\alpha },\ t \leq s,\ u \in X^{\alpha }$.

If the system ( 7.4)–( 7.5) has an exponential dichotomy on R, then the nonhomogeneous equation

$$ \displaystyle\begin{array}{rcl} & & \frac{du} {dt} + (A + A_{1}(t))u = f(t),\quad t\not =\tau _{j},{}\end{array} $$

(7.14)

$$ \displaystyle\begin{array}{rcl} & & \varDelta u\vert _{t=\tau _{j}} = u(\tau _{j} + 0) - u(\tau _{j}) = B_{j}u(\tau _{j}) + g_{j},\quad j \in Z,{}\end{array} $$

(7.15)

has a unique solution bounded on R

$$ \displaystyle{ u_{0}(t) =\int _{ -\infty }^{\infty }G(t,s)f(s)(x)ds +\sum _{ j\in Z}G(t,\tau _{j})g_{j}, } $$

(7.16)

where

$$ \displaystyle\begin{array}{rcl} G(t,s) = \left \{\begin{array}{l} U(t,s)(I - P(s)),\ \ t \geq s,\\ - U(t, s)P(s),\ \ t < s, \end{array} \right.& & {}\\ \end{array} $$

is the Green function such that

$$ \displaystyle\begin{array}{rcl} \|G(t,s)u\|_{\alpha } \leq Me^{-\beta \vert t-s\vert }\|u\|_{\alpha },\ t,s \in R.& &{}\end{array} $$

(7.17)

Analogous to [9], p. 250, it can be proven that a function u(t) is a bounded solution on the semiaxis $[t_{0},+\infty )$ if and only if

$$ \displaystyle\begin{array}{rcl} u(t) = U(t,t_{0})(I - P(t_{0}))u(t_{0}) +\int _{ t_{0}}^{+\infty }G(t,s)f(s)ds +\sum _{ t_{0}\leq \tau _{j}}G(t,\tau _{j})g_{j},\ t \geq t_{0}.& &{}\end{array} $$

(7.18)

A function u(t) is a bounded solution on the semiaxis $(-\infty,t_{0}]$ if and only if

$$ \displaystyle\begin{array}{rcl} u(t) = U(t,t_{0})P(t_{0})u(t_{0}) +\int _{ -\infty }^{t_{0} }G(t,s)f(s)ds +\sum _{t_{0}>\tau _{j}}G(t,\tau _{j})g_{j},\ t \leq t_{0}.& &{}\end{array} $$

(7.19)

Now we estimate $\|G(t,s)u\|_{\alpha }$ for u ∈ X. Let t > s and τ _m−1 < s < τ _m ,  τ _k  < t < τ _k+1. Then

$$ \displaystyle\begin{array}{rcl} & & \|G(t,s)u\|_{\alpha } =\| U(t,s)(I - P(s))u\|_{\alpha } \leq \\ & &\leq \| U(t,\tau _{m})(I - P(\tau _{m}))\|_{\alpha }\|U(\tau _{m},s)u\|_{\alpha } \leq \\ & &\leq Me^{-\beta (t-\tau _{m})}L_{\varTheta }(\tau _{ m} - s)^{-\alpha }\|u\| \leq \tilde{ M}e^{-\beta (t-s)}\vert \tau _{ m} - s\vert ^{-\alpha }\|u\|{}\end{array} $$

(7.20)

and

$$ \displaystyle\begin{array}{rcl} & & \|G(s,t)u\|_{\alpha } =\| U(s,t)P(t)u\|_{\alpha } \leq \\ & &\leq \| U(s,t + 1)P(t + 1)\|_{\alpha }\|A^{\alpha }U(t + 1,t)u\| \leq \tilde{ M}e^{-\beta (t-s)}\|u\|.{}\end{array} $$

(7.21)

If t ₁ and t ₂ belong to the same interval of continuity, then

$$ \displaystyle\begin{array}{rcl} \|P(t_{1})u - P(t_{2})u\|_{\gamma } \leq \tilde{ M}_{1}\|t_{1} - t_{2}\vert ^{\nu }\|u\|_{\gamma +\nu }& &{}\end{array} $$

(7.22)

since as in [9], p. 247,

$$ \displaystyle\begin{array}{rcl} & & \|P(t + h)u - P(t)u\|_{\gamma } \leq \| P(t)u - V (t + h,t)P(t)u\|_{\gamma } + {}\\ & & +\|V (t + h,t)P(t)u - P(t + h)u\|_{\gamma } \leq {}\\ & &\leq \| (I - V (t + h,t))P(t)u\|_{\gamma } +\| P(t + h)(V (t + h,t)u - u)\|_{\gamma }. {}\\ \end{array} $$

Lemma 5.

Let the impulsive system ( 7.4 ) and ( 7.5 ) be exponentially dichotomous with positive constants β and M. Then there exists $\varepsilon > 0$ such that the perturbed systems

$$ \displaystyle\begin{array}{rcl} & & \frac{du} {dt} + (A +\tilde{ A}(t))u = 0,\quad t\not =\tilde{\tau }_{j},{}\end{array} $$

(7.23)

$$ \displaystyle\begin{array}{rcl} & & \varDelta u\vert _{t=\tilde{\tau }_{j}} = u(\tilde{\tau }_{j} + 0) - u(\tilde{\tau }_{j}) =\tilde{ B}_{j}u(\tilde{\tau }_{j}),\quad j \in Z,{}\end{array} $$

(7.24)

with $\sup _{j}\vert \tau _{j} -\tilde{\tau }_{j}\vert \leq \varepsilon,\sup _{j}\|B_{j} -\tilde{ B}_{j}\| \leq \varepsilon,\ \sup _{t}\|A_{1}(t) -\tilde{ A}(t)\|_{L((X^{\alpha },X)} \leq \varepsilon,$ are also exponentially dichotomous with some constants β ₁ ≤β and M ₁ ≥ M.

Proof.

In system ( 7.4) and ( 7.5), we introduce the change of time $t =\vartheta (t')$ such that $\tau _{j} =\vartheta (\tilde{\tau }_{j}),j \in Z,$ and the function $\vartheta$ is continuously differentiable and monotonic on each interval $(\tilde{\tau }_{j},\tilde{\tau }_{j+1}).$

The function $\vartheta$ can be chosen in piecewise linear form:

$$ \displaystyle\begin{array}{rcl} t = a_{j}t' + b_{j},\ a_{j} = \frac{\tau _{j+1} -\tau _{j}} {\tilde{\tau }_{j+1} -\tilde{\tau }_{j}},\ b_{j} = \frac{\tau _{j}\tilde{\tau }_{j+1} -\tau _{j+1}\tilde{\tau }_{j}} {\tilde{\tau }_{j+1} -\tilde{\tau }_{j}} \ \ \mathrm{if}\ \ t' \in (\tilde{\tau }_{j},\tilde{\tau }_{j+1}).& &{}\end{array} $$

(7.25)

The function $\vartheta (t')$ satisfies the conditions

$$ \displaystyle{\vert \vartheta (t') - t'\vert \leq \varepsilon,\ \vert \frac{d\vartheta (t')} {dt'} - 1\vert \leq 2\varepsilon /\theta.} $$

The system ( 7.4) and ( 7.5) in the new coordinates $v(t') = u(\vartheta (t'))$ has the form

$$ \displaystyle\begin{array}{rcl} & & \frac{dv} {dt'} + \frac{d\vartheta (t')} {dt'} \left (A + A_{1}(\vartheta (t')\right )v = 0,\quad t\not =\tilde{\tau }_{j},{}\end{array} $$

(7.26)

$$ \displaystyle\begin{array}{rcl} & & \varDelta v\vert _{t'=\tilde{\tau }_{j}} = v(\tilde{\tau }_{j} + 0) - v(\tilde{\tau }_{j}) = B_{j}v(\tilde{\tau }_{j}),\quad j \in Z.{}\end{array} $$

(7.27)

The system ( 7.26) and ( 7.27) has the evolution operator $U_{1}(t',s') = U(\vartheta (t'),\vartheta (s')).$ If the system ( 7.4) and ( 7.5) has an exponential dichotomy with projector P(t) at point t, then the system ( 7.26) and ( 7.27) has an exponential dichotomy with projector $P_{1}(t') = P(\vartheta (t'))$ at point t′. Really,

$$ \displaystyle\begin{array}{rcl} & & \|U_{1}(t',s')(1 - P_{1}(s'))\|_{\alpha } =\| U(\vartheta (t'),\vartheta (s'))(1 - P(\vartheta (s'))\|_{\alpha } \leq {}\\ & &\leq Me^{-\beta (\vartheta (t')-\vartheta (s'))} \leq Me^{2\varepsilon }e^{-\beta (t'-s')},\ t \geq s. {}\\ \end{array} $$

The inequality for an unstable manifold is proved analogously.

The linear systems ( 7.26), ( 7.27) and ( 7.23), ( 7.24) have the same points of impulsive actions $\tilde{\tau }_{j},j \in Z,$ and

$$ \displaystyle\begin{array}{rcl} & & \|\frac{d\vartheta (t')} {dt'} A_{1}(\vartheta (t')) -\tilde{ A}(t')\| \leq \| \frac{d\vartheta (t')} {dt'} A_{1}(\vartheta (t')) - A_{1}(\vartheta (t'))\| + {}\\ & & +\|A_{1}(\vartheta (t')) - A_{1}(t')\| +\| A_{1}(t') -\tilde{ A}(t')\| \leq K_{2}(\varepsilon ), {}\\ \end{array} $$

where $K_{2}(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0.$

Let $\tilde{U}(t',s')$ be the evolution operator for the system ( 7.23) and ( 7.24). To show that for sufficiently small δ ₀ the system ( 7.23) and ( 7.24) is exponentially dichotomous, we use the following variant of Theorem 7.6.10 [9]:

Assume that the evolution operator U ₁(t′, s′) has an exponential dichotomy on R and satisfies

$$ \displaystyle\begin{array}{rcl} \sup \limits _{0\leq t'-s'\leq d}\|U_{1}(t',s')\|_{\alpha } < \infty & &{}\end{array} $$

(7.28)

for some positive d. Then there exists η > 0 such that

$$ \displaystyle{\|\tilde{U}(t',s') - U_{1}(t',s')\|_{\alpha } <\eta,\ \ \mathrm{whenever}\ \ t - s \leq d;} $$

the evolution operator $\tilde{U}(t',s')$ also has an exponential dichotomy on R with some constants β ₁ ≤ β, M ₁ ≥ M. 

To prove this statement, we set for n ∈ Z

$$ \displaystyle{t_{n} = s' + dn,\quad T_{n} = U_{1}(s' + d(n + 1),s' + dn),\ \tilde{T}_{n} =\tilde{ U}(s' + d(n + 1),s' + dn).} $$

If the evolution operator U ₁(t, s) has an exponential dichotomy, then $\left \{T_{n}\right \}$ has a discrete dichotomy in the sense of [9, Definition 7.6.4].

According to Henry [9], Theorem 7.6.7, there exists η > 0 such that $\{\tilde{T}_{n}\}$ with $\sup _{n}\|T_{n} -\tilde{ T}_{n}\|_{\alpha } \leq \eta$ has a discrete dichotomy.

Now we are in the conditions of [9], Exercise 10, pp. 229–230 (see also a more general statement [5, Theorem 4.1]), which finishes the proof.

Let us estimate the difference $\|\tilde{T}_{k} - T_{k}\|_{\alpha }.$ There exists a positive integer N such that each interval of length d contains no more than N elements of sequence {τ _j }. Let the interval $[\xi _{n},\xi _{n+1}]$ contain points of impulses $\tilde{\tau }_{m},\ldots,\tilde{\tau }_{k}$ where k − m ≤ N. Denote by V ₁(t, s) and $\tilde{V }(t,s)$ the evolution operators of equations without impulses ( 7.26) and ( 7.23), respectively. Then

$$ \displaystyle\begin{array}{rcl} & & \|T_{n} -\tilde{ T}_{n}\|_{\alpha } =\| U_{1}(\xi _{n+1},\xi _{n}) -\tilde{ U}(\xi _{n+1},\xi _{n})\|_{\alpha } \\ & & \leq \| (V _{1}(\xi _{n+1},\tilde{\tau }_{k}) -\tilde{ V }(\xi _{n+1},\tilde{\tau }_{k}))(I + B_{k})V _{1}(\tilde{\tau }_{k},\tilde{\tau }_{k-1})\ldots (I + B_{m})V _{1}(\tilde{\tau }_{m},\xi _{n})\|_{\alpha } + \\ & & +\|\tilde{V }(\xi _{n+1},\tilde{\tau }_{k})(B_{k} -\tilde{ B}_{k})V _{1}(\tilde{\tau }_{k},\tilde{\tau }_{k-1})\ldots (I + B_{m})V _{1}(\tilde{\tau }_{m},\tilde{\xi }_{n})\|_{\alpha } +\ \ldots \ + \\ & & +\|\tilde{V }(\xi _{n+1},\tilde{\tau }_{k})(I +\tilde{ B}_{k})\tilde{V }(\tilde{\tau }_{k},\tilde{\tau }_{k-1})\ldots (I +\tilde{ B}_{m})(V _{1}(\tilde{\tau }_{m},\xi _{n}) -\tilde{ V }(\tilde{\tau }_{m},\xi _{n}))\|_{\alpha }. {}\end{array} $$

(7.29)

Using ( 7.9), we get that

$$ \displaystyle\begin{array}{rcl} \sup _{n}\|T_{n} -\tilde{ T}_{n}\|_{\alpha } \leq K_{3}(\varepsilon )& & {}\\ \end{array} $$

with some $K_{3}(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0.$

The exponentially dichotomous system ( 7.23) and ( 7.24) has Green’s function

$$ \displaystyle\begin{array}{rcl} \tilde{G}(t,s) = \left \{\begin{array}{l} \tilde{U}(t,s)(I -\tilde{ P}(s)),\ t \geq s,\\ -\tilde{ U}(t, s)\tilde{P}(s),\ t < s, \end{array} \right.& & {}\\ \end{array} $$

such that

$$ \displaystyle\begin{array}{rcl} \|\tilde{G}(t,s)u\|_{\alpha } \leq M_{1}e^{-\beta _{1}\vert t-s\vert }\|u\|_{\alpha },\ t,s \in R,\ u \in X^{\alpha }.& & {}\\ \end{array} $$

The sequence of bounded operators T _n : X ^α → X ^α defines the difference equation

$$ \displaystyle\begin{array}{rcl} u_{n+1} = T_{n}u_{n},\ n \in Z,& &{}\end{array} $$

(7.30)

with evolution operator T _n, m  = T _n−1 … T _m ,  n ≥ m,  T _m, m  = I. It is exponentially dichotomous with Green’s function

$$ \displaystyle\begin{array}{rcl} G_{n,m} = \left \{\begin{array}{l} T_{n,m}(I - P_{m}),\ n \geq m, \\ - T_{n,m}P_{m},\ n < m, \end{array} \right.& & {}\\ \end{array} $$

where $P_{m} = P(\xi _{m}).$

The second difference equation

$$ \displaystyle\begin{array}{rcl} u_{n+1} =\tilde{ T}_{n}u_{n},\ n \in Z,& &{}\end{array} $$

(7.31)

has the evolution operator $\tilde{T}_{n,m} =\tilde{ T}_{n-1}\ldots \tilde{T}_{m},\ n \geq m,\ \tilde{T}_{m,m} = I.$

By sufficiently small $\sup _{n}\|T_{n} -\tilde{ T}_{n}\|_{\alpha }$, Eq. ( 7.31) is exponentially dichotomous with Green’s function

$$ \displaystyle\begin{array}{rcl} \tilde{G}_{n,m} = \left \{\begin{array}{l} \tilde{T}_{n,m}(I -\tilde{ P}_{m}),\ n \geq m, \\ -\tilde{ T}_{n,m}\tilde{P}_{m},\ n < m. \end{array} \right.& & {}\\ \end{array} $$

According to Henry [9], p. 233, the difference between two Green’s functions satisfies equality:

$$ \displaystyle\begin{array}{rcl} \tilde{G}_{n,m} - G_{n,m} =\sum _{k\in Z}G_{n,k+1}(\tilde{T}_{k} - T_{k})\tilde{G}_{k,m}& &{}\end{array} $$

(7.32)

and estimation

$$ \displaystyle\begin{array}{rcl} \|\tilde{G}_{n,m} - G_{n,m}\|_{\alpha } = M_{2}e^{-\beta _{2}d\vert n-m\vert }\sup _{ k}\|\tilde{T}_{k} - T_{k}\|_{\alpha },\ n,m \in Z& &{}\end{array} $$

(7.33)

with some constants β ₂ ≤ β ₁, M ₂ ≥ M ₁. 

Now we can consider the difference of two Green’s functions $\tilde{G}(t,s) - G_{1}(t,s).$ Let t = s + nd + t ₁, t ₁ ∈ [0, d). Then

$$ \displaystyle\begin{array}{rcl} & & \|\tilde{G}(t,s) - G_{1}(t,s)\|_{\alpha } = {}\\ & & =\|\tilde{ U}(s + nd + t_{1},s + nd)\tilde{G}(s + nd,s) - U(s + nd + t_{1},s + nd)G(s + nd,s)\|_{\alpha } \leq {}\\ & &\leq \| (\tilde{U}(s + nd + t_{1},s + nd) - U(s + nd + t_{1},s + nd))\tilde{G}(s + nd,s)\|_{\alpha } + {}\\ & & +\|U(s + nd + t_{1},s + nd)(\tilde{G}(s + nd,s) - G(s + nd,s))\|_{\alpha }. {}\\ \end{array} $$

Using ( 7.33) and an estimation of the difference $\tilde{U} - U_{1}$ at a bounded interval as is done in ( 7.29), we get

$$ \displaystyle\begin{array}{rcl} \|\tilde{G}(t,\tau ) - G_{1}(t,\tau )\|_{\alpha } \leq \tilde{ M}_{2}(\varepsilon )e^{-\beta _{2}\vert t-\tau \vert },\quad t,\tau \in R,& &{}\end{array} $$

(7.34)

with $\tilde{M}_{2}(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0.$

By the definition of Green’s function, we have

$$ \displaystyle\begin{array}{rcl} \|\tilde{P}(\tau ) - P_{1}(\tau )\|_{\alpha } \leq \tilde{ M}_{2}(\varepsilon )\ \ \mathrm{for\ all}\ \ \tau \in R.& &{}\end{array} $$

(7.35)

Corollary 1.

