Periodic solutions and their conditional stability for partial neutral functional differential equations

Abstract

We prove the existence and uniqueness of the periodic solution as well as its conditional stability for the semilinear partial neutral function differential equation of the form

$$\begin{aligned} \frac{\partial Fu_t}{\partial t}= A(t)Fu_t +g(t,u_t), t\in (0,\infty );\;u_0=\phi . \end{aligned}$$

Here the operator-valued function $t\mapsto A(t)$ is T-periodic; the nonlinear delay operator $g(t,\psi )$ is T-periodic with respect to t and Lipschitz continuous with respect to $\psi$. In our strategy, on the one hand, we prove a Massera-type theorem for the linearized neutral equation, then pass to the semilinear equation using fixed-point arguments and Neumann series. On the other hand, our abstract results fit perfectly with the case that the family $(A(t))_{t\ge 0}$ generates an evolution family having exponential dichotomy. In such a case, we can apply our abstract results to show the existence, uniqueness, and conditional stability of periodic solution to the above equation.

1 Introduction

We are interested in the partial neutral functional differential equation

$$\begin{aligned} \frac{\partial }{\partial t}Fu_t=A(t)Fu_t +g(t,u_t),\quad t\in (0,+\infty );\; u_0=\phi \end{aligned}$$

(1.1)

where A(t) are a (possibly unbounded) linear operators on a Banach space X for each $t> 0$; $F : {\mathcal {C}} \rightarrow X$ is a bounded linear operator called a difference operator, where ${\mathcal {C}}:=C([-r, 0], X)$; $g : {\mathbb {R}}_+ \times {\mathcal {C}} \rightarrow X$ is a continuous nonlinear operator called a delay operator, and $u_t$ is the history function defined by $u_t(\theta ):=u(t+\theta )$ for $\theta \in [-r,0]$.

Partial neutral functional differential equations (PNFDE) arise from many applications. We refer to Wu [30], Wu and Xia [31] for numerous examples and applications of this type of equations to lossless transmission lines. The theory for PNFDE has then been developed by many other authors (see [1,2,3, 12, 13, 32] and references therein). In [12, 13], the qualitative behavior of solutions to autonomous PNFDE has been investigated yielding important results on stability, attractivity, and bifurcation of solutions around the steady state. One of the important research directions related to the asymptotic behavior of the solutions is to find conditions for the existence, uniqueness of a periodic solution to (1.1) and then to prove their (conditional) stability in case that A(t) and $g(t,\psi )$ are T-periodic functions with respect to t.

Generally, for functional differential equations, there are several methods used such as Massera’s principle [20, 21], special fixed-point methods, e.g., Sadovskii’s [5], Horn’s [14, 23], Chow and Hale’s [4, 7], or Hale and Lunel’s [4] fixed-point theorems. The most popular approaches used in this direction are the ultimate boundedness of solutions and the compactness of Poincaré map realized through some compact embeddings (see, e.g., [6, 22, 28, 29, 33]). However, in some concrete situations, e.g., in the case of partial differential equations in unbounded (in all directions) domains or equations that have unbounded solutions, such compact embeddings do not hold, and the choice of appropriate initial vectors (or conditions) to guarantee the boundedness of the corresponding solutions is not easy. In order to overcome such difficulties, one may invoke to the so-called Massera-type theorem roughly saying that if a differential equation has a bounded solution then it has a periodic one. Actually, in [17] this methodology has been invoked in combination with interpolation functors to show the existence and uniqueness of periodic solutions to fluid flows around rotating obstacles. In that work, the interpolation spaces have been used combined with ergodic methods (see [17]). Furthermore, a general method has been proposed in [10] combining interpolation functors with smoothness of semigroups and topological arguments to obtain the existence of periodic solutions to general fluid flow problems.

Recently, for the case of partial delay functional differential equations (i.e., the special case of (1.1) when $Fu_t=u(t)$), a Massera-type theorem has also been invoked to prove the existence an uniqueness of a periodic solution to delay equations (see [20]) when the family $(A(t))_{t\ge 0}$ generates an evolution family having an exponential dichotomy. In that case, we also obtained the conditional stability of such solutions.

