Semilinear Delay Evolution Equations with Nonlocal Initial Conditions

Abstract

An existence and asymptotic behaviour result for a class of semilinear delay evolution equations subjected to nonlocal initial conditions is established. An application to a semilinear wave equation is also discussed.

In memory of Professor Alfredo Lorenzi

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

19.1 Introduction

In this paper we prove an existence and uniform asymptotic stability result for mild solutions to a semilinear delay differential evolution equation with nonlocal initial data in a Banach space X, i.e.

$$\displaystyle{ \left \{\begin{array}{ll} u'(t) = \mathit{Au}(t) + f(t,u_{t}),&\quad t \in [\,0,+\infty ), \\ u(t) = g(u)(t), &\quad t \in [\,-\tau,0\,].\end{array} \right. }$$

(19.1)

Here $A: D(A) \subseteq X \rightarrow X$ is the infinitesimal generator of a C ₀-semigroup of contractions, $\{S(t): X \rightarrow X;\ t \geq 0\}$, τ ≥ 0, f: [ 0, +∞) × C([ −τ, 0 ]; X) → X is a compact function which is jointly continuous and Lipschitz with respect to its second argument and $g: C_{b}([\,-\tau,+\infty );X) \rightarrow C([\,-\tau,0\,];X)$ is continuous and has affine growth. In the limiting case τ = 0, i.e. when the delay is absent, C([ −τ, 0 ]; X) = X and so, in this case, f: [ 0, +∞) × X → X and g: C _b ([ 0, +∞); X) → X. If I is an unbounded interval, C _b (I; X) denotes the space of all bounded and continuous functions from I to X, equipped with the sup-norm $\|\cdot \|_{C_{b}(I;X)}$, while $\tilde{C}_{b}(I;X)$ stands for the space of all bounded and continuous functions from I to X, endowed with the uniform convergence on compacta topology. Further, C([ a, b ]; X) denotes the space of all continuous functions from [ a, b ] to X endowed with the sup-norm $\|\cdot \|_{C([\,a,b\,];X)}$. As usual, if u ∈ C([ −τ, +∞); X) and t ∈ [ 0, +∞), u _t  ∈ C([ −τ, 0 ]; X) is defined by u _t (s): = u(t + s) for each s ∈ [ −τ, 0 ].

It should be noticed that, as far as the nondelayed case, i.e. τ = 0, is concerned, in many situations, the nonlocal problem (19. 1) has proved more reliable than its classical initial-value counterpart. This is the case, for instance, of long-term weather forecasting in meteorology. See Rabier et al. [23]. In addition, (19. 1), with τ = 0, is nothing but the abstract form of various mathematical models for : wave propagation—see Avalishvili and Avalishvili [3], diffusion processes—see Deng [10], Gordeziani [13] and Olmstead and Roberts [21], fluid dynamics—see Gordeziani et al. [14] and Shelukhin [25, 26], or pharmacokinetics—see McKibben [18, Model II.6, p. 395].

Abstract nondelayed evolution equations subjected to nonlocal initial conditions were investigated by Aizicovici and Lee [1], Aizicovici and McKibben [2], Bryszewski [8], Garcia-Falset and Reich [12] and Paicu and Vrabie [22], to cite only a few.

Since the presence of a delay in the source term of an evolution equation is more realistic than an instantaneous feedback, there is an increasing literature on such kind of problems, i.e. functional evolution equations with delay. This explains why, in recent years, abstract delayed evolution equations or inclusions subjected to nonlocal initial conditions were considered by many authors from which we mention Burlică and Roşu [5], Burlică et al. [6, 7] and Vrabie [28–33] and the references therein. For previous results on initial-value problems for delay evolution equations, see Mitidieri and Vrabie [19, 20]. Some existence, uniqueness and continuity with respect to the data theorems concerning source identification for semilinear delay evolution equations were recently obtained, among others, by Di Blasio and Lorenzi [11] and Lorenzi and Vrabie [16, 17].

19.2 Preliminaries

Definition 19.1

Let X, Y be Banach spaces and Z a subset of Y. We say that a mapping $Q: Z \rightarrow X$ is compact if it carries bounded subsets in Z into relatively compact in X.

Definition 19.2

We say that the Banach space X is C ₀ -compact if there exists a family of linear, compact operators $\{I_{\varepsilon };\ \varepsilon \in (0,1)\} \subseteq \mathcal{L}(X)$ with $\|I_{\varepsilon }\|_{\mathcal{L}(X)} \leq 1$ for each $\varepsilon \in (0,1)$ and $\lim _{\varepsilon \downarrow 0}I_{\varepsilon }x = x$ for each x ∈ X.

Remark 19.1

Let p ∈ [ 1, +∞) and let Ω be a nonempty and bounded domain in $\mathbb{R}^{d}$, d ≥ 1. Then L ^p(Ω) is C ₀-compact. This follows from the fact that, for each p ∈ [ 1, +∞), the Laplace operator subjected to Dirichlet boundary conditions generates a compact C ₀-semigroup on L ^p(Ω). See [27, Theorem 4.1.3, p. 81]. Furthermore, if X is a separable Hilbert space then it is C ₀-compact. Indeed, let $\{e_{k};\ k \in \mathbb{N}\}$ be an orthonormal system, let $\varepsilon \in (0,1)$ and let us define $I_{\varepsilon }: X \rightarrow X$ by:

$$\displaystyle{I_{\varepsilon }(x):=\sum _{ k=0}^{\infty }e^{-\varepsilon k}\langle x,e_{ k}\rangle e_{k}}$$

for each x ∈ X. Since for each $m \in \mathbb{N}$, the operator $I_{\varepsilon }^{m}: X \rightarrow X$, defined by

$$\displaystyle{I_{\varepsilon }^{m}(x):=\sum _{ k=0}^{m}e^{-\varepsilon k}\langle x,e_{ k}\rangle e_{k}}$$

for each x ∈ X, has finite dimensional range and $\lim _{m\rightarrow \infty }I_{\varepsilon }^{m} = I_{\varepsilon }$ uniformly on bounded subsets, it readily follows that $I_{\varepsilon }$ is a compact operator for each $\varepsilon \in (0,1)$. Finally, observing that $\lim _{\varepsilon \downarrow 0}I_{\varepsilon }(x) = x$ for each x ∈ X, we conclude that X is C ₀-compact.

We recall for easy reference the following result due to Schaefer [24].