Let the conditions of Lemma  5 be satisfied. Then for $t \in R,\vert t -\tau _{j}\vert \geq \varepsilon,j \in Z,$ we have

$$ \displaystyle\begin{array}{rcl} \|(P(t) -\tilde{ P}(t))u\|_{\alpha } \leq \tilde{ M}_{3}(\varepsilon )\|u\|_{\alpha +\nu },& &{}\end{array} $$

(7.36)

where ν > 0,α + ν < 1, and $\tilde{M}_{3}(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0.$

Proof.

Using ( 7.22) and ( 7.35), we get

$$ \displaystyle\begin{array}{rcl} & & \|(P(t) -\tilde{ P}(t))u\|_{\alpha } \leq \| (P(t) - P(\vartheta (t)))u\|_{\alpha } + {}\\ & & +\|(P(\vartheta (t)) -\tilde{ P}(\vartheta (t)))u\|_{\alpha } +\| (\tilde{P}(\vartheta (t)) -\tilde{ P}(t))u\|_{\alpha } \leq \tilde{ M}_{3}(\varepsilon )\|u\|_{\alpha +\nu }. {}\\ \end{array} $$

7.4 Almost Periodic Solutions of Equations with Fixed Moments of Impulsive Action

Consider the linear inhomogeneous equation

$$ \displaystyle\begin{array}{rcl} & & \frac{du} {dt} + (A + A_{1}(t))u = f(t),\quad t\not =\tau _{j},{}\end{array} $$

(7.37)

$$ \displaystyle\begin{array}{rcl} & & \varDelta u\vert _{t=\tau _{j}} = u(\tau _{j} + 0) - u(\tau _{j}) = B_{j}u(\tau _{j}) + g_{j},\quad j \in Z.{}\end{array} $$

(7.38)

We assume that

(H7) :

the function f(t): R → X is W-almost periodic and locally Hlder continuous with points of discontinuity at moments t = τ _j , j ∈ Z, at which it is continuous from the left;

(H8) :

the sequence {g _j } of $g_{j} \in X^{\alpha _{1}},\alpha _{1} >\alpha > 0,$ is almost periodic.

Theorem 1.

Assume that Eqs. ( 7.37 ) and ( 7.38 ) satisfy conditions (H1) – (H3) , (H7) , and (H8) and that the corresponding homogeneous equation is exponentially dichotomous.

Then the equation has a unique W-almost periodic solution $u_{0}(t) \in \mathcal{P}\mathcal{C}(R,X^{\alpha }).$

Proof.

We show that an almost periodic solution is given by the formula ( 7.16). For t ∈ (τ _i , τ _i+1], it satisfies

$$ \displaystyle\begin{array}{rcl} & & \|u_{0}(t)\|_{\alpha } \leq \int _{-\infty }^{t}\|A^{\alpha }U(t,s)(I - P(s))f(s)\|ds + \\ & & +\int _{t}^{\infty }\|A^{\alpha }U(t,s)P(s)f(s)\|ds +\sum _{ j\in Z}\|G(t,\tau _{j})g_{j}\|_{\alpha } \leq \\ & &\leq \sum _{j\in Z}\|G(t,\tau _{j})g_{j}\|_{\alpha } +\int _{ \tau _{i}}^{t}\|A^{\alpha }V (t,s)(I - P(s))f(s)\|ds + \\ & & +\sum _{k=0}^{\infty }\int _{ \tau _{i-k-1}}^{\tau _{i-k} }\|U(t,\tau _{i-k})(I - P(\tau _{i-k}))\|_{\alpha }\|A^{\alpha }U(\tau _{i-k},s)f(s)\|ds + \\ & & +\sum _{k=1}^{\infty }\int _{ \tau _{i+k}}^{\tau _{i+k+1} }\|U(t,\tau _{i+k+1})P(\tau _{i+k+1})\|_{\alpha }\|A^{\alpha }U(\tau _{i+k+1},s)f(s)\|ds + \\ & & +\int _{t}^{\tau _{i+1} }\|A^{\alpha }V (t,s)P(s)f(s)\|ds \leq \frac{2M} {1 - e^{-\theta \beta }}\frac{C_{\alpha }\varTheta ^{1-\alpha }} {1-\alpha } \|f\|_{PC} + \\ & & + \frac{2M} {1 - e^{-\theta \beta }}\sup _{j}\|g_{j}\|_{\alpha } \leq \tilde{ M}_{0}\max \{\|f(t)\|_{PC},\|g_{j}\|_{\alpha }\} {}\end{array} $$

(7.39)

with some constant $\tilde{M}_{0} > 0.$

Take an $\varepsilon$-almost period h for the right-hand side of the equation, which satisfies the conditions of Lemma 1; that is, there exists a positive integer q such that τ _j+q  ∈ (s + h, t + h) if τ _j  ∈ (s, t) and $\vert \tau _{j} + h -\tau _{j+q}\vert <\varepsilon,\|B_{j+q} - B_{j}\| <\varepsilon.$

Let $t \in (\tau _{i}+\varepsilon,\tau _{i+1}-\varepsilon ).$ We define points η _k  = (τ _k +τ _k−1)∕2, k ∈ Z. Then

$$ \displaystyle\begin{array}{rcl} & & \|u_{0}(t + h) - u_{0}(t)\|_{\alpha } \leq \sum _{j\in Z}\|G(t + h,\tau _{j+q})g_{j+q} - G(t,\tau _{j})g_{j}\|_{\alpha } + \\ & & +\int _{-\infty }^{\infty }\|G(t + h,s + h)f(s + h) - G(t,s)f(s)\|_{\alpha }ds \leq \\ & &\leq \int _{-\infty }^{\infty }\|(G(t + h,s + h) - G(t,s))f(s + h)\|_{\alpha }ds + \\ & & +\int _{-\infty }^{\infty }\|G(t,s))(f(s + h) - f(s))\|_{\alpha }ds +\sum _{ j\in Z}\|G(t,\tau _{j}))(g_{j+q} - g_{j})\|_{\alpha } + \\ & & +\sum _{j\in Z}\|(G(t + h,\tau _{j+q}) - G(t,\tau _{j}))g_{j+q}\|_{\alpha }. {}\end{array} $$

(7.40)

Denote U ₂(t, s) = U(t + h, s + h). If u(t) = U(t, s)u ₀, u(s) = u ₀, is a solution of the impulsive equations ( 7.4) and ( 7.5), then u ₂(t) = U(t + h, s + h)u ₀, u ₂(s) = u ₀, is a solution of the equation

$$ \displaystyle\begin{array}{rcl} & & \frac{du} {dt} + (A + A_{1}(t + h))u = 0,\quad t\not =\tau _{j+q} - h,{}\end{array} $$

(7.41)

$$ \displaystyle\begin{array}{rcl} & & \varDelta u\vert _{t+h=\tau _{j+q}} = u(\tau _{j+q} + 0) - u(\tau _{j+q}) = B_{j+q}u(\tau _{j+q}),\quad j \in Z.{}\end{array} $$

(7.42)

We will use the notation V ₂(t, s) = V (t + h, s + h) for the evolution operator of an equation without impulses ( 7.41). Denote also $\tilde{\tau }_{n} =\tau _{n+q} - h,\tilde{B}_{n} = B_{n+q}.$ Since Eqs. ( 7.4) and ( 7.5) are exponentially dichotomous, Eqs. ( 7.41) and ( 7.42) are exponentially dichotomous also with projector P ₂(s) = P(s + h). 

The first integral in ( 7.40) is the sum of two integrals:

$$ \displaystyle\begin{array}{rcl} & & \int _{-\infty }^{\infty }\|(G(t + r,s + r) - G(t,s))f(s + r)\|_{\alpha }ds = \\ & & =\int _{ -\infty }^{t}\|(U_{ 2}(t,s)(I - P_{2}(s)) - U(t,s)(I - P(s)))f(s + r)\|_{\alpha }ds + \\ & & +\int _{t}^{\infty }\|(U_{ 2}(t,s)P_{2}(s) - U(t,s)P(s))f(s + r)\|_{\alpha }ds. {}\end{array} $$

(7.43)

We estimate the first integral in ( 7.43); the second integral is considered analogously.

$$ \displaystyle\begin{array}{rcl} & & \int _{-\infty }^{t}\|(U_{ 2}(t,s)(I - P_{2}(s)) - U(t,s)(I - P(s)))f(s + r)\|_{\alpha }ds \leq \\ & &\leq \int _{\tau _{i}+\varepsilon }^{t}\|A^{\alpha }(V _{ 2}(t,s)(I - P_{2}(s)) - V (t,s)(I - P(s)))f(s + r)\|ds + \\ & & +\int _{\tau _{i}-\varepsilon }^{\tau _{i}+\varepsilon }\|A^{\alpha }(U_{ 2}(t,s)(I - P_{2}(s)) - U(t,s)(I - P(s)))f(s + r)\|ds + \\ & & +\int _{\eta _{i}}^{\tau _{i}-\varepsilon }\|A^{\alpha }(U_{ 2}(t,s)(I - P_{2}(s)) - U(t,s)(I - P(s)))f(s + r)\|ds + \\ & & +\sum _{k=1}^{\infty }\int _{ \eta _{i-k}}^{\eta _{i-k+1} }\|A^{\alpha }(U_{2}(t,s)(I - P_{2}(s)) - U(t,s)(I - P(s)))f(s + r)\|ds.\qquad \quad {}\end{array} $$

(7.44)

Let us consider all integrals in ( 7.44) separately. By ( 7.36) and ( 7.11) we have

$$ \displaystyle\begin{array}{rcl} & & I_{11} =\int _{ \tau _{i}+\varepsilon }^{t}\|A^{\alpha }(V _{ 2}(t,s)(I - P_{2}(s)) - V (t,s)(I - P(s)))f(s + r)\|ds = {}\\ & & =\int _{ \tau _{i}+\varepsilon }^{t}\|A^{\alpha }((I - P_{ 2}(t))V _{2}(t,s) - (I - P(t))V (t,s))f(s + r)\|ds \leq {}\\ & &\leq \int _{\tau _{i}+\varepsilon }^{t}\|A^{\alpha }(P_{ 2}(t) - P(t))V _{2}(t,s)f(s + r)\|ds + {}\\ & & +\int _{\tau _{i}+\varepsilon }^{t}\|A^{\alpha }(I - P(t))(V _{ 2}(t,s) - V (t,s))f(s + r)\|ds \leq {}\\ & &\leq \left (\int _{\tau _{i}+\varepsilon }^{t}\frac{\tilde{M}_{3}(\varepsilon )L_{Q}ds} {(t - s)^{\alpha }} +\int _{ \tau _{i}+\varepsilon }^{t} \frac{R_{3}(\varepsilon )ds} {(t - s)^{2\alpha -1}}\right )\|f\|_{PC} \leq \varGamma _{1}(\varepsilon )\|f\|_{PC}. {}\\ \end{array} $$

$$ \displaystyle\begin{array}{rcl} & & I_{12} =\int _{ \tau _{i}-\varepsilon }^{\tau _{i}+\varepsilon }\|A^{\alpha }U(t,s)(I - P(s))f(s + h)\|ds \leq {}\\ & &\leq \int _{\tau _{i}}^{\tau _{i}+\varepsilon }\|A^{\alpha }(I - P(t))V (t,s)f(s + h)\|ds + {}\\ & & +\int _{\tau _{i}-\varepsilon }^{\tau _{i} }\|\|A^{\alpha }(I - P(t))V (t,\tau _{i})(I + B_{i})U(\tau _{i},s)f(s + h)\|ds \leq {}\\ & &\leq \Bigl (\int _{\tau _{i}}^{\tau _{i}+\varepsilon } \frac{C_{\alpha }ds} {(t - s)^{\alpha }} + M\|I + B_{i}\|\int _{\tau _{i}-\varepsilon }^{\tau _{i} } \frac{C_{\alpha }ds} {(s -\tau _{i})^{\alpha }}\Bigl )\|f\|_{PC} \leq {}\\ & &\leq \varGamma _{2}(\varepsilon )\|f\|_{PC}. {}\\ \end{array} $$

Analogously,

$$ \displaystyle\begin{array}{rcl} & & I_{13} =\int _{ \tau _{i}-\varepsilon }^{\tau _{i}+\varepsilon }\|A^{\alpha }U_{ 2}(t,s)(I - P_{2}(s))f(s + h)\|ds \leq \varGamma _{3}(\varepsilon )\|f\|_{PC}, {}\\ \end{array} $$

where $\varGamma _{j}(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0,\ j = 1,2,3.$

Using ( 7.11) and ( 7.36), we get

$$ \displaystyle\begin{array}{rcl} & & I_{14} =\int _{ \eta _{i}}^{\tau _{i}-\varepsilon }\|A^{\alpha }(U_{ 2}(t,s)(I - P_{2}(s)) - U(t,s)(I - P(s)))f(s + r)\|ds = {}\\ & & \quad =\int _{ \eta _{i}}^{\tau _{i}-\varepsilon }\|\Bigl ((I - P_{ 2}(t))V _{2}(t,\tilde{\tau }_{i})(I +\tilde{ B}_{i})V _{1}(\tilde{\tau }_{i},s) - {}\\ & &\qquad - (I - P(t))V (t,\tau _{i})(I + B_{i})V (\tau _{i},s)\Bigl )f(s + h)\|_{\alpha }ds \leq {}\\ & &\quad \leq \int _{\eta _{i}}^{\tau _{i}-\varepsilon }\|(P_{ 2}(t) - P(t))V _{2}(t,\tilde{\tau }_{i})(I + B_{i})V _{2}(\tilde{\tau }_{i},s)f(s + h)\|_{\alpha }ds + {}\\ & & \quad +\int _{ \eta _{i}}^{\tau _{i}-\varepsilon }\|(I - P(t))(V _{ 2}(t,\tilde{\tau }_{i}) - V (t,\tau _{i}))(I + B_{i})V _{2}(\tilde{\tau }_{i},s)f(s + h)\|_{\alpha }ds + {}\\ & & \quad +\int _{ \eta _{i}}^{\tau _{i}-\varepsilon }\|(I - P(t))V (t,\tau _{ i})(\tilde{B}_{i} - B_{i})V _{2}(\tilde{\tau }_{i},s)f(s + h)\|_{\alpha }ds + {}\\ & & \quad +\int _{ \eta _{i}}^{\tau _{i}-\varepsilon }\|(I - P(t))V (t,\tau _{ i})(I - B_{i})(V _{2}(\tilde{\tau }_{i},s) - V (\tau _{i},s))f(s + h)\|_{\alpha }ds \leq {}\\ & &\quad \leq \varGamma _{4}(\varepsilon )\|f\|_{PC}, {}\\ \end{array} $$

where $\varGamma _{4}(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0.$

The last sum in ( 7.44) is transformed as follows:

$$ \displaystyle\begin{array}{rcl} & & \hspace{-6.0pt}I_{15} =\sum _{ k=1}^{\infty }\int _{ \eta _{i-k}}^{\eta _{i-k+1} }\|A^{\alpha }(U_{2}(t,s)(I - P_{2}(s)) - U(t,s)(I - P(s)))f(s + r)\|ds = {}\\ & & \hspace{-6.0pt}=\sum _{ k=1}^{\infty }\int _{ \eta _{i-k}}^{\eta _{i-k+1} }\|(U(t,\eta _{i})(I - P(\eta _{i}))U(\eta _{i},\eta _{i-k+1})U(\eta _{i-k+1},s) - {}\\ & &\hspace{-6.0pt}-U_{2}(t,\eta _{i})(I - P_{2}(\eta _{i}))U_{2}(\eta _{i},\eta _{i-k+1})U_{2}(\xi _{i-k+1},s))f(s + h)\|_{\alpha }ds \leq {}\\ & &\hspace{-6.0pt}\leq \sum _{k=1}^{\infty }\int _{ \eta _{i-k}}^{\eta _{i-k+1} }\Bigl \|\Bigl (U(t,\eta _{i}) - U_{2}(t,\eta _{i}))(I - P(\eta _{i}))U(\eta _{i},\eta _{i-k+1})U(\eta _{i-k+1},s) + {}\\ & & \hspace{-6.0pt}+U_{2}(t,\eta _{i})((I - P(\eta _{i}))U(\eta _{i},\eta _{i-k+1}) - (I - P_{2}(\eta _{i}))U_{2}(\eta _{i},\eta _{i-k+1}))U(\eta _{i-k+1},s) + {}\\ & & \hspace{-6.0pt}+U_{2}(t,\eta _{i})(I - P_{2}(\eta _{i}))U_{2}(\eta _{i},\eta _{i-k+1})(U(\eta _{i-k+1},s) - U_{2}(\eta _{i-k+1},s))\Bigl )f(s + h)\Bigl \|_{\alpha }ds. {}\\ \end{array} $$

As in the proof of Lemma 5, we construct in space X ^α two sequences of bounded operators

$$ \displaystyle{S_{n} = U(\eta _{n+1},\eta _{n}),\quad \tilde{S}_{n} = U_{2}(\eta _{n+1},\eta _{n}),\quad n \in Z,} $$

and corresponding difference equations

$$ \displaystyle\begin{array}{rcl} u_{n+1} = S_{n}u_{n},\quad v_{n+1} =\tilde{ S}_{n}v_{n},\quad n \in Z.& & {}\\ \end{array} $$

Per our assumption, these difference equations are exponentially dichotomous with corresponding evolution operators

$$ \displaystyle{S_{n,m} = S_{n-1}\ldots S_{m},\quad \tilde{S}_{n,m} =\tilde{ S}_{n-1}\ldots \tilde{S}_{m},\quad n \geq m,} $$

and Green’s functions

$$ \displaystyle\begin{array}{rcl} G_{n,m} = \left \{\begin{array}{l} S_{n,m}(I - P_{m}),\ n \geq m, \\ - S_{n,m}P_{m},\ n < m, \end{array} \right.\quad \tilde{G}_{n,m} = \left \{\begin{array}{l} \tilde{S}_{n,m}(I -\tilde{ P}_{m}),\ n \geq m, \\ -\tilde{ S}_{n,m}\tilde{P}_{m},\ n < m, \end{array} \right.& & {}\\ \end{array} $$

where $P_{m} = P(\eta _{m}),\tilde{P}_{m} = P_{2}(\eta _{m}).$

Analogous to ( 7.32) and ( 7.33), we obtain

$$ \displaystyle\begin{array}{rcl} \tilde{G}_{n,m} - G_{n,m} =\sum _{k\in Z}G_{n,k+1}(\tilde{S}_{k} - S_{k})\tilde{G}_{k,m}& & {}\\ \end{array} $$

and

$$ \displaystyle\begin{array}{rcl} \|\tilde{G}_{n,m} - G_{n,m}\|_{\alpha } = M_{1}e^{-\beta _{1}\theta \vert n-m\vert }\sup _{ k}\|\tilde{S}_{k} - S_{k}\|_{\alpha },\ n,m \in Z& &{}\end{array} $$