In the present paper, we consider the existence and uniqueness of periodic solutions to non-autonomous partial neutral functional differential equations (PNFDE) with nonlinear term $g(t,\phi )$ being periodic with respect to t and Lipschitz continuous w.r.t. to $\phi$. The difficulties we face are twofold: On the one hand, we have to differentiate the functional $Fu_t$ instead of u(t), and on the other hand, the variation-of-constant formulae are available only for $Fu_t$. Therefore, one needs conditions on F to obtain u from $Fu_t$. To overcome such difficulties, we first write F in the form $F=\delta _0-(\delta _0-F)$ where $\delta _0$ is the Dirac distribution concentrated at 0. We next combine this representation with Massera’s method and the Neumann’s series. Then, under the “smallness” condition on $\Psi :=\delta _0-F$ we are able to prove the existence and uniqueness of the periodic solution to (1.1)). It is known that, using a renorming procedure, the smallness of $\Psi$ can be replaced by the condition that $\Psi$ has no mass in 0, and, in case that $\Psi$ is represented as an integral operator with a kernel $\eta$ of bounded variation, the property “having no mass in 0” of $\Phi$ is equivalent to the condition that $\eta$ is non-atomic at 0 (see the details in [18]).

It is worth emphasizing that our abstract results fit perfectly with the case when the family $(A(t))_{t\ge 0}$ generates an evolution family $(U(t,s))_{t\ge s\ge 0}$ having an exponential dichotomy (see Definition 4.1 below), since in this case we can choose the initial vector yielding a bounded solution. Moreover, we can also prove conditional stability of the periodic solution. Our main results are Theorems 2.6 and 3.1. The applications of our abstract results to semilinear PNFDE with the exponentially dichotomic linear part are given in Sect. 4 where we also prove the conditional stability of the periodic solution.

2 Periodic solutions to linear partial neutral functional differential equations

Given a function f taking values in a Banach space X, we consider the following linear problem for the unknown function u(t):

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial Fu_t}{\partial t}= A(t)Fu_t +f(t),\quad t\in (0,\infty ),\\ u_0=\phi \in {\mathcal {C}}:=C([-r, 0], X); \end{array}\right. } \end{aligned}$$

(2.1)

with the family of linear partial differential operators $(A(t))_{t\ge 0}$ such that the homogeneous Cauchy problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\mathrm{d}u}{\mathrm{d}t}=A(t)u(t) \hbox { for }t> s\ge 0 \\ u(s)=x\in X \end{array}\right. } \end{aligned}$$

(2.2)

is well-posed in the sense that there exists an evolution family $(U(t, s))_{t\ge s\ge 0}$ such that the solution of the Cauchy problem (2.2) is given by $u(t) = U(t, s)u(s)$. A detailed discussion of the well-posedness for (2.2) can be found in [26, 27]. Furthermore, for ${\mathcal {C}}:=C([-r, 0], X)$ we suppose $F : {\mathcal {C}} \rightarrow X$ to be a bounded linear operator called a difference operator; and $u_t$ is the history function defined by $u_t(\theta ):= u(t+\theta )$ for $\theta \in [-r,0]$. The notion of evolution families is explained in the following definition.

Definition 2.1

A family of bounded linear operators $(U(t, s))_{t\ge s\ge 0}$ on a Banach space X is a (strongly continuous, exponentially bounded) evolution family if

• (i) $U(t,t)=Id$ and $U(t,r)U(r,s)=U(t,s)$ for all $t \ge r\ge s\ge 0$,

• (ii) the map $(t,s)\mapsto U(t,s)x$ is continuous for every $x\in X$, where $(t,s)\in \left\{ (t,s)\in {{\mathbb {R}}}^2 : t\ge s\ge 0\right\} ,$

• (iii) there are constants $K, \alpha \ge 0$ such that $\Vert U(t,s)x\Vert \le Ke^{\alpha (t-s)}\Vert x\Vert$ for all $t\ge s\ge 0$ and $x\in X$.