Theorem 19.1

Let Y be a real Banach space and let $Q: Y \rightarrow Y$ be a continuous, compact operator and let

$$\displaystyle{\mathcal{E}(Q) =\{ x \in Y;\exists \lambda \in [\,0,1\,],\ \text{such that}\ x =\lambda Q(x)\}.}$$

If $\mathcal{E}(Q)$ is bounded, then Q has at least one fixed-point.

Let us denote by $\mathcal{M}(\xi,h)$ the unique mild solution u of the Cauchy problem

$$\displaystyle{\left \{\begin{array}{ll} u'(t) = \mathit{Au}(t) + h(t),&\ \ t \in [\,0,T\,]\\ u(a) =\xi, \end{array} \right.}$$

corresponding to $\xi \in X$ and $h \in L^{1}(0,T;X)$, i.e.

$$\displaystyle{u(t) = S(t)\xi +\int _{ 0}^{t}S(t - s)h(s)\,\mathit{ds}}$$

for $t \in [\,0,T\,]$.

The next slight extension of a compactness result due to Becker [4] is a direct consequence of Vrabie [27, Theorem 2.8.4, p. 194] and Cârjă et al. [9, Lemma 1.5.1, p. 14].

Theorem 19.2

Let $A: D(A) \subseteq X \rightarrow X$ be the infinitesimal generator of a C ₀ -semigroup $\{S(t): X \rightarrow X;\ t \geq 0\}$ , let $\mathcal{D}$ be a bounded subset in $X$ , and $\mathcal{F}$ a subset in L ¹ (0,T;X) for which there exists a compact set $K \subseteq X$ such that f(t) ∈ K for each $f \in \mathcal{F}$ and a.e. for t ∈ [ 0,T ]. Then, for each θ ∈ (0,T), $\mathcal{M}(\mathcal{D},\mathcal{F})$ is relatively compact in C([ θ,T ];X). If, in addition, $\mathcal{D}$ is relatively compact, then $\mathcal{M}(\mathcal{D},\mathcal{F})$ is relatively compact even in C([ 0,T ];X).

19.3 The General Framework

We assume familiarity with the basic concepts and results of the linear semigroup theory, as well as with the theory of delay evolution equations. However, we recall for easy reference some concepts we will frequently use in the sequel and we refer to Vrabie [27] for details concerning linear semigroups and to Hale [15] for details on delay evolution equations.

We recall that a function $u \in C_{b}([\,-\tau,+\infty );X)$ is called a mild solution of the problem (19.1) if it is given by the variation of constants formula, i.e.

$$\displaystyle{ u(t) = \left \{\begin{array}{ll} S(t)g(u)(0) +\int _{ 0}^{t}S(t - s)f(s,u_{ s})\,\mathit{ds},&t \in [\,0,+\infty ) \\ u(t) = g(u)(t), &t \in [\,-\tau,0\,]. \end{array} \right. }$$

(19.2)

Definition 19.3

We say that the function $g: C_{b}([\,-\tau,+\infty );X) \rightarrow C([\,-\tau,0\,];X)$ has affine growth if there exists m ₀ ≥ 0 such that, for each u ∈ C _b ([ −τ, +∞); X), we have

$$\displaystyle{\|g(u)\|_{C([\,-\tau,0\,];X)} \leq \| u\|_{C_{b}([\,-\tau,+\infty );X)} + m_{0}.}$$

We begin with the general assumptions we need in the sequel.

(H _X ):

the Banach space X is C ₀-compact (cf. Definition 19.2) ; 

(H _A ):

the operator $A: D(A) \subseteq X \rightarrow X$ generates a C ₀-semigroup, $\{S(t): X \rightarrow X;\ t \geq 0\}$, and there exists ω > 0 such

$$\displaystyle{\|S(t)\| \leq e^{-\omega t},}$$

for each t ∈ [ 0, +∞) ; 

(H _f ):

the function $f: [\,0,+\infty ) \times C([\,-\tau,0\,];X) \rightarrow X$ is jointly continuous on its domain, compact (cf. Definition 19.1) and:

(f ₁):

there exists ℓ > 0 such that

$$\displaystyle{\|f(t,v) - f(t,\tilde{v})\| \leq \ell\| v -\tilde{ v}\|_{C([\,-\tau,0\,];X)}}$$

for each t ∈ [ 0, +∞) and $v,\tilde{v} \in C([\,-\tau,0\,];X)\,;$

(f ₂):

there exists m > 0 such that

$$\displaystyle{\|f(t,0)\| \leq m}$$

for each t ∈ [ 0, +∞);

(H _c ):

the constants ℓ and ω satisfy the nonresonance condition : 

$$\displaystyle{\ell<\omega;}$$

(H _g ):

the function g: C _b ([ −τ, +∞); X) → C([ −τ, 0 ]; X) satisfies:

(g ₁):

g has affine growth (cf. Definition 19.3) ; 

(g ₂):

there exists a > 0 such that, for each u, v ∈ C _b ([ −τ, +∞); X), from u(t) = v(t) for each t ∈ [ a, +∞), it follows that g(u) = g(v) ; 

(g ₃):

g is continuous from $\tilde{C}_{b}([\,-\tau,+\infty );X)$ to C([ −τ, 0 ]; X).

Remark 19.2

From (g ₁) and (g ₂), we get that for each u ∈ C _b ([ −τ, +∞); X), we have

$$\displaystyle{ \|g(u)\|_{C([\,-\tau,0\,];X)} \leq \| u\|_{C_{b}([\,a,+\infty );X)} + m_{0},\, }$$

(19.3)

where a is given by (g ₂). Indeed, if we assume by contradiction that there exists $u \in C_{b}([\,-\tau,+\infty );X)$ such that $\|u\|_{C_{b}([\,a,+\infty );X)} + m_{0} <\| g(u)\|_{C([\,-\tau,0\,];X)}$, then the function $\tilde{u}:[\,-\tau,+\infty)\to X$, defined by

$$\displaystyle{\tilde{u}(t) = \left \{\begin{array}{ll} u(t), &t \in [\,a,+\infty ),\\ u(a), &t \in [\,-\tau, a), \end{array} \right.}$$

satisfies $u(t) =\tilde{ u}(t)$ for each $t \in [\,a,+\infty )$ and thus $g(u) = g(\tilde{u})$. So,

$$\displaystyle{\|u\|_{C_{b}([\,a,+\infty );X)} + m_{0} <\| g(u)\|_{C([\,-\tau,0\,];X)} =\| g(\tilde{u})\|_{C([\,-\tau,0\,];X)}}$$

$$\displaystyle{\leq \|\tilde{ u}\|_{C_{b}([\,-\tau,+\infty );X)} + m_{0} =\|\tilde{ u}\|_{C_{b}([\,a,+\infty );X)} + m_{0} =\| u\|_{C_{b}([\,a,+\infty );X)} + m_{0}.}$$

This contradiction can be eliminated only if (19. 3) holds true, as claimed.