(7.45)

with some constants β ₁ ≤ β, M ₁ ≥ M. 

$$ \displaystyle\begin{array}{rcl} & & \|S_{n} -\tilde{ S}_{n}\|_{\alpha } =\| U(\eta _{n+1},\eta _{n}) - U_{2}(\eta _{n+1},\eta _{n})\|_{\alpha } = {}\\ & & =\| V (\eta _{n+1},\tau _{n})(I + B_{n})V (\tau _{n},\eta _{n}) - V _{2}(\eta _{n+1},\tilde{\tau }_{n})(I +\tilde{ B}_{n})V _{2}(\tilde{\tau }_{n},\eta _{n})\|_{\alpha } \leq {}\\ & &\leq \| (V (\eta _{n+1},\tau _{n}) - V _{2}(\eta _{n+1},\tilde{\tau }_{n}))(I + B_{n})V (\tau _{n},\eta _{n})\|_{\alpha } + {}\\ & & +\|V _{2}(\eta _{n+1},\tilde{\tau }_{n}))(B_{n} -\tilde{ B}_{n})V (\tau _{n},\eta _{n})\|_{\alpha } + {}\\ & & +\|V _{2}(\eta _{n+1},\tilde{\tau }_{n}))(I +\tilde{ B}_{n})(V (\tau _{n},\eta _{n}) - V _{2}(\tilde{\tau }_{n},\eta _{n}))\|_{\alpha } {}\\ \end{array} $$

Here we assume for definiteness that $\tilde{\tau }_{n} \geq \tau _{n}.$ We have

$$ \displaystyle\begin{array}{rcl} & & \|(V (\eta _{n+1},\tau _{n}) - V _{2}(\eta _{n+1},\tilde{\tau }_{n}))y\|_{\alpha } \leq \| V (\eta _{n+1},\tilde{\tau }_{n})(V (\tilde{\tau }_{n},\tau _{n}) - I)y\|_{\alpha } + {}\\ & & +\|(V (\eta _{n+1},\tilde{\tau }_{n}) - V _{2}(\eta _{n+1},\tilde{\tau }_{n}))y\|_{\alpha } \leq {}\\ & &\leq \varGamma _{5}(\varepsilon )\|y\|_{\alpha } {}\\ \end{array} $$

and

$$ \displaystyle\begin{array}{rcl} & & \|(V _{2}(\tilde{\tau }_{n},\eta _{n}) - V (\tau _{n},\eta _{n}))y\|_{\alpha } \leq \| (V _{2}(\tilde{\tau }_{n},\tau _{n}) - I)V _{2}(\tau _{n},\eta _{n})y\|_{\alpha } + {}\\ & & +\|V _{2}(\tau _{n},\eta _{n}) - V (\tau _{n},\eta _{n})y\|_{\alpha } \leq \varGamma _{6}(\varepsilon )\|y\|_{\alpha }, {}\\ \end{array} $$

where $\varGamma _{5}(\varepsilon ) \rightarrow 0$ and $\varGamma _{6}(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0.$

Now we get

$$ \displaystyle\begin{array}{rcl} & & \|S_{n} -\tilde{ S}_{n}\|_{\alpha } \leq \varGamma _{5}(\varepsilon )\|I + B_{n}\|\|U(\tau _{n},\eta _{n})\|_{\alpha } + {}\\ & & +\varepsilon \|U_{2}(\eta _{n},\tau _{n})\|_{\alpha }\|U(\tau _{n},\eta _{n})\|_{\alpha } +\varGamma _{6}(\varepsilon )\|U_{2}(\eta _{n+1},\tilde{\tau }_{n})\|_{\alpha }\|I +\tilde{ B}_{n}\| \leq \varGamma _{7}(\varepsilon ) {}\\ \end{array} $$

and by ( 7.45)

$$ \displaystyle\begin{array}{rcl} \|U(\eta _{i},\eta _{i-k}) - U_{2}(\eta _{i},\eta _{i-k})\|_{\alpha } \leq M_{1}e^{-\beta _{1}\theta k}\varGamma _{ 7}(\varepsilon ),& &{}\end{array} $$

(7.46)

where $\varGamma _{7}(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0.$

Continuing to evaluate I ₁₅, we can obtain the inequalities

$$ \displaystyle\begin{array}{rcl} & & \|U_{2}(t,\eta _{i})g\|_{\alpha } \leq M_{2}\|g\|_{\alpha }, {}\\ & & \|(U(t,\eta _{i}) - U_{2}(t,\eta _{i}))g\|_{\alpha } \leq \varGamma _{8}(\varepsilon )\|g\|_{\alpha }, {}\\ & & \int _{\xi _{i-k}}^{\eta _{i-k+1} }\|(U(\eta _{i-k+1},s) - U_{2}(\eta _{i-k+1},s))f(s + h)\|_{\alpha }ds \leq \varGamma _{9}(\varepsilon )\|f\|_{PC}, {}\\ \end{array} $$

where $\varGamma _{8}(\varepsilon ) \rightarrow 0$ and $\varGamma _{9}(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0,\ M_{2}$ is some positive constant. Note that as earlier, $t \in (\tau _{i}+\varepsilon,\tau _{i+1}-\varepsilon ).$

Taking into account the last inequalities, we conclude that series I ₁₅ is convergent and there exists $\varGamma _{10}(\varepsilon )$ such that $I_{15} \leq \varGamma _{10}(\varepsilon )\|f\|_{PC}$ and $\varGamma _{10}(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0.$

Using estimations for I ₁₁, …, I ₁₅, we get that there exists $\varGamma _{11}(\varepsilon )$ such that

$$ \displaystyle\begin{array}{rcl} \int _{-\infty }^{\infty }\|(G(t + r,s + r) - G(t,s))f(s + r)\|_{\alpha }ds \leq \varGamma _{ 11}(\varepsilon )\|f\|_{PC}& &{}\end{array} $$

(7.47)

and $\varGamma _{11}(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0.$

By Lemma 1, $\vert \tau _{j+q} -\tau _{j} - h\vert <\varepsilon;$ therefore, $\tau _{j} + h+\varepsilon >\tau _{j+q}$ (we assume that h > 0 for definiteness). The difference G(t, τ _j ) − G(t + h, τ _j+q ) is estimated as follows. Let $t -\tau _{j} \geq \varepsilon.$ Then

$$ \displaystyle\begin{array}{rcl} & & \|(G(t,\tau _{j}) - G(t + h,\tau _{j+q}))g_{j+q}\|_{\alpha } = \\ & & =\| (U(t,\tau _{j})(I - P(\tau _{j})) - U(t + h,\tau _{j+q})(I - P(\tau _{j+q})))g_{j+q}\|_{\alpha } \leq \\ & &\leq \| (U(t,\tau _{j})(I - P(\tau _{j})) - U(t,\tau _{j}+\varepsilon )(I - P(\tau _{j}+\varepsilon )))g_{j+q}\|_{\alpha } + \\ & & +\|(U(t,\tau _{j}+\varepsilon )(I - P(\tau _{j}+\varepsilon )) - U(t + h,\tau _{j} +\varepsilon +h) \times \\ & &\times (I - P(\tau _{j} +\varepsilon +h)))g_{j+q}\|_{\alpha } +\| U(t + h,\tau _{j+q})(I - P(\tau _{j+q})))g_{j+q} - \\ & &-(U(t + h,\tau _{j} +\varepsilon +h)(I - P(\tau _{j} +\varepsilon +h))g_{j+q}\|_{\alpha }. {}\end{array} $$

(7.48)

The first and third differences are small due to the continuity of function U(t, s) at intervals between impulse points:

$$ \displaystyle\begin{array}{rcl} & & \|(U(t,\tau _{j})(I - P(\tau _{j})) - U(t,\tau _{j}+\varepsilon )(I - P(\tau _{j}+\varepsilon )))g_{j+q}\|_{\alpha } \leq {}\\ & &\leq \| U(t,\tau _{j}+\varepsilon )(I - P(\tau _{j}+\varepsilon ))(U(\tau _{j}+\varepsilon,\tau _{j}) - I)g_{j+q}\|_{\alpha } \leq {}\\ & &\leq \| (I - P(t))U(t,\tau _{j}+\varepsilon )\|_{\alpha }\|(U(\tau _{j}+\varepsilon,\tau _{j}) - I)g_{j+q}\|_{\alpha } \leq {}\\ & &\leq Me^{-\beta (t-\tau _{j}-\varepsilon )}C_{ 1-\alpha _{1}+\alpha }\varepsilon ^{\alpha _{1}-\alpha }\|g_{ j+q}\|_{\alpha _{1}}, {}\\ & & \|(U(t + h,\tau _{j} +\varepsilon +h)(I - P(\tau _{j} +\varepsilon +h)) - U(t + h,\tau _{j+q})(I - P(\tau _{j+q})))g_{j+q}\|_{\alpha } = {}\\ & & =\|\| U(t + h,\tau _{j} +\varepsilon +h)(I - P(\tau _{j} +\varepsilon +h))(U(\tau _{j} +\varepsilon +h,\tau _{j+q}) - I)g_{j+q}\|_{\alpha } \leq {}\\ & &\leq Me^{-\beta (t-\tau _{j}-\varepsilon )}C_{ 1-\alpha _{1}+\alpha }\varepsilon ^{\alpha _{1}-\alpha }\|g_{ j+q}\|_{\alpha _{1}}. {}\\ \end{array} $$

The second difference in ( 7.48) is estimated using inequality ( 7.46) and the following transformation:

$$ \displaystyle\begin{array}{rcl} & & \|U(t,\tau _{j}+\varepsilon )(I - P(\tau _{j}+\varepsilon )) - U(t + h,\tau _{j} +\varepsilon +h)(I - P(\tau _{j} +\varepsilon +h))\|_{\alpha } = {}\\ & & =\| U(t,\tau _{j}+\varepsilon )(I - P(\tau _{j}+\varepsilon )) - U_{2}(t,\tau _{j}+\varepsilon )(I - P_{2}(\tau _{j}+\varepsilon ))\|_{\alpha } = {}\\ & & =\| U(t,\eta _{i})(I - P(\eta _{i}))U(\eta _{i},\eta _{j+1})U(\eta _{j+1},\tau _{j}+\varepsilon ) - {}\\ & &-U_{2}(t,\eta _{i})(I - P(\eta _{i})U_{2}(\eta _{i},\eta _{j+1})U_{2}(\eta _{j+1},\tau _{j}+\varepsilon )\|_{\alpha } \leq {}\\ & &\leq \| (U(t,\eta _{i}) - U_{2}(t,\eta _{i}))(I - P(\eta _{i}))U(\eta _{i},\eta _{j+1})U(\eta _{j+1},\tau _{j}+\varepsilon )\|_{\alpha } + {}\\ & & +\|U_{1}(t,\eta _{i})(P(\eta _{i})U(\eta _{i},\eta _{j+1}) - P_{2}(\eta _{i})U_{2}(\eta _{i},\eta _{j+1})U(\eta _{j+1},\tau _{j}+\varepsilon )\|_{\alpha } + {}\\ & & +\|U_{2}(t,\eta _{i})P_{2}(\eta _{i})U_{2}(\eta _{i},\eta _{j+1})(U(\eta _{j+1},\tau _{j}+\varepsilon ) - U_{2}(\eta _{j+1},\tau _{j}+\varepsilon ))\|_{\alpha }. {}\\ \end{array} $$

Therefore,

$$ \displaystyle\begin{array}{rcl} \sum _{j\in Z}\|(G(t + h,\tau _{j+q}) - G(t,\tau _{j}))g_{j+q}\|_{\alpha } \leq \varGamma _{12}(\varepsilon )\sup _{j}\|g_{j}\|_{\alpha _{1}},& &{}\end{array} $$

(7.49)

where $\varGamma _{12}(\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0.$

The second integral and first sum in ( 7.40) are estimated as in ( 7.39):

$$ \displaystyle\begin{array}{rcl} \int _{-\infty }^{\infty }\|G(t,s))(f(s + h) - f(s))\|_{\alpha }ds +\sum _{ j\in Z}\|U(t,\tau _{j})(g_{j+q} - g_{j})\|_{\alpha } \leq M_{3}\varepsilon & & {}\\ \end{array} $$

since h is $\varepsilon$-almost periodic of the right-hand side of the equation.

As a result of these evaluations, we get

$$ \displaystyle{\|u_{0}(t + h) - u_{0}(t)\|_{\alpha } \leq \varGamma (\varepsilon )\ \ \mathrm{for}\ \ t \in R,\ \vert t -\tau _{j}\vert >\varepsilon,\ j \in Z,} $$

with $\varGamma (\varepsilon ) \rightarrow 0$ as $\varepsilon \rightarrow 0.$ The last inequality implies that the function u ₀(t) is W-almost periodic as function R → X ^α. 

Corollary 2.

Assume that Eqs. ( 7.37 ) and ( 7.38 ) satisfy the following:

• i) conditions (H1) – (H3) , (H7);

• ii) the sequence {g _j } of g _j ∈ X ^α is almost periodic;

• iii) the corresponding homogeneous equation is exponentially dichotomous.

Then the equation has a unique W-almost periodic solution $u_{0}(t) \in \mathcal{P}\mathcal{C}(R,X^{\gamma })$ with γ < α.

Now we consider a nonlinear equation with fixed moments of impulsive action:

$$ \displaystyle\begin{array}{rcl} & & \frac{du} {dt} + (A + A_{1}(t))u = f(t,u),\quad t\not =\tau _{j},{}\end{array} $$

(7.50)

$$ \displaystyle\begin{array}{rcl} & & \varDelta u\vert _{t=\tau _{j}} = u(\tau _{j} + 0) - u(\tau _{j}) = B_{j}u(\tau _{j}) + g_{j}(u(\tau _{j})),\quad j \in Z.{}\end{array} $$

(7.51)

Theorem 2.

Let us consider Eqs. ( 7.50 ) and ( 7.51 ) in some domain $U_{\rho }^{\alpha } =\{ x \in X^{\alpha }:\ \| x\|_{\alpha } \leq \rho \}$ of space X ^α . Assume that

• 1) the equation satisfies assumptions (H1) – (H4) , τ _j = τ _j (0);

• 2) the corresponding linear equation is exponentially dichotomous;

• 3) the function $f(t,u):\ R \times U_{\rho }^{\alpha } \rightarrow X$ is continuous in u, W-almost periodic, and Hlder continuous in t uniformly with respect to $u \in U_{\rho }^{\alpha }$ with some ρ > 0, and there exist constants N ₁ > 0 and ν > 0 such that

  $$ \displaystyle\begin{array}{rcl} \|f(t_{1},u_{1}) - f(t_{2},u_{2})\| \leq N_{1}(\vert t_{1} - t_{2}\vert ^{\nu } +\| u_{1} - u_{2}\|_{\alpha });& & {}\\ \end{array} $$

• 4) the sequence {g _j (u)} of continuous functions $U_{\rho }^{\alpha } \rightarrow X^{\alpha _{1}}$ is almost periodic uniformly with respect to $u \in U_{\rho }^{\alpha }$ and

  $$ \displaystyle\begin{array}{rcl} \|g_{j}(u_{1}) - g_{j}(u_{2})\|_{\alpha } \leq N_{1}\|u_{1} - u_{2}\|_{\alpha },\ j \in Z,& & {}\\ \end{array} $$

  for $t_{1},t_{2} \in R,\ u_{1},u_{2} \in U_{\rho }^{\alpha }$ and some α ₁ > α;

• 5) the functions f(t,0) and g _j (0) are uniformly bounded for t ∈ R,j ∈ Z.

  Then in domain $U_{\rho }^{\alpha }$ for sufficiently small N ₁ > 0 there exists a unique W-almost periodic solution u ₀ (t) of Eqs. ( 7.50 ) and ( 7.51 ).

Proof.

Denote by $\mathcal{M}_{\varrho }$ the set of all W-almost periodic functions $\varphi: R \rightarrow X^{\alpha }$ with discontinuity points τ _j , j ∈ Z, satisfying the inequality $\|\varphi \|_{PC} \leq \varrho$. In $\mathcal{M}_{\varrho },$ we define the operator

$$ \displaystyle\begin{array}{rcl} (\mathcal{F}\varphi )(t) =\int _{ -\infty }^{\infty }G(t,s)f(s,\varphi (s))ds +\sum _{ j\in Z}G(t,\tau _{j})g_{j}(\varphi (\tau _{j})).& & {}\\ \end{array} $$

Proceeding in the same way as in the proof of Theorem 1, we prove that $(\mathcal{F}\varphi )(t)$ is a W-almost periodic function and $\mathcal{F}: \mathcal{M}_{\varrho }\rightarrow \mathcal{M}_{\varrho }$ for some $\varrho > 0.$

Next, $\mathcal{F}$ is a contracting operator in $\mathcal{M}_{\varrho }$ by sufficiently small N ₁ > 0. 