Remark 19.3

The class of functions g satisfying (H _g ) is very large and includes several important specific cases. More precisely:

• let $\mathcal{N}: X \rightarrow X$ be a possibly nonlinear operator having linear growth, i.e.

  $$\displaystyle{\|\mathcal{N}(x)\| \leq \| x\|}$$

  for each x ∈ X, let μ is a σ-finite and complete measure on [ 0, +∞), satisfying

  $$\displaystyle{\mathrm{supp}\ \mu = [\,b,+\infty ),}$$

  where b > τ and μ([ 0, +∞)) = 1, and let ψ ∈ C([ −τ, 0 ]; X). Then, the function g, defined by

  $$\displaystyle{ g(u)(t) =\int _{ 0}^{+\infty }\mathcal{N}(u(t+\theta ))\,d\mu (\theta ) +\psi (t), }$$

  (19.4)

  for each $u \in C_{b}([\,-\tau,+\infty );X)$ and t ∈ [ −τ, 0 ], satisfies hypothesis (H _g ) with $m_{0} =\|\psi \| _{C([\,-\tau,0\,];X)}$ and a = b −τ;

• let T > τ and let us consider the T-periodic condition, i.e.

  $$\displaystyle{g(u)(t) = u(t + T),}$$

  for $u \in C_{b}([\,-\tau,+\infty );X)$ and t ∈ [ −τ, 0 ]. Clearly g satisfies (H _g ), with m ₀ = 0 and a = T −τ;

• let T > τ and let us consider the T-anti-periodic condition, i.e.

  $$\displaystyle{g(u)(t) = -u(t + T),}$$

  for u ∈ C _b ([ −τ, +∞); X) and t ∈ [ −τ, 0 ]. Also in this case g satisfies (H _g ), with m ₀ = 0 and a = T −τ;

• let us consider the multi-point discrete mean condition, i.e.

  $$\displaystyle{g(u)(t) =\sum _{ i=1}^{n}\alpha _{ i}u(t + t_{i})}$$

  for u ∈ C _b ([ −τ, +∞); X) and t ∈ [ −τ, 0 ], where α _i  ∈ (0, 1), for $i = 1,2,\ldots,n$, $\sum _{i=1}^{n}\alpha _{ i} \leq 1$ and $0 < t_{1} < t_{2} <\ldots < t_{n}$ are arbitrary but fixed. In this case, g satisfies (H _g ) with m ₀ = 0 and a = t ₁.

19.4 The Main Result and Some Auxiliary Lemmas

The main result of this paper is:

Theorem 19.3

If (H _X ), (H _A ), (H _f ), (H _g ) and (H _c ) hold true, then the problem (19.1) has at least one mild solution, u ∈ C _b ([ −τ,+∞);X). Moreover, for each mild solution of (19.1) , we have

$$\displaystyle{ \|u\|_{C_{b}([\,-\tau,+\infty );X)} \leq \frac{m} {\omega -\ell} + \left [ \frac{\omega } {\omega -\ell}\cdot \left ( \frac{1} {e^{\omega a} - 1} + \frac{\ell} {\omega }\right ) + 1\right ] \cdot m_{0}. }$$

(19.5)

If, in addition, instead of (H _c ), the stronger nonresonance condition ℓe ^ωτ < ω is satisfied, then each mild solution of (19.1) is globally asymptotically stable.

The proof of Lemma 19.1 below can be found in Vrabie [33, Lemma 6.2].

Lemma 19.1

If (H _c ) is satisfied and $u \in C_{b}([\,-\tau,+\infty );X)$ is such that

$$\displaystyle{\|u(t)\| \leq e^{-\omega t}\|u(0)\| + (1 - e^{-\omega t})\frac{\ell} {\omega }\left [\|u\|_{C_{b}([\,-\tau,+\infty );X)} + \frac{m} {\ell} \right ]}$$

for each t ∈ [ 0,+∞) and

$$\displaystyle{\|u\|_{C([\,-\tau,0\,];X)} \leq \| u\|_{C_{b}([\,a,+\infty );X)} + m_{0},}$$

then u satisfies (19.5) .

The lemma below is a specific form of general result in Vrabie [32, Lemma 4.3], where X is a general Banach space, A is a nonlinear m-dissipative operator and f is globally Lipschitz.

Lemma 19.2

Let us assume that (H _A ), (H _f ) and (H _c ) are satisfied. Then, for each $\varphi \in C([\,-\tau,0\,];X)$ , the problem

$$\displaystyle{ \left \{\begin{array}{ll} u'(t) = Au(t) + f(t,u_{t}),&\quad t \in [\,0,+\infty ), \\ u(t) =\varphi (t), &\quad t \in [\,-\tau,0\,], \end{array} \right. }$$

(19.6)

has a unique mild solution $u \in C_{b}([\,-\tau,+\infty );X)$ .

We conclude this section with a Bellman Lemma for integral inequalities with delay proved in Burlică and Roşu [5].

Lemma 19.3

Let $y: [\,-\tau,+\infty ) \rightarrow \mathbb{R}_{+}$ and $\alpha _{0},\beta: [\,0,+\infty ) \rightarrow \mathbb{R}_{+}$ be continuous functions with α ₀ nondecreasing. If

$$\displaystyle{ y(t) \leq \alpha _{0}(t) +\int _{ 0}^{t}\beta (s)\|y_{ s}\|_{C([\,-\tau,0\,]:\mathbb{R})}\,\mathit{ds} }$$

(19.7)

for each t ∈ [ 0,+∞), then

$$\displaystyle{ y(t) \leq \alpha (t) +\int _{ 0}^{t}\alpha (s)\beta (s)e^{\int _{s}^{t}\beta (\sigma )\,d\sigma }\mathit{ds}, }$$

(19.8)

for each t ∈ [ 0,+∞), where $\alpha (t):=\| y_{0}\|_{C([\,-\tau,0\,];\mathbb{R})} +\alpha _{0}(t)$ for each t ∈ [ 0,+∞).

19.5 Proof of the Main Result

19.5.1 The Approximate Problem

We shall use an interplay of two fixed point arguments and an approximation procedure. Let $\varepsilon > 0$, let $g_{\varepsilon } = I_{\varepsilon }g$, where $I_{\varepsilon }$ is given by (H _X ), and let us consider the $\varepsilon$-approximate problem

$$\displaystyle{ \left \{\begin{array}{ll} u'(t) = \mathit{Au}(t) + f(t,u_{t}),&\quad t \in [\,0,+\infty ), \\ u(t) = g_{\varepsilon }(u)(t), &\quad t \in [\,-\tau,0\,]. \end{array} \right. }$$

(19.9)

Remark 19.4

In view of (g ₂), for each $v \in C_{b}([\,-\tau,+\infty );X)$, g(v) depends only on the values of v on [ a, +∞). In fact we have

$$\displaystyle{ g(v) = g(\tilde{v}_{\vert [\,0,+\infty )}) = g(\tilde{w}_{\vert [\,a,+\infty )}), }$$

(19.10)

where a > 0 is given by (g ₂), $\tilde{v}$ is any function in C _b ([ −τ, +∞); X) which coincides with v on [ 0, +∞) and $\tilde{w}$ is any function in C _b ([ −τ, +∞); X) which coincides with v on [ a, +∞). This explains why, in that follows, we will assume with no loss of generality that g is defined merely on C _b ([ 0, +∞); X), or even on C _b ([ a, +∞); X). From (g ₃) and (19. 10), we deduce that g is continuous from both $\tilde{C}_{b}([\,0,+\infty );X)$ and $\tilde{C}_{b}([\,a,+\infty );X)$ to C([ −τ, 0 ]; X).

Lemma 19.4

Let us assume that (H _X ), (H _A ), (H _f ) and (H _g ) are satisfied, let $\varepsilon > 0$ be arbitrary and let $g_{\varepsilon }$ be defined as above. Then the approximate problem (19.9) has at least one mild solution $u_{\varepsilon } \in C_{b}([\,-\tau,+\infty );X)$ .

Proof

Let us first consider the problem

$$\displaystyle{ \left \{\begin{array}{ll} u'(t) = \mathit{Au}(t) + f(t,u_{t}),&\quad t \in [\,0,+\infty ), \\ u(t) = g_{\varepsilon }(v)(t), &\quad t \in [\,-\tau,0\,]. \end{array} \right. }$$

(19.11)

By Lemma 19.2 and Remark 19.4, it follows that, for each $v \in C_{b}([\,0,+\infty );X)$, (19.11) has a unique mild solution $u \in C_{b}([\,-\tau,+\infty );X)$. Thus, we can define the operator

$$\displaystyle{\mathcal{S}_{\varepsilon }: C_{b}([\,0,+\infty );X) \rightarrow C_{b}([\,0,+\infty );X)}$$

by

$$\displaystyle{\mathcal{S}_{\varepsilon }(v):= u_{\vert [\,0,+\infty )},}$$

where u is the unique mild solution of (19. 11) corresponding to v. We will complete the proof, by showing that the operator $\mathcal{S}_{\varepsilon }$ defined as above satisfies the hypotheses of Schaefer Fixed Point Theorem 19.1 and thus (19.9) has at least one mild solution. So, we have to check out that $\mathcal{S}_{\varepsilon }$ is continuous with respect to norm topology of C _b ([ 0, +∞); X), is compact and

$$\displaystyle{\mathcal{E}(\mathcal{S}_{\varepsilon }) =\{ u \in C_{b}([\,0,+\infty );X);\exists \lambda \in [\,0,1\,],\ \text{such that}\ u =\lambda \mathcal{S}_{\varepsilon }(u)\}}$$

is bounded.

To prove the continuity of $\mathcal{S}_{\varepsilon }$ let $v,\tilde{v} \in C_{b}([\,0,+\infty );X)$, set $u(t) = \mathcal{S}_{\varepsilon }(v)(t)$ and $\tilde{u}(t) = \mathcal{S}_{\varepsilon }(\tilde{v})(t)$ for t ∈ [ 0, +∞) and let us observe that

$$\displaystyle{\|u(t) -\tilde{ u}(t)\| \leq e^{-\omega t}\|g_{\varepsilon }(v)(0) - g_{\varepsilon }(\tilde{v})(0)\| +\int _{ 0}^{t}e^{-\omega (t-s)}\|f(s,u_{ s}) - f(s,\tilde{u}_{s})\|\,\mathit{ds}}$$

$$\displaystyle{\leq e^{-\omega t}\|g(v)(0) - g(\tilde{v})(0)\| +\ell\int _{ 0}^{t}e^{-\omega (t-s)}\|u_{ s} -\tilde{ u}_{s}\|_{C([\,-\tau,0\,];X)}\,\mathit{ds}}$$

for each t ∈ [ 0, +∞). So, we have

$$\displaystyle{\|u(t) -\tilde{ u}(t)\| \leq e^{-\omega t}\|g(v)(0) - g(\tilde{v})(0)\| + (1 - e^{-\omega t})\frac{\ell} {\omega }\|u -\tilde{ u}\|_{C_{b}([\,-\tau,+\infty );X)}}$$

for each t ∈ [ 0, +∞).

But

$$\displaystyle{\|u -\tilde{ u}\|_{C_{b}([\,-\tau,+\infty );X)} \leq \| u -\tilde{ u}\|_{C([\,-\tau,0\,];X)} +\| u -\tilde{ u}\|_{C_{b}([\,0,+\infty );X)}.}$$

On the other hand, by the nonlocal initial condition, we have

$$\displaystyle{\|u -\tilde{ u}\|_{C([\,-\tau,0\,];X)} =\| g_{\varepsilon }(v) - g_{\varepsilon }(\tilde{v})\|_{C([\,-\tau,0\,];X)} \leq \| g(v) - g(\tilde{v})\|_{C([\,-\tau,0\,];X)}.}$$

Thus

$$\displaystyle{\|u -\tilde{ u}\|_{C_{b}([\,0,+\infty );X)} \leq \frac{\omega +\ell} {\omega -\ell}\|g(v) - g(\tilde{v})\|_{C([\,-\tau,0\,];X)}}$$

for each $v,\tilde{v} \in C_{b}([\,0,+\infty );X)$. So, from (g ₃) in (H _g ) and Remark 19.4, we conclude that $\mathcal{S}_{\varepsilon }$ is continuous on C _b ([ 0, +∞); X) in the norm topology.