Hence, there exists $\varphi _{0} \in \mathcal{M}_{\varrho }$ such that

$$ \displaystyle\begin{array}{rcl} \varphi _{0}(t) =\int _{ -\infty }^{\infty }G(t,s)f(s,\varphi _{ 0}(s))ds +\sum _{j\in Z}G(t,\tau _{j})g_{j}(\varphi _{0}(\tau _{j})).& & {}\\ \end{array} $$

The function $\varphi _{0}(t)$ is locally Hölder continuous on every interval (τ _j , τ _j+1), j ∈ Z. Actually,

$$ \displaystyle\begin{array}{rcl} & & \varphi _{0}(t+\delta ) -\varphi _{0}(t) =\int _{ -\infty }^{\infty }G(t+\delta,s)f(s,\varphi _{ 0}(s))ds -\int _{-\infty }^{\infty }G(t,s)f(s,\varphi _{ 0}(s))ds + {}\\ & & +\sum _{j\in Z}G(t+\delta,\tau _{j})g_{j}(\varphi _{0}(\tau _{j})) -\sum _{j\in Z}G(t,\tau _{j})g_{j}(\varphi _{0}(\tau _{j})) = {}\\ & & =\int _{ -\infty }^{t}(V (t+\delta,t) - I)U(t,s)(I - P(s))f(s,\varphi _{ 0}(s))ds - {}\\ & &-\int _{t+\delta }^{\infty }(V (t+\delta,t) - I)U(t,s)P(s)f(s,\varphi _{ 0}(s))ds + {}\\ & & +\int _{t}^{t+\delta }V (t+\delta,s)(I - P(s))f(s,\varphi _{ 0}(s))ds +\int _{ t}^{t+\delta }V (t,s)P(s)f(s,\varphi _{ 0}(s))ds {}\\ & & +\sum _{\tau _{j}<t}(V (t+\delta,t) - I)U(t,\tau _{j})(I - P(\tau _{j}))g_{j}(\varphi _{0}(\tau _{j})) + {}\\ & & +\sum _{\tau _{j}>t}(V (t+\delta,t) - I)U(t,\tau _{j})P(\tau _{j})g_{j}(\varphi _{0}(\tau _{j})). {}\\ \end{array} $$

Applying ( 7.7), ( 7.20), ( 7.21), and ( 7.39), we conclude that for every interval t ∈ (t′, t″) not containing impulse points τ _j , there exists a positive constant C such that $\|\varphi _{0}(t+\delta ) -\varphi _{0}(t)\|_{\alpha } \leq C\delta ^{\alpha _{1}-\alpha }.$

The local Hölder continuity of $f(t,\varphi _{0}(t))$ follows from

$$ \displaystyle\begin{array}{rcl} & & \|f(t,\varphi _{0}(t)) - f(s,\varphi _{0}(s))\| \leq N_{1}\left (\vert t - s\vert ^{\nu } +\|\varphi _{0}(t) -\varphi _{0}(s)\|_{\alpha }\right ) \leq {}\\ & &\leq C_{1}\left (\vert t - s\vert ^{\nu } + \vert t - s\vert ^{\alpha _{1}-\alpha }\right ). {}\\ \end{array} $$

By Lemma 37, [19], p. 214, if $\varphi _{0}(t)$ is W-almost periodic and $\inf _{k}(\tau _{k+1} -\tau _{k}) > 0$, then $\{\varphi _{0}(\tau _{k})\}$ is an almost periodic sequence.

The linear inhomogeneous equation

$$ \displaystyle\begin{array}{rcl} & & \frac{du} {dt} + (A + A_{1}(t))u = f(t,\varphi _{0}(t)),\quad t\not =\tau _{j},{}\end{array} $$

(7.52)

$$ \displaystyle\begin{array}{rcl} & & \varDelta u\vert _{t=\tau _{j}} = u(\tau _{j} + 0) - u(\tau _{j}) = B_{j}u(\tau _{j}) + g_{j}(\varphi _{0}(\tau _{j})),\quad j \in Z,{}\end{array} $$

(7.53)

has a unique W-almost periodic solution in the sense of Definition 4. Due to the uniqueness, it coincides with $\varphi _{0}(t).$

Hence, the W-almost periodic function $\varphi _{0}(t): R \rightarrow X^{\alpha }$ satisfies Eq. ( 7.50) for t ∈ (τ _j , τ _j+1) and the difference equation ( 7.51) for t = τ _j . 

Now we study the stability of the almost periodic solution assuming exponential stability of the linear equation. First, using ideas in [17], we prove the following generalized Gronwall inequality for impulsive systems.

Lemma 6.

Assume that {t _j } is an increasing sequence of real numbers such that $Q \geq t_{j+1} - t_{j} \geq \theta > 0$ for all j, M ₁ ,M ₂ , and M ₃ are positive constants, and α ∈ (0,1). Then there exists a positive constant $\tilde{C}$ such that the positive piecewise continuous function u: [t ₀ ,t] → R satisfying

$$ \displaystyle\begin{array}{rcl} & & z(t) \leq M_{1}z_{0} + M_{2}\sum _{j=1}^{m}\int _{ t_{j-1}}^{t_{j} }(t_{j} - s)^{-\alpha }z(s)ds + M_{ 2}\int _{t_{m}}^{t}(t - s)^{-\alpha }z(s)ds + \\ & & +M_{3}\sum _{j=1}^{m}z(t_{ j})\quad \mathrm{for}\quad t \in (t_{m},t_{m+1}] {}\end{array} $$

(7.54)

also satisfies

$$ \displaystyle\begin{array}{rcl} z(t) \leq M_{1}z_{0}\tilde{C}\left (1 + M_{2}\tilde{C}\frac{Q^{1-\alpha }} {1-\alpha } + M_{3}\tilde{C}\right )^{m}.& &{}\end{array} $$

(7.55)

Proof.

We apply the method of mathematical induction. At the interval t ∈ [t ₀, t ₁] the inequality ( 7.54) has the form

$$ \displaystyle{z(t) \leq M_{1}z_{0} + M_{2}\int _{t_{0}}^{\tau _{1} }(\tau _{1} - s)^{-\alpha }z(s)ds.} $$

By Lemma 2 there exists $\tilde{C}$ such that

$$ \displaystyle\begin{array}{rcl} 0 \leq z(t) \leq M_{1}z_{0}\tilde{C},\ \ t \in [t_{0},t],\tilde{C} =\tilde{ C}(M_{1},M_{2},Q).& &{}\end{array} $$

(7.56)

Hence, ( 7.55) is true for t ∈ [t ₀, t ₁]. Assume ( 7.55) is true for t ∈ [t ₀, t _n ] and prove it for t ∈ (t _n , t _n+1]. Hence, for t ∈ (t _n , t _n+1] we have

$$ \displaystyle\begin{array}{rcl} & & z(t) \leq M_{1}z_{0} + M_{2}\int _{t_{0}}^{t_{1} }(t_{1} - s)^{-\alpha }z(s)ds + M_{ 3}z(t_{1}) + {}\\ & & +M_{2}\sum _{j=2}^{n}\int _{ t_{j-1}}^{t_{j} }(t_{j} - s)^{-\alpha }z(s)ds + M_{ 3}\sum _{j=1}^{n}z(t_{ j}) + M_{2}\int _{t_{n}}^{t}(t - s)^{-\alpha }z(s)ds \leq {}\\ & &\leq M_{1}z_{0} + M_{2}\frac{Q^{1-\alpha }} {1-\alpha } M_{1}z_{0}\tilde{C} + M_{3}M_{1}z_{0}\tilde{C} + M_{2}\int _{t_{n}}^{t}(t - s)^{-\alpha }z(s)ds + {}\\ & & +\sum _{j=2}^{n}\left (1 + M_{ 2}\tilde{C}\frac{Q^{1-\alpha }} {1-\alpha } + M_{3}\tilde{C}\right )^{j}\left (M_{ 2}\tilde{C}\frac{Q^{1-\alpha }} {1-\alpha } + M_{3}\tilde{C}\right )M_{1}z_{0} = {}\\ & & = M_{1}z_{0} + M_{2}\frac{Q^{1-\alpha }} {1-\alpha } M_{1}z_{0}\tilde{C} + M_{3}M_{1}z_{0}\tilde{C} + M_{2}\int _{t_{n}}^{t}(t - s)^{-\alpha }z(s)ds + {}\\ & & +\sum _{j=2}^{n}\left (1 + M_{ 2}\tilde{C}\frac{Q^{1-\alpha }} {1-\alpha } + M_{3}\tilde{C}\right )^{j-1}\left [\left (1 + M_{ 2}\tilde{C}\frac{Q^{1-\alpha }} {1-\alpha } + M_{3}\tilde{C}\right ) - 1\right ]M_{1}z_{0} = {}\\ & & \leq M_{1}z_{0}\left (1 + M_{2}\frac{Q^{1-\alpha }} {1-\alpha } \tilde{C} + M_{3}\tilde{C}\right )^{n} + M_{ 2}\int _{t_{n}}^{t}(t - s)^{-\alpha }z(s)ds. {}\\ \end{array} $$

Hence, for t ∈ [t _n , t _n+1), the function z(t) satisfies the inequality

$$ \displaystyle{z(t) \leq C_{1} + M_{2}\int _{t_{n}}^{t}(t - s)^{-\alpha }z(s)ds,} $$

where $C_{1} = M_{1}z_{0}\left (1 + M_{2}\frac{Q^{1-\alpha }} {1-\alpha } \tilde{C} + M_{3}\tilde{C}\right )^{n}.$ Applying ( 7.56) at the interval (t _n , t _n+1], we obtain ( 7.55). The lemma is proved.

Theorem 3.

Let Eqs. ( 7.50 ) and ( 7.51 ) satisfy assumptions of Theorem  2 and let the corresponding linear equation be exponentially stable.

Then for sufficiently small N ₁ > 0, the equation has a unique W-almost periodic solution u ₀ (t), and this solution is exponentially stable.

Proof.

The existence and uniqueness of the W-almost periodic solution u ₀(t) follows from Theorem 2. We prove its asymptotic stability. Let u(t) be an arbitrary solution of the equation satisfying $\|u(t_{0}) - u_{0}(t_{0})\|_{\alpha } \leq \delta,$ where δ is a small positive number.

Then by t ≥ t ₀ the difference of these solutions satisfies

$$ \displaystyle\begin{array}{rcl} & & u(t) - u_{0}(t) = U(t,t_{0})(u(t_{0}) - u_{0}(t_{0})) +\int _{ t_{0}}^{t}U(t,s)\Bigl (f(s,u(s)) - {}\\ & &-f(s,u_{0}(s))\Bigl )ds +\sum _{t_{0}\leq \tau _{k}<t}U(t,\tau _{k})\left (g_{k}(u(\tau _{k})) - g_{k}(u_{0}(\tau _{k}))\right ). {}\\ \end{array} $$

Then for t ₀ ∈ (τ ₀, τ ₁) and t ∈ (τ _j , τ _j+1] we have

$$ \displaystyle\begin{array}{rcl} & & \|u(t) - u_{0}(t)\|_{\alpha } \leq \| U(t,t_{0})\|_{\alpha }\|u(t_{0}) - u_{0}(t_{0})\|_{\alpha } + {}\\ & & +\int _{t_{0}}^{\tau _{1} }\|U(t,\tau _{1})\|_{\alpha }\|V (\tau _{1},s)(f(s,u(s)) - f(s,u_{0}(s)))\|_{\alpha }ds + \cdots + {}\\ & & +\int _{\tau _{j-1}}^{\tau _{j} }\|U(t,\tau _{j})\|_{\alpha }\|V (\tau _{j},s)(f(s,u(s)) - f(s,u_{0}(s)))\|_{\alpha }ds + {}\\ & & +\int _{\tau _{j}}^{t}\|V (t,s)(f(s,u(s)) - f(s,u_{ 0}(s)))\|_{\alpha }ds + {}\\ & & +\sum _{t_{0}\leq \tau _{k}<t}\|U(t,\tau _{k})\left (g_{k}(u(\tau _{k})) - g_{k}(u_{0}(\tau _{k}))\right )\|_{\alpha } \leq {}\\ & &\leq Me^{-\beta (t-t_{0})}\|u(t_{ 0}) - u_{0}(t_{0})\|_{\alpha } + Me^{-\beta (t-\tau _{1})}\int _{ t_{0}}^{\tau _{1} } \frac{L_{Q}N_{1}} {(\tau _{1} - s)^{\alpha }}\|u(s) - u_{0}(s)\|_{\alpha }ds + {}\\ & & +\cdots + Me^{-\beta (t-\tau _{j})}\int _{ \tau _{j-1}}^{\tau _{j} } \frac{L_{Q}N_{1}} {(\tau _{j} - s)^{\alpha }}\|u(s) - u_{0}(s)\|_{\alpha }ds + {}\\ & & +\int _{\tau _{j}}^{t} \frac{L_{Q}N_{1}} {(t - s)^{\alpha }}\|u(s) - u_{0}(s)\|_{\alpha }ds +\sum _{t_{0}\leq \tau _{k}<t}Me^{-\beta (t-\tau _{k})}N_{ 1}\|u(\tau _{k}) - u_{0}(\tau _{k})\|_{\alpha }. {}\\ \end{array} $$

Denote $v(t) = e^{\beta t}\|u(t) - u_{0}(t)\|_{\alpha },M_{2} = e^{\beta Q}ML_{Q}N_{1},M_{3} = MN_{1}.$ Then

$$ \displaystyle\begin{array}{rcl} v(t) \leq Mv(t_{0}) + M_{2}\int _{t_{0}}^{\tau _{1} } \frac{v(s)ds} {(\tau _{1} - s)^{\alpha }} + \cdots + M_{2}\int _{t_{j}}^{t} \frac{v(s)ds} {(\tau _{j} - s)^{\alpha }} + M_{3}\sum _{k=1}^{j}v(\tau _{ k}).& & {}\\ \end{array} $$

Then by Lemma 6 we get

$$ \displaystyle\begin{array}{rcl} \|u(t) - u_{0}(t)\|_{\alpha } \leq M\tilde{C}e^{-\beta (t-t_{0})}\left (1 + M_{ 2}\tilde{C}\frac{Q^{1-\alpha }} {1-\alpha } + M_{3}\tilde{C}\right )^{i(t,t_{0})}\|u(t_{ 0}) - u_{0}(t_{0})\|_{\alpha }.& & {}\\ \end{array} $$

Therefore, if

$$ \displaystyle\begin{array}{rcl} \beta > p\ln \left (1 + M_{2}\tilde{C}\frac{Q^{1-\alpha }} {1-\alpha } + M_{3}\tilde{C}\right ),& & {}\\ \end{array} $$

where p is defined by ( 7.3), then the W-almost periodic solution u ₀(t) of Eqs. ( 7.50) and ( 7.51) is asymptotically stable. This can be achieved by sufficiently small N ₁. 

7.5 Almost Periodic Solutions of Equations with Nonfixed Moments of Impulsive Action

We consider the following equation with points of impulsive action depending on solutions

$$ \displaystyle\begin{array}{rcl} & & \frac{du} {dt} + Au = f(t,u),\quad t\not =\tau _{j}(u),{}\end{array} $$

(7.57)

$$ \displaystyle\begin{array}{rcl} & & u(\tau _{j}(u) + 0) - u(\tau _{j}(u)) = B_{j}u + g_{j}(u),\quad j \in Z.{}\end{array} $$

(7.58)

Definition 6 ([11]).

A solution u ₀(t) of Eqs. ( 7.57) and ( 7.58) defined for all t ≥ t ₀, is called Lyapunov stable in space X ^α if, for an arbitrary $\varepsilon > 0$ and η > 0, there exists such a number $\delta =\delta (\varepsilon,\eta )$ that, for any other solution u(t) of the system, $\|u_{0}(t_{0}) - u(t_{0})\|_{\alpha } <\delta$ implies that $\|u_{0}(t) - u(t)\|_{\alpha } <\varepsilon$ for all t ≥ t ₀ such that $\vert t -\tau _{j}^{0}\vert >\eta,$ where $\tau _{j}^{0}$ are the times during which the solution u ₀(t) intersects the surfaces t = τ _j (u), j ∈ Z. 

A solution u ₀(t) is said to be attractive if for each $\varepsilon > 0,\eta > 0,$ and t ₀ ∈ R, there exist $\delta _{0} =\delta _{0}(t_{0})$ and $T = T(\delta _{0},\varepsilon,\eta ) > 0$ such that for any other solution u(t) of the system, $\|u_{0}(t_{0}) - u(t_{0})\| <\delta _{0}$ implies $\|u_{0}(t) - u(t)\|_{\alpha } <\varepsilon$ for t ≥ t ₀ + T and $\vert t -\tau _{k}^{0}\vert >\eta.$

A solution u ₀(t) is called asymptotically stable if it is stable and attractive.

Theorem 4.

Assume that in some domain $U_{\rho }^{\alpha } =\{ u \in X^{\alpha },\|u\|_{\alpha } \leq \rho \}$ , Eqs. ( 7.57 ) and ( 7.58 ) satisfy conditions (H1), (H3)–(H6), and

• 1) all solutions in domain $U_{\rho }^{\alpha }$ intersect each surface t = τ _j (u) no more than once;

• 2) $\|f(t_{1},u) - f(t_{2},u)\| \leq H_{1}\vert t_{1} - t_{2}\vert ^{\nu },\ \nu > 0,\ H_{1} > 0;$

• 3) $\|f(t,u_{1}) - f(t,u_{2})\| +\| g_{j}(u_{1}) - g_{j}(u_{2})\|_{\alpha } + \vert \tau _{j}(u_{1}) -\tau _{j}(u_{2})\vert \leq N_{1}\|u_{1} - u_{2}\|_{\alpha },$ uniformly to t ∈ R,j ∈ Z,

• 4) $AB_{j} = B_{j}A,\|f(t,0)\| \leq M_{0},\|g_{j}(0)\|_{1} \leq M_{0},j \in Z$

• 5) the linear homogeneous equation

  $$ M_{{\ast}} = \frac{M_{1}} {1 - e^{-\beta _{1}\theta }} \left (1 + \frac{C_{\alpha }Q^{1-\alpha }} {1-\alpha } \right ). $$
  $$ \displaystyle\begin{array}{rcl} & & \frac{du} {dt} + Au = 0,\quad t\not =\tau _{j}, {}\end{array} $$

  (7.59)
  $$ \displaystyle\begin{array}{rcl} & & \varDelta u\vert _{t=\tau _{j}} = u(\tau _{j} + 0) - u(\tau _{j}) = B_{j}u(\tau _{j}),\quad j \in Z, {}\end{array} $$

  (7.60)

  is exponentially stable in space X ^α

  $$ \displaystyle\begin{array}{rcl} \|U(t,s)u\|_{\alpha } \leq Me^{-\beta (t-s)}\|u\|_{\alpha },\ t \geq s,u \in X^{\alpha }& & {}\end{array} $$

  (7.61)

  where τ _j = τ _j (0), β > 0 and M ≥ 1.

• 6) N ₁ M _∗ < 1 and ρ ≥ρ ₀ = M ₀ M _∗ ∕(1 − N ₁ M _∗ ), where

Then for sufficiently small values of the Lipschitz constant N ₁ , Eqs. ( 7.57 ) and ( 7.58 ) have in $U_{\rho }^{\alpha }$ a unique W-almost periodic solution and this solution is exponentially stable.

Proof.