The next step is to show that, for each bounded set $\mathcal{K}$ in C _b ([ 0, +∞); X), $\mathcal{S}_{\varepsilon }(\mathcal{K})$ is relatively compact in C _b ([ 0, +∞); X). To this aim, let $\mathcal{K}$ be a bounded set in C _b ([ 0, +∞); X) and let $(v_{k})_{k}$ be an arbitrary sequence in $\mathcal{K}$. We show first that $u_{k} = \mathcal{S}_{\varepsilon }(v_{k})$, $k = 1,2,\ldots$, is bounded. Indeed, from (H _A ), (H _f ) and (19. 2), we get

$$\displaystyle{\|u_{k}(t)\| \leq e^{-\omega t}\|u_{ k}(0)\| + (1 - e^{-\omega t})\frac{\ell} {\omega }\left [\|u_{k}\|_{C_{b}([\,-\tau,+\infty );X)} + \frac{m} {\ell} \right ]}$$

for each t ∈ (0, +∞). For t ∈ [ −τ, 0 ], we have

$$\displaystyle{\|u_{k}(t)\| =\| g_{\varepsilon }(v_{k})(t)\| \leq \| v_{k}\|_{C_{b}([\,0,+\infty );X)} + m_{0}}$$

for each $k \in \mathbb{N}$ and, since $(v_{k})_{k}$ is bounded, there exists m ₁ > 0 such that

$$\displaystyle{\|u_{k}\|_{C([\,-\tau,0\,];X)} \leq m_{1}}$$

for each $k \in \mathbb{N}$. On the other hand

$$\displaystyle{\|u_{k}\|_{C_{b}([\,-\tau,+\infty );X)} =\max \{\| u_{k}\|_{C([\,-\tau,0\,];X)},\|u_{k}\|_{C_{b}([\,0,+\infty );X)}\}}$$

$$\displaystyle{\leq m_{1} +\| u_{k}\|_{C_{b}([\,0,+\infty );X)}.}$$

Accordingly

$$\displaystyle{\|u_{k}(t)\| \leq e^{-\omega t}m_{ 1} + (1 - e^{-\omega t})\frac{\ell} {\omega }\left [\|u_{k}\|_{C_{b}([\,0,+\infty );X)} + m_{1} + \frac{m} {\ell} \right ]}$$

$$\displaystyle{\leq m_{1} + \frac{\ell} {\omega }\left [\|u_{k}\|_{C_{b}([\,0,+\infty );X)} + m_{1} + \frac{m} {\ell} \right ]}$$

for each $k \in \mathbb{N}$ and t ∈ (0, +∞). Hence

$$\displaystyle{\left (1 -\frac{\ell} {\omega }\right )\|u_{k}\|_{C_{b}([\,0,+\infty );X)} \leq m_{1}\left (1 + \frac{\ell} {\omega }\right ) + \frac{m} {\omega } }$$

for each $k \in \mathbb{N}$. Finally, we deduce that

$$\displaystyle{\|u_{k}\|_{C_{b}([\,0,+\infty );X)} \leq \frac{\omega +\ell} {\omega -\ell}\cdot m_{1} + \frac{m} {\omega -\ell}}$$

for each $k \in \mathbb{N}$. So $(\mathcal{S}_{\varepsilon }(v_{k}))_{k}$ is bounded and consequently, for each T > 0, the set

$$\displaystyle{\{f(t,\mathcal{S}_{\varepsilon }(v_{k})_{t});\ k \in \mathbb{N},\ t \in [\,0,T\,]\}}$$

is relatively compact. By virtue of Theorem 19.2, it follows that $\{\mathcal{S}_{\varepsilon }(v_{k});\ k \in \mathbb{N}\}$ is relatively compact in C([ δ, T ]; X) for each T > 0 and δ ∈ (0, T).

By (H _X ), it follows that $\{\mathcal{S}_{\varepsilon }(v_{k})(0);\ k \in \mathbb{N}\} =\{ g_{\varepsilon }(v_{k})(0);\ k \in \mathbb{N}\}$ is relatively compact in X, simply because $(v_{k})_{k}$ is bounded and $I_{\varepsilon }$ is compact. Using once again Theorem 19.2, we conclude that $\{\mathcal{S}_{\varepsilon }(v_{k}));\ k \in \mathbb{N}\}$ is relatively compact in C([ 0, T ]; X) for each T > 0 and thus in $\tilde{C}_{b}([\,0,+\infty );X)$.

For the sake of simplicity, let us denote also by $(u_{k})_{k} = (\mathcal{S}_{\varepsilon }(v_{k}))_{k}$ a convergent subsequence of $(\mathcal{S}_{\varepsilon }(v_{k}))_{k}$ in $\tilde{C}_{b}([\,0,+\infty );X)$ to some function u. Then

$$\displaystyle{\|u_{k}(t) - u(t)\| \leq e^{-\omega (t-\tau )}\|u_{ k}(\tau ) - u(\tau )\| +\int _{ \tau }^{t}\ell e^{-\omega (t-s)}\|u_{}{ k}_{s} - u_{s}\|_{C([\,-\tau,0\,];X)}\,\mathit{ds}}$$

for each t ∈ [ τ, +∞). Taking $y(t) = e^{\omega (t-\tau )}\|u_{k}(t) - u(t)\|$, $\alpha _{0}(t) =\| u_{k}(\tau ) - u(\tau )\|$ and β = ℓ in Lemma 19.3 applied on the shifted intervals [ 0, +∞) and [ τ, +∞), after some simple calculations and recalling that ℓ < ω, we deduce

$$\displaystyle{\|u_{k}(t) - u(t)\| \leq e^{(\ell-\omega )(t-\tau )}[\|u_{ k}(\tau ) - u(\tau )\| +\| u_{k} - u\|_{C([\,0,\tau \,];X)}]}$$

for each t ∈ [ τ, +∞). Since $(u_{k})_{k}$ is bounded, this shows that $\lim _{k}u_{k} = u$ in C _b ([ τ, +∞); X). Furthermore, from $\lim _{k}u_{k} = u$ in $\tilde{C}_{b}([\,0,+\infty );X)$, it follows that $(u_{k})_{k} = (\mathcal{S}_{\varepsilon }(v_{k}))_{k}$ is convergent even in $C_{b}([\,0,+\infty );X)$. Thus $\mathcal{S}_{\varepsilon }(\mathcal{K})$ is relatively compact in C _b ([ 0, +∞); X), as claimed.

It remains merely to prove that $\mathcal{E}(\mathcal{S}_{\varepsilon })$ is bounded. To this aim, let $u \in \mathcal{E}(\mathcal{S}_{\varepsilon })$, i.e.

$$\displaystyle{u =\lambda \mathcal{S}_{\varepsilon }(u)}$$

for some λ ∈ [ 0, 1 ]. Consequently

$$\displaystyle{\|u(t)\| \leq \|\mathcal{S}_{\varepsilon }(u)(t)\|}$$

for each t ∈ [ 0, +∞). From (H _A ) and (19. 2), it follows that

$$\displaystyle{\|\mathcal{S}_{\varepsilon }(u)(t)\| \leq e^{-\omega t}\|\mathcal{S}_{\varepsilon }(u)(0)\| +\int _{ 0}^{t}e^{-\omega (t-s)}\|f(s,u_{ s})\|\,\mathit{ds}}$$

for each t ∈ [ 0, +∞) and thus, by (H _f ) and (g ₁) in (H _g ), we get

$$\displaystyle{\|\mathcal{S}_{\varepsilon }(u)(t)\| \leq e^{-\omega t}\|\mathcal{S}_{\varepsilon }(u)(0)\| + (1 - e^{-\omega t})\frac{\ell} {\omega }\left [\|\mathcal{S}_{\varepsilon }(u)\|_{C_{b}([\,-\tau,+\infty );X)} + \frac{m} {\ell} \right ]}$$

for each t ∈ [ 0, +∞). Since, by (g ₁), (g ₂) in (H _g ), (H _X ) and Remark 19.2, we have

$$\displaystyle{\|\mathcal{S}_{\varepsilon }(u)\|_{C([\,-\tau,0\,];X)} \leq \| g_{\varepsilon }(u)\|_{C([\,-\tau,0\,];X)}}$$

$$\displaystyle{\leq \| u\|_{C_{b}([\,a,+\infty );X)} + m_{0} \leq \|\mathcal{S}_{\varepsilon }(u)\|_{C_{b}([\,a,+\infty );X)} + m_{0},}$$

we are in the hypotheses of Lemma 19.1 which shows that $\mathcal{E}(\mathcal{S}_{\varepsilon })$ is bounded. Thus Schaefer Fixed Point Theorem 19.1 applies implying that $\mathcal{S}_{\varepsilon }$ has at least one fixed point which v = u _{| [ 0, +∞)}, where u ∈ C _b ([ −τ, +∞); X) is a mild solution of the problem (19. 9). This concludes the proof of Lemma 19.4. □ 

19.5.2 Proof of the Main Result: Continued

We can now proceed with the final part of the proof of Theorem 19.3.

Proof

For each $\varepsilon \in (0,1)$ let us fix a mild solution $u_{\varepsilon }$ of the problem (19. 9) whose existence is ensured by Lemma 19.4. We first prove that the set $\{u_{\varepsilon };\ \varepsilon \in (0,1)\}$ is relatively compact in $\tilde{C}_{b}([\,-\tau,+\infty );X)$. As a consequence, there exists a sequence $\varepsilon _{n} \downarrow 0$ such that the corresponding sequence $(u_{\varepsilon _{n}})_{n}$ converges in $\tilde{C}_{b}([\,-\tau,+\infty );X)$ to a function u ∈ C _b ([ −τ, +∞); X) which turns out to be a mild solution of (19. 1).

From Lemma 19.1, we know that $\{u_{\varepsilon };\ \varepsilon \in (0,1)\}$ is bounded in C _b ([ −τ, +∞); X). From Theorem 19.2, we conclude that $\{u_{\varepsilon };\ \varepsilon \in (0,1)\}$ is relatively compact in C([ δ, T]; X) for each T > 0 and each δ ∈ (0, T). Then, it is relatively compact in $\tilde{C}_{b}([\,a,+\infty );X)$. In view of (g ₂) and (g ₃) and Remark 19.4, it follows that $\{g_{\varepsilon }(u_{\varepsilon });\ \varepsilon \in (0,1)\}$ is relatively compact in C([ −τ, 0 ]; X). This means that, there exists a subsequence of $(\varepsilon _{n})_{n}$, denoted again by $(\varepsilon _{n})_{n}$, such that $(u_{\varepsilon _{n}})_{n}$—denoted for simplicity also by (u _n ) _n —converges in $\tilde{C}_{b}([\,a,+\infty );X)$ to some function u ∈ C _b ([ a, +∞); X) and the restriction of $(u_{n})_{n}$ to [ −τ, 0], i.e. $(g_{\varepsilon _{n}}(u_{n}))_{n}$, converges in C([ −τ, 0 ]; X) to some element v ∈ C([ −τ, 0 ]; X), i.e.

$$\displaystyle{\left \{\begin{array}{ll} \lim _{n\rightarrow \infty }u_{n} = u &\text{in}\ \tilde{C}_{b}([\,a,+\infty );X) \\ \lim _{n\rightarrow \infty }u_{n} =\lim _{n\rightarrow \infty }g_{\varepsilon _{n}}(u_{n}) =\lim _{n\rightarrow \infty }I_{\varepsilon _{n}}g(u_{n}) = v&\text{in}\ C([\,-\tau,0\,];X). \end{array} \right.}$$

We have

$$\displaystyle{\|u_{n}(t) - u_{p}(t)\| \leq e^{-\omega t}\|u_{ n}(0) - u_{p}(0)\| + (1 - e^{-\omega t})\frac{\ell} {\omega }\|u_{n} - u_{p}\|_{C_{b}([\,-\tau,+\infty );X)}}$$

for each $n,p \in \mathbb{N}$ and each t ∈ [ 0, +∞). Since

$$\displaystyle{\|u_{n} - u_{p}\|_{C_{b}([\,-\tau,+\infty );X)} \leq \| u_{n} - u_{p}\|_{C([\,-\tau,0\,];X)} +\| u_{n} - u_{p}\|_{C_{b}([\,0,+\infty );X)}}$$

$$\displaystyle{ =\| g_{\varepsilon _{n}}(u_{n}) - g_{\varepsilon _{p}}(u_{p})\|_{C([\,-\tau,0\,];X)} +\| u_{n} - u_{p}\|_{C_{b}([\,0,+\infty );X)}, }$$