• 1. First, using the method proposed in [6], we prove the existence of the W-almost periodic solution. Let y = { y _j } be an almost periodic sequence of elements $y_{j} \in X^{\alpha },\|y_{j}\|_{\alpha } \leq \varrho.$ We consider the equation with fixed moments of impulsive action

  $$ \displaystyle\begin{array}{rcl} & & \frac{du} {dt} + Au = f(t,u),\quad t\not =\tau _{j}(y), {}\end{array} $$

  (7.62)
  $$ \displaystyle\begin{array}{rcl} & & u(\tau _{j}(y_{j}) + 0) - u(\tau _{j}(y_{j})) = B_{j}u(\tau _{j}(y_{j})) + g_{j}(y_{j}),\quad j \in Z. {}\end{array} $$

  (7.63)

By Lemma 5, if a constant N ₁ sufficiently small, then corresponding to ( 7.62) and ( 7.63) the linear impulsive equation [if $f \equiv 0,g_{j}(y_{j}) \equiv 0,j \in Z,$] is exponentially stable. Its evolution operator U(t, τ, y) satisfies estimate

$$ \displaystyle\begin{array}{rcl} \|U(t,\tau,y)u\|_{\alpha } \leq M_{1}e^{-\beta _{1}(t-\tau )}\|u\|_{\alpha },\ t \geq \tau,& &{}\end{array} $$

(7.64)

with some positive constants M ₁ ≥ M, β ₁ ≤ β. 

Equations ( 7.62) and ( 7.63) have a unique solution bounded on the axis which satisfies the integral equation

$$ \displaystyle\begin{array}{rcl} \tilde{u}(t,y) =\int _{ -\infty }^{t}U(t,\tau,y)f(\tau,\tilde{u}(\tau,y))d\tau +\sum _{\tau _{ j}(y_{j})<t}U(t,\tau _{j}(y_{j}),y)g_{j}(y_{j}).& &{}\end{array} $$

(7.65)

We choose u ₀(t, y) ≡ 0 and construct the sequence of W-almost periodic functions

$$ \displaystyle{u_{n+1}(t,y)\,=\,\int _{-\infty }^{t}U(t,\tau,y)f(\tau,u_{ n}(\tau,y))d\tau +\!\sum _{\tau _{j}(y_{j})<t}U(t,\tau _{j}(y_{j}),y)g_{j}(y_{j}),\ n\,=\,0,1,\ldots.} $$

The proof of the W-almost periodicity of u _n+1(t, y) in space X ^α is similar to the proof of Theorem 1.

One can verify that for sufficiently small N ₁ > 0 the sequence {u _n (t, y)} converges to the W-almost periodic solution u ^∗(t, y): R → X ^α of Eq. ( 7.65). As in the proof to Theorem 2, we prove that u ^∗(t, y) is the W-almost periodic solution of impulsive equations ( 7.62) and ( 7.63).

Let $t \in (\tilde{\tau }_{i},\tilde{\tau }_{i+1}),$ where $\tilde{\tau }_{i} =\tau _{i}(y_{i}).$ As in ( 7.39), we obtain

$$ \displaystyle\begin{array}{rcl} & & \|u^{{\ast}}(t,y)\|_{\alpha } \leq \int _{ -\infty }^{t}\|A^{\alpha }U(t,s,y)(f(s,0) + f(s,u^{{\ast}}(s,y)) - f(s,0))\|ds + {}\\ & & +\sum _{\tau _{j}(y_{j})<t}\|U(t,\tilde{\tau }_{j},y)(g_{j}(0) + g_{j}(y_{j}) - g_{j}(0))\|_{\alpha } \leq {}\\ & &\leq \frac{M_{1}} {1 - e^{-\beta _{1}\theta }} \left (\frac{C_{\alpha }\varTheta ^{1-\alpha }} {1-\alpha } \left (M_{0} + N_{1}\sup _{t}\|u^{{\ast}}(t,y)\|_{\alpha }\right ) + M_{ 0} + N_{1}\sup _{j}\|y_{j}\|_{\alpha }\right ). {}\\ \end{array} $$

Hence, by sufficiently small N ₁ > 0

$$ \displaystyle\begin{array}{rcl} \sup _{t}\|u^{{\ast}}(t,y)\| \leq \rho _{ 0}.& &{}\end{array} $$

(7.66)

If we choose the almost periodic sequence $y^{{\ast}} =\{ y_{j}^{{\ast}}\},y_{j}^{{\ast}}\in X^{\alpha },$ such that

$$ \displaystyle{u^{{\ast}}(\tau _{ j}(y_{j}^{{\ast}}),y^{{\ast}}) = y_{ j}^{{\ast}}} $$

for all j ∈ Z, then the function u ^∗(t, y ^∗) will be exactly the W-almost periodic solution of Eqs. ( 7.57) and ( 7.58).

We consider the space $\mathcal{N}$ of sequences y = { y _j },  y _j  ∈ X ^α, with norm $\|y\|_{S} =\sup _{j}\|y_{j}\|_{\alpha }$ and map $S:\ \mathcal{N} \rightarrow \mathcal{N},$

$$ \displaystyle{S(y) =\{ u^{{\ast}}(\tau _{ j}(y_{j}),y)\}_{j\in Z}.} $$

By ( 7.66), S maps the domain $U_{\varrho }^{\alpha } \subset \mathcal{N}$ onto itself for ρ = ρ ₀. 

Now we prove that S is a contraction:

$$ \displaystyle\begin{array}{rcl} & & \|S(y)_{j} - S(z)_{j}\|_{\alpha } =\| u^{{\ast}}(\tau _{ j}(y_{j}),y) - u^{{\ast}}(\tau _{ j}(z_{j}),z)\|_{\alpha } \leq \\ & &\leq \| u^{{\ast}}(\tilde{\tau }_{ j}^{1},y) - u^{{\ast}}(\tilde{\tau }_{ j}^{1},z)\|_{\alpha } +\| u^{{\ast}}(\tilde{\tau }_{ j}^{1},z) - u^{{\ast}}(\tilde{\tau }_{ j}^{2},z)\|_{\alpha },{}\end{array} $$

(7.67)

where $\tilde{\tau }_{j}^{1} =\tau _{j}(y_{j}),\ \tilde{\tau }_{j}^{2} =\tau _{j}(z_{j}).$

Denote $\mathcal{J} = \cup \mathcal{J}_{j},$

$$ \displaystyle{\mathcal{J}_{j} = (\max \{\tilde{\tau }_{j-1}^{1},\tilde{\tau }_{ j-1}^{2}\},\min \{\tilde{\tau }_{ j}^{1},\tilde{\tau }_{ j}^{2}\}] = (\tau ''_{ j-1},\tau '_{j}].} $$

Denote also $\xi _{i} = (\tau '_{i} -\tau ''_{j-1})/2,\ i \in Z.$

To estimate the difference $\|u^{{\ast}}(\tilde{\tau }_{j}^{1},y) - u^{{\ast}}(\tilde{\tau }_{j}^{1},z)\|_{\alpha }$, we apply iteration on n. Put u ₀(t, y) = u ₀(t, z) = 0. Then for $t \in (\tilde{\tau }''_{i},\tilde{\tau }'_{i+1}]$ we get

$$ \displaystyle\begin{array}{rcl} & & \|u_{1}(t,y) - u_{1}(t,z)\|_{\alpha } = \\ & & =\|\sum _{k\leq i}A^{\alpha }U(t,\tilde{\tau }_{k}^{1},y)g_{ k}(y_{k}) -\sum _{k\leq i}A^{\alpha }U(t,\tilde{\tau }_{ k}^{2},z)g_{ k}(z_{k})\| \leq \\ & &\leq \sum _{k\leq i}\|A^{\alpha }U(t,\tilde{\tau }_{k}^{1},y)\left (g_{ k}(y_{k}) - g_{k}(z_{k})\right )\| +\| A^{\alpha }\left (U(t,\tilde{\tau }_{ i}^{1},y) - U(t,\tilde{\tau }_{ i}^{2},z)\right )g_{ i}(z_{i})\| + \\ & & +\sum _{k<i}\|A^{\alpha }\left (U(t,\tilde{\tau }_{k}^{1},y) - U(t,\tilde{\tau }_{ k}^{2},z)\right )g_{ k}(z_{k})\| \leq \\ & &\leq \sum _{k<i}M_{1}e^{-\beta _{1}\vert t-\tilde{\tau }_{k}^{1}\vert }N_{1}\|y_{k} - z_{k}\|_{\alpha } +\| A^{\alpha }e^{-A(t-\tilde{\tau }''_{i})}(e^{-A(\tilde{\tau }''_{i}-\tilde{\tau }'_{i})} - I)g_{ i}(z_{i})\| + \\ & & +\sum _{k<i}\|(U(t,\xi _{i},y)(U(\xi _{i},\xi _{k+1},y)U(\xi _{k+1},\tilde{\tau }_{k}^{1},y) - \\ & &-U(t,\xi _{i},z)(U(\xi _{i},\xi _{k+1},z)U(\xi _{k+1},\tilde{\tau }_{k}^{2},z))g_{ k}(z_{k})\|_{\alpha } \leq \\ & &\leq \frac{M_{1}N_{1}} {1 - e^{-\beta _{1}\theta }} \|y - z\|_{S} + C_{\alpha }C_{0}(t -\tilde{\tau }''_{i})^{-\alpha }\vert \tilde{\tau }''_{ k} -\tilde{\tau }'_{k}\vert \|g_{i}(z_{i})\|_{1} + \\ & & +\sum _{k<i}\|A^{\alpha }(U(t,\xi _{i},y) - U(t,\xi _{i},z))U(\xi _{i},\xi _{k+1},y)U(\xi _{k+1},\tilde{\tau }_{k}^{1},y)g_{ k}(z_{k})\| + \\ & & +\sum _{k<i}\|A^{\alpha }U(t,\xi _{i},z)(U(\xi _{i},\xi _{k+1},y) - U(\xi _{i},\xi _{k+1},z))U(\xi _{k},\tilde{\tau }_{k}^{1},y)g_{ k}(z_{k})\| + \\ & & +\sum _{k<i}\|A^{\alpha }U(t,\xi _{i},z)U(\xi _{i},\xi _{k+1},z)(U(\xi _{k+1},\tilde{\tau }_{k}^{1},y) - U(\xi _{ k+1},\tilde{\tau }_{k}^{2},z))g_{ k}(z_{k})\|. {}\end{array} $$

(7.68)

To evaluate the difference $U(\xi _{i},\xi _{k+1},y) - U(\xi _{i},\xi _{k+1},z)$ we construct two sequences of bounded operators X ^α → X ^α defined by

$$ \displaystyle{T_{n} = U(\xi _{n+1},\xi _{n},y),\ \tilde{T}_{n} = U(\xi _{n+1},\xi _{n},z),\ n \in Z.} $$

The corresponding difference equations u _n+1 = T _n u _n and $u_{n+1} =\tilde{ T}_{n}u_{n}$ are exponentially stable. Their evolution operators

$$ \displaystyle{T_{n,m} = T_{n-1}\ldots T_{m},\ n \geq m,\ T_{m,m} = I,} $$

and

$$ \displaystyle{\tilde{T}_{n,m} =\tilde{ T}_{n-1}\ldots \tilde{T}_{m},\ n \geq m,\ \tilde{T}_{m,m} = I,} $$

satisfy equality

$$ \displaystyle\begin{array}{rcl} \tilde{T}_{n,m} - T_{n,m} =\sum _{k<n}T_{n,k+1}(\tilde{T}_{k} - T_{k})\tilde{T}_{k,m},\ n \geq m.& &{}\end{array} $$

(7.69)

Analogous to ( 7.32) and ( 7.33), we obtain

$$ \displaystyle\begin{array}{rcl} \|\tilde{T}_{n,m} - T_{n,m}\|_{\alpha } \leq M_{2}e^{-\beta _{2}\theta (n-m)}\sup _{ k}\|\tilde{T}_{k} - T_{k}\|_{\alpha },\ n \geq m,& &{}\end{array} $$

(7.70)

with some β ₂ ≤ β ₁, M ₂ ≥ M ₁. 

Now we estimate the difference $\|\tilde{T}_{n} - T_{n}\|_{\alpha }:$

$$ \displaystyle\begin{array}{rcl} & & \|T_{n} -\tilde{ T}_{n}\|_{\alpha } =\| U(\xi _{n+1},\xi _{n},y) - U(\xi _{n+1},\xi _{n},z)\|_{\alpha } = \\ & & =\| e^{-A(\xi _{n+1}-\tilde{\tau }_{n}^{1}) }(I + B_{n})e^{-A(\tilde{\tau }_{n}^{1}-\xi _{ n})} - e^{-A(\xi _{n+1}-\tilde{\tau }_{n}^{2}) }(I + B_{n})e^{-A(\tilde{\tau }_{n}^{2}-\xi _{ n})}\|_{\alpha } \leq \\ & &\leq \| (e^{-A(\xi _{n+1}-\tilde{\tau }_{n}^{1}) } - e^{-A(\xi _{n+1}-\tilde{\tau }_{n}^{2}) })(I + B_{n})e^{-A(\tilde{\tau }_{n}^{1}-\xi _{ n})}\|_{\alpha } + \\ & & +\|e^{-A(\xi _{n+1}-\tilde{\tau }_{n}^{2}) }(I + B_{n})(e^{-A(\tilde{\tau }_{n}^{1}-\xi _{ n})} - e^{-A(\tilde{\tau }_{n}^{2}-\xi _{ n})})\|_{\alpha } \leq \\ & &\leq 2C_{\alpha }C_{1}(\theta /2)^{-1-\alpha }\vert \tilde{\tau }_{ n}^{1} -\tilde{\tau }_{ n}^{2}\vert. {}\end{array} $$

(7.71)

Therefore,

$$ \displaystyle\begin{array}{rcl} & & \|(\tilde{T}_{n,m} - T_{n,m})u\|_{\alpha } =\| (U(\xi _{n},\xi _{m},y) - U(\xi _{n},\xi _{m},z))u\|_{\alpha } \leq \\ & &\leq M_{2}e^{-\beta _{2}\theta (n-m)}2C_{\alpha }C_{ 1}(\theta /2)^{-1-\alpha }\sup _{ j}\vert \tilde{\tau }_{i}^{1} -\tilde{\tau }_{ i}^{2}\vert \|u\|_{\alpha },\ n \geq m.{}\end{array} $$

(7.72)

To finish the estimation of ( 7.68), we consider the following two differences:

$$ \displaystyle\begin{array}{rcl} & & \|(U(t,\xi _{i},y) - U(t,\xi _{i},z)))u\|_{\alpha } \leq \| A^{\alpha }(e^{-A(t-\tau '_{i})}(I + B_{ i})e^{-A(\tau '_{i}-\xi _{i})} - \\ & &-e^{-A(t-\tau ''_{i})}(I + B_{ i})e^{-A(\tau ''_{i}-\xi _{i})})u\| \leq \frac{4C_{0}C_{1-\alpha }} {\theta (t -\tau ''_{i})^{\alpha }} \vert \tau ''_{i} -\tau '_{i}\vert \|u\|_{\alpha }. {}\end{array} $$

(7.73)

$$ \displaystyle\begin{array}{rcl} & & \|(U(\xi _{k},\tilde{\tau }_{k}^{1},y) - U(\xi _{ k},\tilde{\tau }_{k}^{2},z))u\|_{\alpha } =\| A^{\alpha }(I - e^{-A(\tau ''_{k}-\tau '_{k})})e^{-A(\xi _{k+1}-\tau ''_{k})}u\| \leq \\ & &\leq C_{0}C_{1}(\theta /2)^{-\alpha }\vert \tau ''_{ i} -\tau '_{i}\vert \|u\|_{\alpha }. {}\end{array} $$

(7.74)

Taking into account ( 7.70), ( 7.73), and ( 7.74), by ( 7.68) we obtain for t ∈ (τ″ _i , τ′ _i+1]

$$ \displaystyle\begin{array}{rcl} \|u_{1}(t,y) - u_{1}(t,z)\|_{\alpha } \leq N_{1}\|y - z\|_{S}\left (K'_{1} + K''_{2}(t -\tau ''_{i})^{-\alpha }\right ),& &{}\end{array} $$

(7.75)

where the positive constants K′₁ and K″₂ don’t depend on i. 

Now we consider the (n + 1)st iteration

$$ \displaystyle\begin{array}{rcl} & & \|u_{n+1}(t,y) - u_{n+1}(t,z)\|_{\alpha } = \\ & & =\|\int _{ -\infty }^{t}A^{\alpha }U(t,\tau,y)f(\tau,u_{ n}(\tau,y))d\tau +\sum _{k\leq i}A^{\alpha }U(t,\tilde{\tau }_{ k}^{1},y)g_{ k}(y_{k}) - \\ & &-\int _{-\infty }^{t}A^{\alpha }U(t,\tau,z)f(\tau,u_{ n}(\tau,z))d\tau -\sum _{k\leq i}A^{\alpha }U(t,\tilde{\tau }_{ k}^{2},z)g_{ k}(z_{k})\| \leq \\ & &\leq \int _{-\infty }^{t}\|A^{\alpha }U(t,\tau,y)\left (f(\tau,u_{ n}(\tau,y)) - f(\tau,u_{n}(\tau,z))\right )\|d\tau + \\ & & +\int _{-\infty }^{t}\|A^{\alpha }(U(t,\tau,y) - U(t,\tau,z))f(\tau,u_{ n}(\tau,z))\|d\tau + \\ & & +\sum _{k\leq i}\|A^{\alpha }U(t,\tilde{\tau }_{k}^{1},y)\left (g_{ k}(y_{k}) - g_{k}(z_{k})\right )\| + \\ & & +\sum _{k\leq i}\|A^{\alpha }\left (U(t,\tilde{\tau }_{k}^{1},y) - U(t,\tilde{\tau }_{ k}^{2},z)\right )g_{ k}(z_{k})\|. {}\end{array} $$

(7.76)

Similar to ( 7.39), we get

$$ \displaystyle\begin{array}{rcl} & & \int _{\tau ''_{i}}^{t}\|A^{\alpha }e^{-A(t-s)}\left (f(\tau,u_{ n}(\tau,y)) - f(\tau,u_{n}(\tau,z))\right )\|d\tau + \\ & & \sum _{k<i}\int _{\tau ''_{k}}^{\tau '_{k+1} }\|A^{\alpha }U(t,\tau,y)\left (f(\tau,u_{n}(\tau,y)) - f(\tau,u_{n}(\tau,z))\right )\|d\tau + \\ & & \leq \frac{M_{1}} {1 - e^{-\theta \beta _{1}}} \frac{C_{\alpha }\varTheta ^{1-\alpha }} {1-\alpha } N_{1}\sup _{\tau \in \mathcal{J}}\|u_{n}(\tau,y) - u_{n}(\tau,z)\|, \\ & & \sum _{k\leq i}\|A^{\alpha }U(t,\tilde{\tau }_{k}^{1},y)\left (g_{ k}(y_{k}) - g_{k}(z_{k})\right )\| \leq \frac{M_{1}} {1 - e^{-\theta \beta _{1}}} N_{1}\|y - z\|_{\alpha }.{}\end{array} $$

(7.77)

If $\|u_{n}(\tau,y)\|_{\alpha } \leq \rho$ and $\|u_{n}(\tau,z)\|_{\alpha } \leq \rho$, then for t ∈ (τ″ _i , τ′ _i+1]

$$ \displaystyle\begin{array}{rcl} & & \sum _{k\leq i}\int _{\tau '_{k}}^{\tau ''_{k} }\|A^{\alpha }U(t,s,y)\left (f(s,u_{n}(s,y)) - f(s,u_{n}(s,z))\right )\|ds \leq \\ & &\leq \sum _{k\leq i}\int _{\tau '_{k}}^{\tau ''_{k} }\|U(t,s,y)f(s,u_{n}(s,y))\|_{\alpha }ds +\sum _{k\leq i}\int _{\tau '_{k}}^{\tau ''_{k} }\|U(t,s,y)f(s,u_{n}(s,z))\|_{\alpha }ds \leq \\ & &\leq 2\sum _{k<i}M_{1}e^{-\beta _{1}\vert t-\tau ''_{k}\vert }(M_{ 0} + N_{1}\rho ) + 2\int _{\tau '_{i}}^{\tau ''_{i} }\|A^{\alpha }U(t,s,y)\|(M_{0} + N_{1}\rho )ds \leq \\ & &\leq \left ( \frac{2M_{1}} {1 - e^{-\beta _{1}\theta }} + \frac{2M_{1}} {1-\alpha } (t -\tau ''_{i})^{-\alpha }\right )(M_{ 0} + N_{1}\rho )N_{1}\|y - z\|_{S}, {}\end{array} $$

(7.78)

since for t > τ ₂ > τ ₁

$$ \displaystyle{\int _{\tau _{1}}^{\tau _{2} } \frac{ds} {(t - s)^{\alpha }} \leq \frac{\tau _{2} -\tau _{1}} {(1-\alpha )((t -\tau _{2})^{\alpha }}.} $$

The second integral in ( 7.76) satisfies the following inequality:

$$ \displaystyle\begin{array}{rcl} & & I_{2} =\int _{ -\infty }^{t}\|A^{\alpha }(U(t,s,y) - U(t,s,z))f(s,u_{ n}(s,z))\|ds \leq \\ & &\leq \int _{\tau ''_{i}}^{t}\|A^{\alpha }(e^{-A(t-s)} - e^{-A(t-s)})f(s,u_{ n}(s,z))\|ds + \\ & & +\int _{\tau '_{i}}^{\tau ''_{i} }\|A^{\alpha }(U(t,s,y) - U(t,s,z))f(s,u_{n}(s,z))\|ds + \\ & & +\int _{\xi _{i}}^{\tau '_{i} }\|A^{\alpha }(U(t,s,y) - U(t,s,z))f(s,u_{n}(s,z))\|ds + \\ & & +\sum _{k<i}\int _{\xi _{k}}^{\xi _{k+1} }\|A^{\alpha }(U(t,s,y) - U(t,s,z))f(s,u_{n}(s,z))\|ds.{}\end{array} $$

(7.79)

We consider all integrals in ( 7.79) separately.

$$ \displaystyle\begin{array}{rcl} & & I_{21} =\int _{ \tau '_{i}}^{\tau ''_{i} }\|A^{\alpha }U(t,s,y)f(s,u_{n}(s,z))\|ds \leq \frac{C_{\alpha }\|I + B_{i}\|(M_{0} + N_{1}\rho )} {(1-\alpha )(t -\tau ''_{i})^{\alpha }} \vert \tau ''_{i} -\tau '_{i}\vert, {}\\ & & I_{22} =\int _{ \tau '_{i}}^{\tau ''_{i} }\|A^{\alpha }U(t,s,z)f(s,u_{n}(s,z))\|ds \leq \frac{C_{\alpha }\|I + B_{i}\|(M_{0} + N_{1}\rho )} {(1-\alpha )(t -\tau ''_{i})^{\alpha }} \vert \tau ''_{i} -\tau '_{i}\vert, {}\\ & & I_{23} =\int _{ \xi _{i}}^{\tau '_{i} }\|A^{\alpha }(U(t,s,y) - U(t,s,z))f(s,u_{n}(s,z))\|ds = {}\\ & & =\int _{ \xi _{i}}^{\tau '_{i} }\|A^{\alpha }(U(t,\tilde{\tau }_{i}^{1},y)U(\tilde{\tau }_{ i}^{1},s,y) - U(t,\tilde{\tau }_{ i}^{2},z)U(\tilde{\tau }_{ i}^{2},s,z))f(s,u_{ n}(s,z))\|ds \leq {}\\ & &\leq \int _{\xi _{i}}^{\tau '_{i} }\|A^{\alpha }\Bigl ((e^{-A(t-\tilde{\tau }_{i}^{1}) } - e^{-A(t-\tilde{\tau }_{i}^{2}) })(I + B_{i})e^{-A(\tilde{\tau }_{i}^{1}-s) } - {}\\ & &-A^{\alpha }e^{-A(t-\tilde{\tau }_{i}^{2}) })(I + B_{i})(e^{-A(\tilde{\tau }_{i}^{1}-s) } - e^{-A(\tilde{\tau }_{i}^{2}-s) })\Bigl )f(s,u_{n}(s,z))\|ds \leq {}\\ & &\leq \frac{2C_{0}C_{\alpha _{1}}C_{1+\alpha -\alpha _{1}}} {(t -\tau ''_{i})^{\alpha _{1}}} \|I + B_{i}\|\frac{(\tau '_{i} -\xi _{i})^{\alpha _{1}-\alpha }} {\alpha _{1}-\alpha } \vert \tau ''_{i} -\tau '_{i}\vert. {}\\ \end{array} $$

The last sum in ( 7.79) is transformed as follows:

$$ \displaystyle\begin{array}{rcl} & & I_{24} =\sum _{k<i}\int _{\xi _{k}}^{\xi _{k+1} }\|A^{\alpha }(U(t,s,y) - U(t,s,z))f(s,u_{n}(s,z))\|ds = {}\\ & & =\sum _{k<i}\int _{\xi _{k}}^{\xi _{k+1} }\|(U(t,\xi _{i},y)U(\xi _{i},\xi _{k+1},y)U(\xi _{k+1},s,y) - {}\\ & &-U(t,\xi _{i},z)U(\xi _{i},\xi _{k+1},z)U(\xi _{k+1},s,z))f(s,u_{n}(s,z))\|_{\alpha }ds \leq {}\\ & &\leq \sum _{k<i}\int _{\xi _{k}}^{\xi _{k+1} }\Biggl (\|(U(t,\xi _{i},y) - U(t,\xi _{i},z))U(\xi _{i},\xi _{k+1},y)U(\xi _{k+1},s,y)f(s,u_{n}(s,z))\|_{\alpha } + {}\\ & & +\|U(t,\xi _{i},z)(U(\xi _{i},\xi _{k+1},y) - U(\xi _{i},\xi _{k+1},z))U(\xi _{k+1},s,y)f(s,u_{n}(s,z))\|_{\alpha } + {}\\ & & +\|U(t,\xi _{i},z)U(\xi _{i},\xi _{k+1},z)(U(\xi _{k+1},s,y) - U(\xi _{k+1},s,z))f(s,u_{n}(s,z))\|_{\alpha }\Biggl )ds. {}\\ \end{array} $$

To finish the estimation of integral I ₂₄ we use ( 7.72), ( 7.73), and ( 7.74):

$$ \displaystyle\begin{array}{rcl} & & \int _{\xi _{k}}^{\xi _{k+1} }\|A^{\alpha }(U(\xi _{k+1},s,y) - U(\xi _{k+1},s,z))f\|_{\alpha }ds \leq {}\\ & &\leq \int _{\tau ''_{k}}^{\xi _{k+1} }\|A^{\alpha }(e^{-A(\xi _{k+1}-s)} - e^{-A(\xi _{k+1}-s)})f\|ds + {}\\ & & +\int _{\tau '_{k}}^{\tau ''_{k} }\|A^{\alpha }(e^{-A(\xi _{k+1}-\tau ''_{k})}(I + B_{ k})e^{-A(\tau ''_{k}-s)} - e^{-A(\xi _{k+1}-s)})f\|ds + {}\\ & & +\int _{\xi _{k}}^{\tau '_{k} }\|(e^{-A(\xi _{k+1}-\tau '_{k})}(I + B_{ k})e^{-A(\tau '_{k}-s)} - e^{-A(\xi _{k+1}-\tau ''_{k})}(I + B_{ k})e^{-A(\tau ''_{k}-s)})f\|_{\alpha }ds \leq {}\\ & &\leq \tilde{ K}C_{\alpha _{1}}(\xi _{k+1} -\tau ''_{k})^{-\alpha _{1} }\|I + B_{k}\|\vert \tau ''_{k} -\tau '_{k}\vert \|f\| {}\\ \end{array} $$

with some positive constant $\tilde{K}.$ Therefore,

$$ \displaystyle\begin{array}{rcl} I_{2} \leq \left (K'_{2}N_{1} + \frac{K''_{2}N_{1}} {(t -\tau ''_{i})^{\alpha _{1}}} \right )\|y - z\|_{S}& &{}\end{array} $$

(7.80)

with α ₁ > α and positive constants K′₁ and K″₂ independent of i, k. 

By ( 7.75), ( 7.79), and ( 7.80) we obtain for t ∈ (τ″ _i , τ′ _i+1]

$$ \displaystyle\begin{array}{rcl} & & \|u_{n+1}(t,y) - u_{n+1}(t,z)\|_{\alpha } \leq \\ & &\leq \sum _{k<i}\int _{\tau ''_{k}}^{\tau '_{k+1} }\|A^{\alpha }U(t,\tau,y)(f(\tau,u_{n}(\tau,y)) - f(\tau,u_{n}(\tau,z)))\|d\tau + \\ & & +\int _{\tau ''_{i}}^{t}\|A^{\alpha }U(t,\tau,y)(f(\tau,u_{ n}(\tau,y)) - f(\tau,u_{n}(\tau,z)))\|d\tau + \\ & & +\left (K'_{3} + \frac{K''_{3}} {(t -\tau ''_{i})^{\alpha _{1}}} \right )N_{1}\|y - z\|_{S}, {}\end{array} $$

(7.81)

where the constants K′₃ and K″₃ don’t depend on n. 

Let the nth iteration satisfy the inequality

$$ \displaystyle{\|u_{n}(t,y) - u_{n}(t,z)\|_{\alpha } \leq \left (L'_{n} + \frac{L''_{n}} {(t -\tau ''_{i})^{\alpha _{1}}} \right )N_{1}\|y - z\|_{S},\ t \in (\tau ''_{i},\tau '_{i+1}],} $$

with positive constants L′ _n and L″ _n . We estimate the (n + 1)st iteration.

$$ \displaystyle\begin{array}{rcl} & & \|u_{n+1}(t,y) - u_{n+1}(t,z)\|_{\alpha } \leq \left (K'_{3} + \frac{K''_{3}} {(t -\tilde{\tau }''_{i})^{\alpha _{1}}} \right )N_{1}\|y - z\|_{S} + \\ & & +N_{1}^{2}\|y - z\|_{ S}\sum _{k<i}\int _{\tau ''_{k}}^{\tau '_{k+1} }\|A^{\alpha }U(t,s)\|\left (L'_{n} + \frac{L''_{n}} {(s -\tau ''_{k})^{\alpha _{1}}} \right )ds + \\ & & +N_{1}^{2}\|y - z\|_{ S}\int _{\tau ''_{i}}^{t}\|A^{\alpha }U(t,s)\|\left (L'_{ n} + \frac{L''_{n}} {(s -\tau ''_{i})^{\alpha _{1}}} \right )ds \leq \\ & &\leq N_{1}^{2}\|y - z\|_{ S}\Biggl (\sum _{k\leq i}\int _{\tau ''_{k}}^{\tau '_{k+1} }M_{1}e^{-\beta _{1}\vert t-s\vert }\left (L'_{ n} + \frac{L''_{n}} {(s -\tau ''_{k})^{\alpha _{1}}} \right )ds + \\ & & +\int _{\tau ''_{i}}^{t}M_{ 1}(t - s)^{-\alpha _{1} }\left (L'_{n} + \frac{L''_{n}} {(s -\tau ''_{i})^{\alpha _{1}}} \right )ds\Biggl ) + \\ & & +\left (K'_{3} + \frac{K''_{3}} {(t -\tau ''_{i})^{\alpha _{1}}} \right )N_{1}\|y - z\|_{S} \leq \\ & &\leq \Biggl ( \frac{M_{1}} {1 - e^{-\beta _{1}\theta }} \left (L'_{n}Q + \frac{L''_{n}Q^{1-\alpha _{1}}} {1 -\alpha _{1}} \right ) + \frac{L''_{n}M_{1}2^{2\alpha }} {1 -\alpha _{1}} (t -\tau ''_{i})^{1-2\alpha _{1} } + \\ & & +\frac{L'_{n}M_{1}} {1 -\alpha _{1}} (t -\tau ''_{i})^{1-\alpha _{1} }\Biggl )N_{1}^{2}\|y - z\|_{ S} + \left (K'_{3} + \frac{K''_{3}} {(t -\tau ''_{i})^{\alpha _{1}}} \right )N_{1}\|y - z\|_{S} \leq \\ & & = \left (L'_{n+1} + \frac{L''_{n+1}} {(t -\tau ''_{i})^{\alpha _{1}}} \right )N_{1}\|y - z\|_{S}. {}\end{array} $$

(7.82)

One can verify that for sufficiently small N ₁ the sequences L′ _n and L″ _n are uniformly bounded by some constants L′_∗ and L″_∗. 

Since the sequences u _n (t, y) and u _n (t, z) tend to limit the functions u _∗(t, y) and u _∗(t, z), respectively, we conclude by ( 7.82) for t ∈ (τ″ _i , τ′ _i+1] that

$$ \displaystyle\begin{array}{rcl} \|u_{{\ast}}(t,y) - u_{{\ast}}(t,z)\|_{\alpha } \leq \left (L'_{{\ast}} + \frac{L''_{{\ast}}} {(t -\tau ''_{i+1})^{\alpha _{1}}} \right )N_{i+1}\|y - z\|_{S}& & {}\\ \end{array} $$

and

$$ \displaystyle\begin{array}{rcl} \|u^{{\ast}}(\tau '_{ i+1},y) - u^{{\ast}}(\tau '_{ i+1},z)\|_{\alpha } \leq \left (L'_{{\ast}} + \frac{L''_{{\ast}}} {\theta ^{\alpha _{1}}} \right )N_{1}\|y - z\|_{S}.& &{}\end{array} $$

(7.83)

Now we estimate the second summand in ( 7.67). Note that by our assumption $\tilde{\tau }_{j}^{1} <\tilde{\tau }_{ j}^{2}.$

$$ \displaystyle\begin{array}{rcl} \|u^{{\ast}}(\tilde{\tau }_{ j}^{1},z) - u^{{\ast}}(\tilde{\tau }_{ j}^{2},z)\|_{\alpha } =\Bigl \|\int _{ \tilde{\tau }_{ j}^{1}}^{\tilde{\tau }_{j}^{2} } \frac{d} {ds}u^{{\ast}}(s,z)ds\Bigl \|_{\alpha }.& & {}\\ \end{array} $$

By Theorem 3.5.2, [9], at the interval $(\tilde{\tau }_{j-1}^{2},\tilde{\tau }_{j}^{2})$ the derivative satisfies

$$ \displaystyle{\Bigl \| \frac{d} {ds}u^{{\ast}}(s,z)\Bigl \|_{\gamma } \leq \tilde{ K}_{ 1}(s -\tilde{\tau }_{j-1}^{2})^{\alpha -\gamma -1}} $$

with some positive constant $\tilde{K}_{1}$ independent of j and initial value from $U_{\rho }^{\alpha }.$

Then for $t \in (\tilde{\tau }_{j}^{1},\tilde{\tau }_{j}^{2})$

$$ \displaystyle{\Bigl \| \frac{d} {ds}u^{{\ast}}(s,z)\Bigl \|_{\gamma } \leq \tilde{ K}_{ 1}\left ( \frac{\theta } {2}\right )^{\alpha -\gamma -1} =\tilde{ K}_{ 2}} $$

and

$$ \displaystyle\begin{array}{rcl} \|u^{{\ast}}(\tilde{\tau }_{ j}^{1},z) - u^{{\ast}}(\tilde{\tau }_{ j}^{2},z)\|_{\alpha } \leq \tilde{ K}_{ 2}\vert \tilde{\tau }_{j}^{1} -\tilde{\tau }_{ j}^{2}\vert \leq \tilde{ K}_{ 2}N_{1}\|y - z\|_{S}.& &{}\end{array} $$

(7.84)

By ( 7.83) and ( 7.84) we have

$$ \displaystyle\begin{array}{rcl} & & \|u^{{\ast}}(\tilde{\tau }_{ j}^{1},z) - u^{{\ast}}(\tilde{\tau }_{ j}^{2},z)\|_{\alpha } =\varGamma _{ 9}\|y - z\|_{S},{}\end{array} $$

(7.85)

where Γ ₉ < 1 uniformly for j and $y,z \in \mathcal{N}_{\varrho _{0}}.$

By ( 7.67), ( 7.83), and ( 7.85) we conclude that the map $S:\ \mathcal{N}_{\varrho _{0}} \rightarrow \mathcal{N}_{\varrho _{0}}$ is a contraction. Therefore, there exists a unique almost periodic sequence $y^{{\ast}} =\{ y_{j}^{{\ast}}\}$ such that $u^{{\ast}}(\tau _{j}(y_{j}^{{\ast}}),y^{{\ast}}) = y_{j}^{{\ast}}$ for all j ∈ Z. The function u ^∗(t, y ^∗) is the W-almost periodic solution of Eqs. ( 7.57) and ( 7.58).

• 2. Now we prove the stability of the almost periodic solution. Fix arbitrary $\varepsilon > 0$ and η > 0. Let t ₀ ∈ [τ ₀(0) +η, τ ₁(0) −η]. 