(19.12)

it follows that

$$\displaystyle{\|u_{n}(t) - u_{p}(t)\| \leq e^{-\omega t}\|g_{\varepsilon _{ n}}(u_{n}) - g_{\varepsilon _{p}}(u_{p})\|_{C([\,-\tau,0\,];X)}}$$

$$\displaystyle{+(1 - e^{-\omega t})\frac{\ell} {\omega }\left [\|g_{\varepsilon _{n}}(u_{n}) - g_{\varepsilon _{p}}(u_{p})\|_{C([\,-\tau,0\,];X)} +\| u_{n} - u_{p}\|_{C_{b}([\,0,+\infty );X)}\right ]}$$

for each $n,p \in \mathbb{N}$ and t ∈ [ 0, +∞). Hence

$$\displaystyle{\|u_{n} - u_{p}\|_{C_{b}([\,0,+\infty );X)} \leq \frac{\omega +\ell} {\omega -\ell}\|g_{\varepsilon _{n}}(u_{n}) - g_{\varepsilon _{p}}(u_{p})\|_{C([\,-\tau,0\,];X)}}$$

for each $n,p \in \mathbb{N}$. As $(g_{\varepsilon _{n}}(u_{n}))_{n}$ is fundamental in C([ −τ, 0 ]; X) being convergent, from the last inequality, if follows that (u _n ) _n is fundamental in C _b ([ 0, +∞); X). By virtue of (19. 12), we deduce that (u _n ) _n is fundamental even in C _b ([ −τ, +∞); X) and so it is convergent in this space.

Now, from Remark 19.2, it follows that Lemma 19.1 applies and thus we get (19. 5). Since the global asymptotic stability, in the case when ℓ e ^{τ ω} < ω, follows very similar arguments as those in the last part of the proof of Burlică and Roşu [5, Theorem 3.1], this completes the proof of Theorem 19.3. □ 

19.6 The Damped Wave Equation

Let Ω be a nonempty bounded and open subset in $\mathbb{R}^{d}$, d ≥ 1, with C ¹ boundary Γ, let Q ₊ = [ 0, +∞) ×Ω, Q _τ  = [ −τ, 0 ] ×Ω, Σ ₊ = [ 0, +∞) ×Γ, let ω > 0 and let us consider the following damped wave equation with delay, subjected to nonlocal initial conditions:

$$\displaystyle{ \left \{\begin{array}{ll} \frac{\partial ^{2}u} {\partial t^{2}} (t,x) =\varDelta u(t,x) - 2\omega \frac{\partial u} {\partial t} (t,x) -\omega ^{2}u(t,x) + h\left (t,u(t-\tau,\cdot )\right ) &\text{in}\ Q_{ +}, \\ u(t,x) = 0 &\text{on}\ \varSigma _{+}, \\ u(t,x) =\int _{ 0}^{+\infty }\alpha (s)u(t + s,x)\,\mathit{ds} +\psi _{ 1}(t,x) &\text{in}\ Q_{\tau }, \\ \frac{\partial u} {\partial t} (t,x) =\int _{ 0}^{+\infty }\mathcal{N}\left (s,u(t + s,x), \frac{\partial u} {\partial t} (t + s,x)\right )\,\mathit{ds} +\psi _{2}(t,x)&\text{in}\ Q_{\tau }, \end{array} \right. }$$

(19.13)

$h: [\,0,+\infty ) \times H_{0}^{1}(\varOmega ) \rightarrow L^{2}(\varOmega )$, $\alpha \in L^{2}([\,0,+\infty ); \mathbb{R})$, $\mathcal{N}: [\,0,+\infty ) \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$, while $\psi _{1} \in C([\,-\tau,0\,];H_{0}^{1}(\varOmega ))$ and $\psi _{2} \in C([\,-\tau,0\,];L^{2}(\varOmega ))$.

Theorem 19.4

Let Ω be a nonempty bounded and open subset in $\mathbb{R}^{d}$ , d ≥ 1, with C ¹ boundary Γ, let τ ≥ 0, $\psi _{1} \in C([\,-\tau,0\,];H_{0}^{1}(\varOmega ))$ , $\psi _{2} \in C([\,-\tau,0\,];L^{2}(\varOmega ))$ and let us assume that $h: [\,0,+\infty ) \times \mathbb{R} \rightarrow \mathbb{R}$ , $\alpha \in L^{2}([\,0,+\infty ); \mathbb{R})$ and $\mathcal{N}: [\,0,+\infty ) \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ are continuous and satisfy :

(h₁ ) :

there exists ℓ > 0 such that $\|h(t,w) - h(t,y)\|_{L^{2}(\varOmega )} \leq \ell\| w - y\|_{H_{0}^{1}(\varOmega )}$ for each t ∈ [ 0,+∞) and $w,y \in H_{0}^{1}(\varOmega )\,;$

(h₂):

there exists m > 0 such that $\|h(t,0)\|_{L^{2}(\varOmega )} \leq m$ for each t ∈ [ 0,+∞) ;

(n ₁):

there exists a nonnegative continuous function $\eta \in L^{2}(\mathbb{R}_{+}; \mathbb{R}_{+})$ such that

$$\displaystyle{\vert \mathcal{N}(t,u,v)\vert \leq \eta (t)(\vert u\vert + \vert v\vert ),}$$

for each $t \in \mathbb{R}_{+}$ and $u,v \in \mathbb{R}.$

Let λ ₁ be the first eigenvalue of −Δ and let us assume that

(n₂):

$\left \{\begin{array}{ll} \|\eta \|_{L^{2}([\,0,+\infty );\mathbb{R})} \leq 1, \\ (1 +\lambda _{ 1}^{-1}\omega )\|\alpha \|_{L^{2}([\,0,+\infty );\mathbb{R})} +\lambda _{ 1}^{-1}(1+\omega )\|\eta \|_{L^{2}([\,0,+\infty );\mathbb{R})} \leq 1\,;\end{array} \right.$

(n₃):

there exists b > τ such that α(t) = η(t) = 0 for each t ∈ [ 0,b ];

(c₁):

ℓ < ω.

Then the problem(19.13)has at least one mild solution $u \in C_{b}([\,-\tau,+\infty );H_{0}^{1}(\varOmega ))$ with $\frac{\partial u} {\partial t} \in C_{b}([\,-\tau,+\infty );L^{2}(\varOmega ))$ . In addition, u satisfies

$$\displaystyle{\|u\|_{C_{b}([\,-\tau,+\infty );H_{0}^{1}(\varOmega ))} + \left \|\frac{\partial u} {\partial t} \right \|_{C_{b}([\,-\tau,+\infty );L^{2}(\varOmega ))}}$$

$$\displaystyle{\leq \frac{m} {\omega -\ell} + \left [ \frac{\omega } {\omega -\ell}\cdot \left ( \frac{1} {e^{\omega a} - 1} + \frac{\ell} {\omega }\right ) + 1\right ] \cdot m_{0},}$$

where

$$\displaystyle{m_{0} =\|\psi _{1}\|_{C([\,-\tau,0\,];H_{0}^{1}(\varOmega ))} +\|\omega \psi _{1} +\psi _{2}\|_{C([\,-\tau,0\,];L^{2}(\varOmega ))}.}$$

If, instead of (c ₁ ), the stronger nonresonance condition ℓe ^τω < ω is satisfied, then each mild solution of(19.13)is globally asymptotically stable.

Proof

Let us observe that (19. 13) can be equivalently rewritten in the form (19. 1) in the Hilbert space $X = \left (\begin{array}{cc} H_{0}^{1}(\varOmega ) \\ \times \\ L^{2}(\varOmega ) \end{array} \right ),$ endowed with the usual inner product

$$\displaystyle{\left \langle \left (\begin{array}{cc} u\\ v \end{array} \right ),\left (\begin{array}{cc} \tilde{u}\\ \tilde{v} \end{array} \right )\right \rangle =\int _{\varOmega }\nabla u(x)\cdot \nabla \tilde{u}(x)\,\mathit{dx}+\int _{\varOmega }v(x)\tilde{v}(x)\,\mathit{dx}}$$

for each $\left (\begin{array}{cc} u\\ v \end{array} \right ),\left (\begin{array}{cc} \tilde{u}\\ \tilde{v} \end{array} \right ) \in X$, where A, f, and g are defined as follows. First, let us define the linear operator $A: D(A) \subseteq X \rightarrow X$ by

$$\displaystyle{D(A) = \left (\begin{array}{cc} H_{0}^{1}(\varOmega ) \cap H^{2}(\varOmega ) \\ \times \\ H_{0}^{1}(\varOmega ) \end{array} \right ),}$$

$$\displaystyle{A\left (\begin{array}{cc} u\\ v \end{array} \right ):= \left (\begin{array}{cc} -\omega u + v\\ \varDelta u -\omega v \end{array} \right )}$$

for each $\left (\begin{array}{cc} u\\ v \end{array} \right ) \in D(A)$. Second, let us define f: [ 0, +∞) × C([ −τ, 0 ]; X) → X by

$$\displaystyle{f\left (t,\left (\begin{array}{cc} z\\ y \end{array} \right )\right )(x) = \left (\begin{array}{cc} 0\\ h(t, z(-\tau )) \end{array} \right )}$$

for each t ∈ [ 0, +∞), x ∈ Ω and $\left (\begin{array}{cc} z\\ y \end{array} \right ) \in C([\,-\tau,0\,];X)$. Third, let the nonlocal constraint $g: C_{b}([\,-\tau,+\infty );X) \rightarrow C([\,-\tau,0\,];X)$ be given by

$$\displaystyle{\left [g\left (\begin{array}{cc} u\\ v \end{array} \right )(t)\right ](x) = \left (\begin{array}{cc} \int _{0}^{+\infty }\alpha (s)u(t + s,x)\,\mathit{\mathit{ds}} +\psi _{ 1}(t,x) \\ \int _{0}^{+\infty }\mathcal{M}\left (s,x,u(t + s,x),w(t + s,x)\right )\,\mathit{ds} +\psi _{ 3}(t,x) \end{array} \right )}$$

for each $\left (\begin{array}{cc} u\\ v \end{array} \right ) \in C_{b}([\,-\tau,+\infty );X)$, each t ∈ [ −τ, 0 ] and a.e. x ∈ Ω, w = v −ω u, $\mathcal{M}(t,u,w) = \mathcal{N}(t,u,w) +\omega \alpha (t)u$ and $\psi _{3} =\psi _{2} +\omega \psi _{1}$.

We begin by observing that, in view of Remark 19.1, X satisfies (H _X ). Clearly, the linear operator B: = A +ω I for each u ∈ D(B) = D(A), where A is defined as above and I is the identity on X, is the infinitesimal generator of a C ₀-group of unitary operators $\{G(t): X \rightarrow X;\ t \in \mathbb{R}\}$ in X. See Vrabie [27, Theorem 4.6.2, p. 93]. Consequently, A generates a C ₀-semigroup of contractions {S(t): X → X;  t ≥ 0}, defined by

$$\displaystyle{S(t)\xi = e^{-\omega t}G(t)\xi }$$

for each t ≥ 0 and each ξ ∈ X, i.e. A is m-dissipative. So A satisfies (H _A ). Next, let us observe that f is compact. Indeed, if $C = \left \{\left (\begin{array}{cc} u_{\alpha }\\ v_{\alpha } \end{array} \right );\ \alpha \in \varLambda \right \}$ is a bounded subset in C([ −τ, 0 ]; X), then $\{u_{\alpha }(-\tau );\ \alpha \in \varLambda \}$ is bounded in $H_{0}^{1}(\varOmega )$ which is compactly embedded in L ²(Ω). So, $\{u_{\alpha }(-\tau );\ \alpha \in \varLambda \}$ is relatively compact in L ²(Ω) and since h is continuous and has linear growth, it follows that, for each T > 0, f([ 0, T ] × C) is relatively compact in X. But this shows that f is compact. Moreover, in view of (h ₁), it follows that the function f satisfies (H _f ), while from (n ₁), (n ₂) and (n ₃), we deduce that g, which is of the form (19. 4) in Remark 19.3 with

$$\displaystyle{\psi (t) = \left (\begin{array}{ll} \psi _{1}(t) \\ \psi _{3}(t) \end{array} \right ),}$$

for t ∈ [ −τ, 0 ], and satisfies (H _g ) with $m_{0} =\|\psi \| _{C([\,-\tau,0\,];X)}$ and a = b −τ > 0. Hence, the conclusion of Theorem 19.4 follows from Theorem 19.3. □