The W-almost periodic solution u ₀(t) satisfies the integral equation

$$ \displaystyle\begin{array}{rcl} u_{0}(t) = U_{0}(t,t_{0})u_{0} +\int _{ t_{0}}^{t}U_{ 0}(t,s)f(s,u_{0}(s))ds +\sum _{t_{0}<\tau _{j}^{0}<t}U_{0}(t,\tau _{j}^{0})g_{ j}(\tau _{j}^{0}),& &{}\end{array} $$

(7.86)

where $\tau _{j}^{0} =\tau _{j}(u_{0}(\tau _{j}^{0}))$ and U ₀(t, s) is the evolution operator of the linear equation

$$ \displaystyle\begin{array}{rcl} \frac{du} {dt} + Au = 0,\ u(\tau _{j}^{0} + 0) - u(\tau _{ j}^{0}) = B_{ j}u(\tau _{j}^{0}),\ j = 1,2,\ldots.& & {}\\ \end{array} $$

Let u ₁ ∈ X ^α such that $\|u_{0} - u_{1}\|_{\alpha } <\delta.$ The solution u ₁(t) with initial value $u_{1}(t_{0}) = u_{1}$ satisfies equation

$$ \displaystyle\begin{array}{rcl} u_{1}(t) = U_{1}(t,t_{0})u_{1} +\int _{ t_{0}}^{t}U_{ 1}(t,s)f(s,u_{1}(s))ds +\sum _{t_{0}<\tau _{j}^{1}<t}U_{1}(t,\tau _{j}^{1})g_{ j}(\tau _{j}^{1}),& &{}\end{array} $$

(7.87)

where $\tau _{j}^{1} =\tau _{j}(u_{1}(\tau _{j}^{1}))$ and U ₁(t, s) is the evolution operator of the linear equation

$$ \displaystyle\begin{array}{rcl} \frac{du} {dt} + Au = 0,\ u(\tau _{j}^{1} + 0) - u(\tau _{ j}^{1}) = B_{ j}u(\tau _{j}^{1}),\ j = 1,2,\ldots.& & {}\\ \end{array} $$

By Lemma 5, for a sufficiently small Lipschitz constant N ₁ the evolution operator U ₀(t, s) satisfies the inequality

$$ \displaystyle\begin{array}{rcl} \|U_{0}(t,s)u\|_{\alpha } \leq M_{1}e^{-\beta _{1}(t-s)}\|u\|_{\alpha },\ t \geq s,& &{}\end{array} $$

(7.88)

with some positive constants β ₁ ≤ β, M ₁ ≥ M. Moreover, one can verify that for some domain $U_{\tilde{\rho }}^{\alpha },\tilde{\rho }\leq \rho,$ and N ₁ ≤ N ₀ the evolution operator satisfies

$$ \displaystyle\begin{array}{rcl} \|U_{1}(t,s)u\|_{\alpha } \leq M_{1}e^{-\beta _{1}(t-s)}\|u\|_{\alpha },\ t \geq s,\ t,s \in [t_{ 0},t_{0} + T],& &{}\end{array} $$

(7.89)

if the values u ₁(t) belong to $U_{\tilde{\rho }}^{\alpha }$ for $\tau _{j}^{1} \in [t_{0},t_{0} + T]$.

At the interval without impulses, the difference between solutions u ₀(t) − u ₁(t) satisfies the inequality

$$ \displaystyle\begin{array}{rcl} & & \|u_{1}(t) - u_{0}(t)\|_{\alpha } \leq \| e^{-A(t-t_{1})}(u_{ 0}(t_{1}) - u_{1}(t_{1}))\|_{\alpha } + {}\\ & & +\int _{t_{1}}^{t}\|A^{\alpha }e^{-A(t-t_{1})}(f(s,u_{ 1}(s)) - f(s,u_{0}(s)))\|ds \leq {}\\ & &\leq M_{1}e^{-\beta _{1}(t-t_{1})}\|u_{ 0}(t_{1}) - u_{1}(t_{1})\|_{\alpha } +\int _{ t_{1}}^{t}\frac{M_{1}N_{1}e^{-\beta _{1}(t-s)}} {(t - s)^{\alpha }} \|u_{1}(s) - u_{0}(s)\|_{\alpha }ds. {}\\ \end{array} $$

Then by Lemma 2,

$$ \displaystyle\begin{array}{rcl} \|u_{1}(t) - u_{0}(t)\|_{\alpha } \leq M_{1}\tilde{C}e^{-\beta _{1}(t-t_{1})}\|u_{ 1}(t_{1}) - u_{0}(t_{1})\|_{\alpha },\ t - t_{1} \leq Q.& &{}\end{array} $$

(7.90)

Hence, if initial values belong to the bounded domain from X ^α, then the corresponding solutions are uniformly bounded for t from the bounded interval.

Assume for definiteness that $\tau _{j}^{0} \geq \tau _{j}^{1}$ and estimate $\vert \tau _{j}^{1} -\tau _{j}^{0}\vert$ by $(u_{1}(\tau _{j}^{1}) - u_{0}(\tau _{j}^{1})).$

$$ \displaystyle\begin{array}{rcl} & & \|(u_{1}(\tau _{j}^{1}) - u_{ 0}(\tau _{j}^{0})\|_{\alpha } \leq \| (u_{ 0}(\tau _{j}^{1}) - u_{ 0}(\tau _{j}^{0})\|_{\alpha } +\| u_{ 0}(\tau _{j}^{1}) - u_{ 1}(\tau _{j}^{1})\|_{\alpha } \leq {}\\ & &\leq \Bigl \|\int _{\tau _{j}^{1}}^{\tau _{j}^{0} } \frac{d} {d\xi }u_{0}(\xi )d\xi \Bigl \|_{\alpha } +\| u_{0}(\tau _{j}^{1}) - u_{ 1}(\tau _{j}^{1})\|_{\alpha } \leq {}\\ & &\leq \tilde{ K}_{2}\vert \tau _{j}^{1} -\tau _{ j}^{0}\vert +\| u_{ 0}(\tau _{j}^{1}) - u_{ 1}(\tau _{j}^{1})\|_{\alpha }. {}\\ \end{array} $$

Hence,

$$ \displaystyle\begin{array}{rcl} \vert \tau _{j}^{1} -\tau _{ j}^{1}\vert \leq \| u_{ 0}(\tau _{j}^{0}) - u_{ 1}(\tau _{j}^{1})\|_{\alpha } \leq \frac{N_{1}} {1 -\tilde{ K}_{2}N_{1}}\|u_{0}(\tau _{j}^{1}) - u_{ 1}(\tau _{j}^{1})\|_{\alpha }.& &{}\end{array} $$

(7.91)

We assume that $t \in (\tau ''_{i},\tau '_{i+1}]$ and estimate the difference

$$ \displaystyle\begin{array}{rcl} & & \|u_{0}(t) - u_{1}(t)\|_{\alpha } =\| U_{0}(t,t_{0})(u_{0} - u_{1})\|_{\alpha } +\| (U_{0}(t,t_{0}) - U_{1}(t,t_{0}))u_{1}\|_{\alpha } \\ & & +\int _{t_{0}}^{t}\|U_{ 0}(t,s)f(s,u_{0}(s)) - U_{1}(t,s)f(s,u_{1}(s))\|_{\alpha }ds + \\ & & +\|\sum _{t_{0}<\tau _{j}^{1}<t}U(t,\tau _{j}^{1})g_{ j}(\tau _{j}^{1}) -\sum _{ t_{0}<\tau _{j}^{0}<t}U(t,\tau _{j}^{0})g_{ j}(\tau _{j}^{0})\|_{\alpha } \leq \\ & &\leq \| U_{0}(t,t_{0})(u_{0} - u_{1})\|_{\alpha } +\| (U_{0}(t,t_{0}) - U_{1}(t,t_{0}))u_{1}\|_{\alpha } + \\ & & +\int _{t_{0}}^{\tau '_{1} }\|U_{0}(t,s)f(s,u_{0}(s)) - U_{1}(t,s)f(s,u_{1}(s))\|_{\alpha }ds + \\ & & +\sum _{j=1}^{i-1}\int _{ \tau ''_{j}}^{\tau '_{j+1} }\|U_{0}(t,s)(f(s,u_{0}(s)) - f(s,u_{1}(s)))\|_{\alpha }ds + \\ & & +\sum _{j=1}^{i-1}\int _{ \tau ''_{j}}^{\tau '_{j+1} }\|(U_{0}(t,s) - U_{1}(t,s))f(s,u_{1}(s))\|_{\alpha }ds + \\ & & +\sum _{j=1}^{i}\int _{ \tau '_{j}}^{\tau ''_{j} }\|U_{0}(t,s)f(s,u_{0}(s)) - U_{1}(t,s)f(s,u_{1}(s))\|_{\alpha }ds + \\ & & +\int _{\tau ''_{i}}^{t}\|U_{ 0}(t,s)f(s,u_{0}(s)) - U_{1}(t,s)f(s,u_{1}(s))\|_{\alpha }ds + \\ & & +\sum _{j=1}^{i}\|U_{ 0}(t,\tau _{j}^{0})g_{ j}(\tau _{j}^{0}) - U_{ 1}(t,\tau _{j}^{1})g_{ j}(\tau _{j}^{1})\|_{\alpha }. {}\end{array} $$

(7.92)

Denote $v(t) =\| u_{0}(t) - u_{1}(t)\|_{\alpha }.$ Assume that for t ∈ [t ₀, τ′ _i ] the values u(t) belong to $U_{\tilde{\rho }}^{\alpha };$ hence, the evolution operators U ₀(t, τ) and U ₁(t, τ) satisfy ( 7.88) and ( 7.89) at this interval. By ( 7.92), analogous to the proof of ( 7.75), ( 7.79), and ( 7.80), we conclude that there exist positive constants M ₂ and P ₁ independent of i such that for $t \in \mathcal{J}_{i+1}$

$$ \displaystyle\begin{array}{rcl} & & v(t) \leq M_{1}e^{-\beta _{1}(t-t_{0})}v(t_{ 0}) +\int _{ t_{0}}^{\tau '_{1} } \frac{M_{2}N_{1}} {(\tau '_{1} - s)^{\alpha }}e^{-\beta _{1}(t-\tau ''_{1})}v(s)ds + \\ & & +\sum _{j=2}^{i-1}\int _{ \tau ''_{j-1}}^{\tau '_{j} }M_{2}N_{1}e^{-\beta _{1}(t-\tau ''_{j})}v(s)ds +\sum _{ j=1}^{i-1}P_{ 1}N_{1}e^{-\beta _{1}(t-\tau ''_{j})}v(\tau '_{ j}) + \\ & & + \frac{1} {(t -\tau ''_{i})^{\alpha _{1}}} \left (\int _{\tau ''_{i-1}}^{\tau '_{i} }M_{2}N_{1}e^{-\beta _{1}(t-\tau ''_{i})}v(s)ds + P_{ 1}N_{1}e^{-\beta _{1}(t-\tau ''_{i})}v(\tau '_{ i})\right ) + \\ & & +\int _{\tau ''_{i}}^{t}M_{ 2}N_{1}e^{-\beta _{1}(t-s)}(t - s)^{-\alpha _{1} }v(s)ds {}\end{array} $$

(7.93)

with α ₁ > α. By ( 7.90), at the interval [t ₀, τ′₁] v(t) satisfies

$$ \displaystyle\begin{array}{rcl} v(t) \leq M_{1}\tilde{C}e^{-\beta (t-t_{0})}v(t_{ 0}),\ t \in [t_{0},\tau '_{1}].& &{}\end{array} $$

(7.94)

By ( 7.93) and ( 7.94), for t ∈ (τ″₁, τ′₂] we get

$$ \displaystyle\begin{array}{rcl} & & v(t) \leq M_{1}e^{-\beta _{1}(t-t_{0})}v(t_{ 0}) + \frac{1} {(t -\tau ''_{1})^{\alpha _{1}}} \int _{t_{0}}^{\tau '_{1} }M_{2}N_{1}e^{-\beta _{1}(t-\tau ''_{1})}v(s)ds + {}\\ & & +P_{1}N_{1}e^{-\beta _{1}(t-\tau ''_{1})}(t -\tau ''_{ 1})^{-\alpha _{1} }v(\tau '_{1}) +\int _{ \tau ''_{1}}^{t}M_{ 2}N_{1}e^{-\beta _{1}(t-s)}(t - s)^{-\alpha _{1} }v(s)ds. {}\\ \end{array} $$

Hence, for $M_{3} = M_{2}e^{\beta _{1}Q},\tilde{C}_{1} =\tilde{ C}/(1-\alpha ),$ $v_{1}(t) = e^{\beta _{1}t}v(t)$ and $P_{2} = P_{1}e^{\beta _{1}\sup _{j}\vert \tau ''_{j}-\tau '_{j}\vert }$

$$ \displaystyle\begin{array}{rcl} v_{1}(t) \leq M_{1}v_{1}(t_{0})\left (1 + \frac{N_{1}\tilde{C}(M_{3}\tilde{Q} + P_{2})} {(t -\tau ''_{1})^{\alpha _{1}}} \right ) +\int _{ \tau ''_{1}}^{t}M_{ 2}N_{1}(t - s)^{-\alpha _{1} }v_{1}(s)ds.& & {}\\ \end{array} $$

By Lemma 2

$$ \displaystyle\begin{array}{rcl} v(t) \leq M_{1}\tilde{C}_{1}v(t_{0})e^{-\beta _{1}(t-t_{0})}\left (1 + \frac{N_{1}\tilde{C}(M_{3}\tilde{Q} + P_{2})} {(t -\tau ''_{1})^{\alpha _{1}}} \right ),\ t \in (\tau ''_{1},\tau '_{2}].& &{}\end{array} $$

(7.95)

Denote $\tilde{Q} =\max _{j}\{1,(\tau '_{j+1} -\tau ''_{j})\}$ and $\tilde{\theta }=\min _{j}\{1,(\tau '_{j+1} -\tau ''_{j})\}.$ Let us prove that

$$ \displaystyle\begin{array}{rcl} v(t)\leq M_{1}\tilde{C}_{1}v(t_{0})e^{-\beta _{1}(t-t_{0})}\left (1+\frac{N_{1}\tilde{C}_{1}(M_{3}\tilde{Q}+P_{2})} {(t-\tau ''_{j})^{\alpha _{1}}} \right )\left (1+\frac{N_{1}\tilde{C}_{1}(M_{3}\tilde{Q}+P_{2})} {(1-\alpha _{1})\tilde{\theta }^{\alpha _{1}}} \right )^{i-1}\qquad & &{}\end{array} $$

(7.96)

for t ∈ (τ″ _i , τ′ _i+1]i ≥ 2. We apply the method of mathematical induction. Assume that ( 7.96) is true for t ∈ [τ″ _i−1, τ′ _i ] and prove it for t ∈ [τ″ _i , τ′ _i+1]. Really, by ( 7.93) for t ∈ [τ″ _i , τ′ _i+1] we have

$$ \displaystyle\begin{array}{rcl} & & v(t) \leq M_{1}e^{-\beta _{1}(t-t_{0})}v(t_{ 0})\Biggl (\left (1 + (M_{3}\tilde{Q} + P_{2})N_{1}\tilde{C}\right ) + {}\\ & & +\sum _{j=2}^{i-1}\mathcal{A}^{j}M_{ 3}N_{1}\tilde{Q}\tilde{C}_{1} +\sum _{ j=2}^{i-1}\mathcal{A}^{j-1}\left (1 + \frac{N_{1}\tilde{C}_{1}(M_{3}\tilde{Q} + P_{2})} {\tilde{\theta }^{\alpha _{1}}} \right )N_{1}P_{2}\tilde{C}_{1} + {}\\ & & +\mathcal{A}^{i-2}\Biggl (N_{ 1}M_{3}\tilde{C}_{1}\Bigl ((\tau '_{i} -\tau ''_{i-1}) + \frac{N_{1}\tilde{C}_{1}(M_{3}\tilde{Q} + P_{2})(\tau '_{i} -\tau ''_{i-1})} {(1 -\alpha _{1})(\tau '_{i} -\tau ''_{i-1})^{\alpha _{1}}} \Bigl ) + {}\\ & & +N_{1}P_{2}\tilde{C}_{1}\Bigl (1 + \frac{N_{1}\tilde{C}_{1}(M_{3}\tilde{Q} + P_{2})} {(\tau '_{i} -\tau ''_{i-1})^{\alpha _{1}}} \Bigl )\Biggl ) + \mathcal{B}(t) \leq {}\\ & &\leq \mathcal{A} +\sum _{ j=2}^{i-1}\mathcal{A}^{j-1}(1 + N_{ 1}\tilde{C}_{1}(M_{3}\tilde{Q} + P_{2}) - 1) + \frac{\mathcal{A}^{i-1}N_{1}\tilde{C}_{1}(M_{3}\tilde{Q} + P_{2})} {(t -\tau ''_{i})^{\alpha _{1}}} + {}\\ & & +\mathcal{B}(t) \leq \mathcal{A}^{i-1}\left (1 + \frac{N_{1}\tilde{C}_{1}(M_{3}\tilde{Q} + P_{2})} {(t -\tau ''_{i})^{\alpha _{1}}} \right ) + \mathcal{B}(t). {}\\ \end{array} $$

where

$$ \displaystyle{\mathcal{A} = \left (1 + \frac{N_{1}\tilde{C}_{1}(M_{3}\tilde{Q} + P_{2})} {(1-\alpha )\tilde{\theta }^{\alpha _{1}}} \right ),\ \mathcal{B}(t) =\int _{ \tau ''_{i}}^{t} \frac{M_{2}N_{1}} {(t - s)^{\alpha _{1}}} e^{-\beta _{1}(t-s)}v(s)ds.} $$

Hence, for t ∈ (τ″ _i , τ′ _i+1], the function $v_{1}(t) = e^{\beta _{1}t}v(t)$ satisfies the inequality

$$ \displaystyle{v_{1}(t) \leq \mathcal{A}^{i-1}\left (1 + \frac{N_{1}\tilde{C}_{1}(M_{3}\tilde{Q} + P_{2})} {(t -\tau ''_{i})^{\alpha _{1}}} \right ) + M_{2}N_{1}\int _{\tau ''_{i}}^{t}(t - s)^{-\alpha _{1} }v_{1}(s)ds.} $$

Applying Lemma 2, we obtain ( 7.96).

Let N ₁ > 0 be such that $\mathcal{A}^{i(t_{0},t)}e^{-\beta _{1}(t-t_{0})} < e^{-\delta _{1}(t-t_{0})}$ for some positive δ ₁. For the given $\varepsilon > 0$ and η > 0 we choose v(t ₀) = v ₀ such that

$$ \displaystyle{M_{1}\tilde{C}_{1}v_{0}\left (1 + \frac{N_{1}\tilde{C}_{1}(M_{3}\tilde{Q} + P_{2})} {\eta ^{\alpha _{1}}} \right ) <\varepsilon.} $$

This proves the asymptotic stability of solution u ₀. 

Example 1.

Let us consider the parabolic equation with impulses in variable moments of time:

$$ \displaystyle\begin{array}{rcl} & & u_{t} = u_{xx} + a(t)u_{x} + b(t,x),{}\end{array} $$

(7.97)

$$ \displaystyle\begin{array}{rcl} & & \Delta u\Biggl |_{t=\tau _{j}(u)} = u(\tau _{j}(u) + 0,x) - u(\tau _{j}(u),x) = -a_{j}u(\tau _{j}(u),x),{}\end{array} $$

(7.98)

with boundary conditions

$$ \displaystyle\begin{array}{rcl} u(t,0) = u(t,\pi ) = 0,& &{}\end{array} $$

(7.99)

where the sequence of hypersurfaces τ _j is defined by

$$ \displaystyle\begin{array}{rcl} \tau _{j}(u) =\theta _{j} + b_{j}\int _{0}^{\pi }u^{2}(\xi )d\xi,& &{}\end{array} $$

(7.100)

where the sequence of real numbers $\{\theta _{j}\}$ has uniformly almost periodic sequences of differences and $\theta _{j+1} -\theta _{j} \geq \theta \geq 1/2,$

{a _j } and {b _j } are almost periodic sequences of positive numbers,

a(t) is a Bohr almost periodic function,

b(t, x) is a Bohr almost periodic function in t uniformly with respect to x ∈ [0, π]. 

Denote

$$ \displaystyle{X = L_{2}(0,\pi ),\ A = - \frac{\partial ^{2}} {\partial x^{2}},\ X^{1} = D(A) = H^{2}(0,\pi ) \cap H_{ 0}^{1}(0,\pi ).} $$

The operator A is sectorial with simple eigenvalues $\lambda _{k} = k^{2}$ and corresponding eigenfunctions

$$ \displaystyle{\varphi _{k}(x) = \left (\frac{2} {\pi } \right )^{1/2}\sin kx,\ k = 1,2,\ldots.} $$

Operator − A generates an analytic semigroup e ^−At. 

Let $u =\sum _{ k=1}^{\infty }a_{k}\sin kx,\ a_{k} = \frac{1} {\pi } \int _{0}^{\pi }u(x)\sin kxdx.$ Then

$$ \displaystyle{Au =\sum _{ k=1}^{\infty }k^{2}a_{ k}\sin kx,\ A^{\alpha }u =\sum _{ k=1}^{\infty }k^{2\alpha }a_{ k}\sin kx,\ e^{-At} =\sum _{ k=1}^{\infty }e^{-k^{2}t }a_{k}\sin kx.} $$

Hence,

$$ \displaystyle{X^{1/2} = D(A^{1/2}) = H_{ 0}^{1}(0,\pi ).} $$

Let us consider Eqs. ( 7.97)–( 7.99) in space $X^{1/2} = D(A^{1/2}) = H_{0}^{1}(0,\pi ):$

$$ \displaystyle{\frac{du} {dt} + Au = f(t,u),\ u(\tau _{j}(u) + 0) = (1 - a_{j})u(\tau _{j}(u)),\ j = 0,\ldots,} $$

where f(t, u): R × X ^1∕2 → X,  f(t, u)(x) = a(t)u _x + b(t, x). 

We verify that in some domain $\mathcal{D} =\{ u \geq 0,\|u\| \leq \rho \}$ solutions of ( 7.97)–( 7.99) don’t have beating at the surfaces t = τ _j (u). Assume to the contrary that solution u(t) intersects the surface t = τ _j (u) at two points $t_{j}^{1}$ and $t_{j}^{2}$, $t_{j}^{1} < t_{j}^{2}.$

Denote $u(t_{j}^{1}) = u_{1},u(t_{j}^{2}) = u_{2},\tilde{u} = e^{-A(t_{j}^{2}-t_{ j}^{1}) }u(t_{j}^{1} + 0).$ Then $u(t_{j}^{1} + 0) = (1 - a_{j})u_{1},\ \tau _{j}(u_{1})) = t_{j}^{1},\ \tau _{j}(u_{2})) = t_{j}^{2},$ and

$$ \displaystyle{u_{2} = e^{-A(t_{j}^{2}-t_{ j}^{1}) }u(t_{j}^{1} + 0) +\int _{ t_{j}^{1}}^{t_{j}^{2} }e^{-A(t_{j}^{2}-s) }f(s,u(s))ds.} $$

We have

$$ \displaystyle\begin{array}{rcl} & & \vert \tau _{j}(u_{2}) -\tau _{j}(\tilde{u})\vert \leq b_{j}\vert \int _{0}^{t}(u_{ 2}(t,x) -\tilde{ u}(t,x))(u_{2}(t,x) +\tilde{ u}(t,x))dx \leq {}\\ & &\leq b_{j}\|u_{2}(t,x) -\tilde{ u}(t,x)\|_{L_{2}}\|u_{2}(t,x) +\tilde{ u}(t,x)\|_{L_{2}} \leq {}\\ & &\leq b_{j}\|\int _{t_{j}^{1}}^{t_{j}^{2} }e^{-A(t_{j}^{2}-s) }f(s,u(s))ds\|_{L_{2}}\|u_{2}(t,x) +\tilde{ u}(t,x)\|_{L_{2}}. {}\\ \end{array} $$

The function f(t, u) satisfies $\|f(t,u)\|_{X} \leq K(1 +\| u\|_{X^{1/2}});$ hence, solutions of the equation without impulses exist for all t ≥ t ₀ and there exist positive constants M ₁ and M ₂ such that $M_{2} \geq \sup _{u\in \mathcal{D}}\|f(t,u)\|_{L_{2}},\ M_{3} \geq \sup _{u\in \mathcal{D}}\|u_{2}(t,x) +\tilde{ u}(t,x)\|_{L_{2}}.$ Therefore, $\tau _{j}(u_{2}) -\tau _{j}(\tilde{u}) \leq b_{j}\vert t_{j}^{2} - t_{j}^{1}\vert M_{2}M_{3}.$ By sufficiently small $b =\sup _{j}b_{j}$ we have bM ₂ M ₃ < 1 and

$$ \displaystyle\begin{array}{rcl} & & 0 < t_{j}^{2} - t_{ j}^{1} =\tau _{ j}(u_{2}) -\tau _{j}(u_{1}) \leq \tau _{j}(u_{2}) -\tau _{j}(\tilde{u}) +\tau _{j}(\tilde{u}) -\tau _{j}(u_{1}), {}\\ & & t_{j}^{2} - t_{ j}^{1} \leq \frac{1} {1 - bM_{2}M_{3}}(\tau _{j}(\tilde{u}) -\tau _{j}(u_{1})) \leq \frac{b_{j}((1 - a_{j})^{2} - 1)} {1 - bM_{2}M_{3}} \|u_{1}\|_{L_{2}}^{2} < 0. {}\\ \end{array} $$

This contradicts our assumption.

Corresponding to ( 7.97)–( 7.99), the linear impulsive equation is exponentially stable in space X ^1∕2. By Theorem 4, for sufficiently small $b =\sup _{j}b_{j}$ and $a =\sup _{t}\vert a(t)\vert$ the equation has an asymptotically stable W-almost periodic solution.

7.6 Equations with Unbounded Operators B _j

Many results in our chapter remain true if operators B _j in linear parts of impulsive action are unbounded. We refer to [27], where the following semilinear impulsive differential equation

$$ \displaystyle\begin{array}{rcl} & & \frac{du} {dt} = Au + f(t,u),\quad t\not =\tau _{j},{}\end{array} $$

(7.101)

$$ \displaystyle\begin{array}{rcl} & & \varDelta u\vert _{t=\tau _{j}} = u(\tau _{j}) - u(\tau _{j} - 0) = B_{j}u(\tau _{j} - 0) + g_{j}(u(t_{j} - 0)),\quad j \in Z,{}\end{array} $$

(7.102)

was studied. Here u: R → X,  X is a Banach space, A is a sectorial operator in X, {B _j } is a sequence of some closed operators, and {τ _j } is an unbounded and strictly increasing sequence of real numbers. Assume that the equation satisfies conditions (H1), (H3), (H5), (H6), and

(H4u) the sequence {B _j } of closed linear operators B _j  ∈ L(X ^α+γ, X ^α) is almost periodic in the space L(X ^α+γ, X ^α), for α ≥ 0 and some γ ≥ 0. 

As in [17], we assume that solutions u(t) of ( 7.1), ( 7.2) are right-hand-side continuous; hence, u(τ _j ) = u(τ _j + 0) at all points of impulsive action. Due to such a selection we avoid considering operators $e^{-A(t-\tau _{j})}(I + B_{j})$ with unbounded operator B _j and can work with the family of bounded operators $e^{-A(t-\tau _{j})}.$

Since the operator A is sectorial and operators B _j are subordinate to A, an evolution operator of a corresponding linear impulsive equation is constructed correctly. Now analogs of the theorems 2 and 3 can be proven.

Example 2 ([27]).

We consider the following parabolic equation with impulsive action:

$$ \displaystyle\begin{array}{rcl} & & u_{t} = u_{xx} + f(t,x),{}\end{array} $$

(7.103)

$$ \displaystyle\begin{array}{rcl} & & \varDelta u\Bigl |_{t=\tau _{j}} = u(\tau _{j},x) - u(\tau _{j} - 0,x) = b_{k}(\sin x)u_{x} + c_{k}x(\pi -x),{}\end{array} $$

(7.104)

with boundary conditions

$$ \displaystyle\begin{array}{rcl} u(t,0) = u(t,\pi ) = 0,& &{}\end{array} $$

(7.105)

where {τ _j } is a sequence of real numbers with uniformly almost periodic sequences of differences, $\tau _{j+1} -\tau _{j} \geq \theta \geq 1/2,$

{b _j } and {c _j } are almost periodic sequences of real numbers,

f(t, x) is almost periodic and locally Hlder continuous with respect to t and for every fixed t belongs to L ₂(0, π). 

As in Example 1, denote

$$ \displaystyle{X = L_{2}(0,\pi ),\ A = - \frac{\partial ^{2}} {\partial x^{2}},\ X^{1} = D(A) = H^{2}(0,\pi ) \cap H_{ 0}^{1}(0,\pi ).} $$

The operator A is sectorial with simple eigenvalues $\lambda _{k} = k^{2}$ and corresponding eigenfunctions $\varphi _{k}(x) =\sin kx,k = 1,2,\ldots.$

Operators B _j have form $B_{j} = b_{j}\sin x \frac{\partial } {\partial x}.$

If $u =\sum _{ k=1}^{\infty }a_{k}\sin kx,\ a_{k} = \frac{1} {\pi } \int _{0}^{\pi }u(x)\sin kxdx,$ then

$$ \displaystyle\begin{array}{rcl} B_{j}u = b_{j}\sin xu_{x} = b_{j}\sin x\sum _{k=1}^{\infty }a_{ k}k\cos kx = \frac{b_{j}} {2} (R - L)A^{1/2}u = b_{ j}TA^{1/2}u,& & {}\\ \end{array} $$

where $Ru =\sum _{ k=1}^{\infty }a_{k}\sin (k - 1)x$ and $Lu =\sum _{ k=1}^{\infty }a_{k}\sin (k + 1)x$ are bounded shift operators in X. Hence, operators B _j : X ^α+1∕2 → X ^α are linear continuous, α ≥ 0. 

By ( 7.13), the evolution operator for homogeneous equations ( 7.103) and ( 7.104) is

$$ \displaystyle{U(t,s) = e^{-A(t-s)},\ \mathrm{if}\ \tau _{ k} \leq s \leq t <\tau _{k+1},} $$

and

$$ \displaystyle{U(t,s) = e^{-A(t-\tau _{k})}(I + B_{ k})e^{-A(\tau _{k}-\tau _{k-1})}\ldots (I + B_{ m})e^{-A(\tau _{m}-s)}} $$

if τ _m−1 ≤ s < τ _m  < τ _m+1 … τ _k  ≤ t < τ _k+1,  m < k,  k, m ∈ Z. 

Theorem 5.

Let $p\,\ln (1 + b) < 1,$ where p is defined by ( 7.3 ) and $b =\sup _{j}\vert b_{j}\vert.$ Then Eqs. ( 7.103 ) and ( 7.104 ) with boundary conditions ( 7.105 ) have a unique W-almost periodic solution which is asymptotically stable.

Proof.

We show that the unique almost periodic solution of ( 7.103) and ( 7.104) is given as function R → L ₂(0, π) by formula

$$ \displaystyle\begin{array}{rcl} u_{0}(t) =\int _{ -\infty }^{t}U(t,s)\tilde{f}(s)ds +\sum _{\tau _{ j}\leq t}U(t,\tau _{j})\tilde{g}_{j},& &{}\end{array} $$

(7.106)

where $\tilde{f}(t) \equiv f(t,.): R \rightarrow L_{2}(0,\pi ),g_{j}(x) = c_{j}x(\pi -x),\tilde{g}_{j} = g_{j}(.): Z \rightarrow L_{2}(0,\pi ).$

First, u ₀(t) is bounded in space X ^α: 

$$ \displaystyle\begin{array}{rcl} & & \int _{-\infty }^{t}\|U(t,s)\tilde{f}(s)\|_{\alpha }ds \leq \int _{\tau _{ i}}^{t}\|A^{\alpha }e^{-A(t-s)}\tilde{f}(s)\|ds + \\ & & +\int _{\tau _{i-1}}^{\tau _{i} }\|A^{\alpha }e^{-A(t-\tau _{i})}(I + B_{ i})e^{-A(\tau _{i}-s)}\tilde{f}(s)\|ds + \\ & & +\sum _{k=2}^{\infty }\int _{ \tau _{i-k}}^{\tau _{i-k+1} }\|A^{\alpha }e^{-A(t-\tau _{i})}(I + B_{ i})e^{-A(\tau _{i}-\tau _{i-1})}\| \times \\ & &\times \prod _{j=i-1}^{i-k+2}\|(I + B_{ j})e^{-A(\tau _{j}-\tau _{j-1})}\|\|(I + B_{ i-k+1})e^{-A(\tau _{i-k+1}-s)}\tilde{f}(s)\|ds,{}\end{array} $$

(7.107)

where t ∈ [τ _i , τ _i+1). The first integral in ( 7.107) has upper bound

$$ \displaystyle{\int _{\tau _{i}}^{t}\|A^{\alpha }e^{-A(t-s)}\tilde{f}(s)\|ds \leq \frac{C_{\alpha }} {1-\alpha }\|\tilde{f}\|_{PC}.} $$

Next, we need the following inequality (see [17], p. 35):

$$ \displaystyle\begin{array}{rcl} \|A^{\alpha }TA^{\beta }e^{-At}\| = \frac{1} {2}\|A^{\alpha }(R - L)A^{\beta }e^{-At}\| \leq \frac{4^{\alpha } + 1} {2} \|A^{\alpha +\beta }e^{-At}\|.& &{}\end{array} $$

(7.108)

Then by ( 7.108),

$$ \displaystyle\begin{array}{rcl} \|A^{\alpha }e^{-A(t-\tau _{i})}(I + B_{ i})e^{-A(\tau _{i}-s)}\| \leq \| A^{\alpha }e^{-A(t-s)}\| + \frac{5} {2}\|A^{\alpha +1/2}e^{-A(t-s)}\| \leq & & \\ \leq \left (C_{\alpha }(t - s)^{-\alpha } + \frac{5} {2}C_{\alpha +1/2}(t - s)^{-(\alpha +1/2)}\right )e^{-\delta (t-s)}.& &{}\end{array} $$

(7.109)

From Henry [9], p. 25, we have

$$ \displaystyle{\|A^{\alpha }e^{-At}\psi \| < b_{\alpha }(t)\|\psi \|,} $$

where b _α (t) = (te∕α)^−α if $0 < t \leq \alpha /\lambda _{1},$ and $b_{\alpha }(t) =\lambda _{ 1}^{\alpha }e^{-\lambda _{1}t}$ if $t \geq \alpha /\lambda _{1}.$ Since $\|T\| = 1$ and $\lambda _{1} = 1$, we have

$$ \displaystyle\begin{array}{rcl} & & \|(I + B_{j})e^{-A(\tau _{j}-\tau _{j-1})}\| \leq \| e^{-A(\tau _{j}-\tau _{j-1})}\| + \vert b_{ j}\vert \|A^{1/2}e^{-A(\tau _{j}-\tau _{j-1})}\| \leq \\ & &\leq (1 + \vert b_{j}\vert )e^{-(\tau _{j}-\tau _{j-1})} {}\end{array} $$

(7.110)

if $\theta \geq 1/2.$

Let $0 <\varepsilon _{1} < 1 - p\ln (1 + b).$ Then there exists a positive integer k ₁ such that for k ≥ k ₁

$$ \displaystyle{\frac{i(\tau _{i-k},\tau _{i})} {\tau _{i} -\tau _{i-k}} \ln (1 + b) - 1 < -\varepsilon _{1}.} $$

Denote

$$ \displaystyle{N_{1} =\max _{1\leq k\leq k_{1}}\exp \left (i(\tau _{i-k},\tau _{i})\ln (1 + b) - (\tau _{i} -\tau _{i-k})\right ).} $$

Then

$$ \displaystyle\begin{array}{rcl} & & \prod _{j=i-k+1}^{i}\|(I + B_{ j})e^{-A(\tau _{j}-\tau _{j-1})}\| \leq (1 + b)^{i(\tau _{i-k},\tau _{i})}e^{-(\tau _{i}-\tau _{i-k})} \leq \\ & &\leq N_{1}e^{-\varepsilon _{1}(\tau _{i}-\tau _{i-k})} \leq N_{ 1}e^{-\varepsilon _{1}\theta k}. {}\end{array} $$

(7.111)

For t ∈ (τ _i , τ _i+1), by ( 7.109) and ( 7.111) we get

$$ \displaystyle\begin{array}{rcl} & & \|U(t,\tau _{i-k})\|_{\alpha } \leq \| A^{\alpha }e^{-A(t-\tau _{i})}(I + B_{ i})e^{-A(\tau _{i}-\tau _{i-1})}\| \times {}\\ & &\times \prod _{j=i-k+1}^{i-1}\|(I + B_{ j})e^{-A(\tau _{j}-\tau _{j-1})}\| \leq K_{ 1}e^{-\varepsilon _{1}(t-\tau _{i-k})} {}\\ \end{array} $$

with constant K ₁ independent of t and τ _i−k . 

Using the last inequality, we obtain the boundedness of $\|u_{0}(t)\|_{\alpha }.$ We can now proceed analogously to the proof of Theorem 1 and show the almost periodicity of u ₀(t).