The existence of the evolution family $(U(t, s))_{t\ge s\ge 0}$ allows us to define a mild solution to (2.1) as a function u(t) satisfying the integral equation

$$\begin{aligned} Fu_t= U(t,0)Fu_0+\int ^{t}_{0}U(t,\tau )f(\tau )\mathrm{d}\tau ,\; x \in X \hbox { for all }t\ge 0. \end{aligned}$$

(2.3)

From properties of the evolution family $(U(t, s))_{t\ge s\ge 0}$, we have

$$\begin{aligned} \begin{aligned} Fu_t&=U(t,s)U(s,0)Fu_0+\int ^{s}_{0}U(t,s)U(s,\tau )f(\tau )\mathrm{d}\tau +\int ^{t}_{s}U(t,\tau )f(\tau )\mathrm{d}\tau \\&=U(t,s)\left[ U(s,0)Fu_0+\int ^{s}_{0}U(s,\tau )f(\tau )\mathrm{d}\tau \right] +\int ^{t}_{s}U(t,\tau )f(\tau )\mathrm{d}\tau . \end{aligned} \end{aligned}$$

It implies

$$\begin{aligned} Fu_t=U(t,s)Fu_s +\int ^{t}_{s}U(t,\tau )f(\tau )\mathrm{d}\tau \hbox { for all }t\ge s\ge 0. \end{aligned}$$

(2.4)

Assumption 2.2

We suppose that the Banach space X possesses a separable pre-dual, i.e., $X = Y'$ where Y is a separable Banach space. We assume that A(t) is T-periodic, i.e., $A(t + T ) = A(t)$ for a fixed constant $T>0$ and all $t\in {\mathbb {R}}_+$. Then $(U(t, s))_{t\ge s\ge 0}$ becomes T -periodic in the sense that

$$\begin{aligned} U(t+T, s+T )= U(t, s) \text{ for } \text{ all } t\ge s\ge 0. \end{aligned}$$

(2.5)

We also assume that the space Y considered as a subspace of $Y''$ (through the canonical embedding) is invariant under the operator $U'(T,0)$ which is the dual of U(T, 0).

Furthermore, for the difference operator F, we suppose the following.

Assumption 2.3

Let the difference operator $F:{\mathcal {C}}\rightarrow X$ be of the form $F\phi =\phi (0)-\Psi \phi$ for all $\phi \in {\mathcal {C}}$ where $\Psi \in {\mathcal {L}}({\mathcal {C}}, X)$ satisfies $\Vert \Psi \Vert <1$.

Remark 2.4

Note that, if $\Psi$ has no mass in 0, i.e., for every $\epsilon >0$, there exists a positive number $\delta \le r$ such that

$$\begin{aligned} \Vert \Psi (\phi )\Vert \le \epsilon \Vert \phi \Vert \hbox { for all }\phi \in {\mathcal {C}} \hbox { satisfying supp}\phi \subseteq [-\delta , 0], \end{aligned}$$

then by renorming in the space ${\mathcal {C}}$ we can pass to an equivalent norm such that we have $\Vert \Psi \Vert <1$ (see the details on [18]). Note also that if $\Psi$ is represented in the form $\Psi (\phi )=\int _{-r}^0[d\eta (\tau )]\phi (\tau )$ for some function $\eta (\cdot )$ of bounded variation, then the property of “having no mass in 0” of $\Psi$ is equivalent to the fact that the function $\eta (\cdot )$ is non-atomic at 0 in the sense of Hale and Lunel [11, Chap. 9.2] or Wu [30, Chapt. 2.3].

To show the existence and uniqueness of the periodic mild solution to (2.1), we need the following space of bounded continuous functions with values in the Banach space X (with norm $\Vert \cdot \Vert$) defined as

$$\begin{aligned} C_b ({\mathbb {R}}_+,X):= \{v : {\mathbb {R}}_+\rightarrow X \mid v \text{ is } \text{ continuous } \text{ and } \sup _{t \in {\mathbb {R}}_+} \Vert v(t) \Vert <\infty \} \end{aligned}$$

(2.6)

endowed with the norm

$$\begin{aligned} \Vert v\Vert _{C_b}:=\sup _{t \in {\mathbb {R}}_+} \Vert v(t)\Vert . \end{aligned}$$

Lemma 2.5

For the Banach X and an evolution family $(U(t,s))_{t\ge s \ge 0}$ satisfying Assumption 2.2, we assume that the following condition holds true: For $f\in C_b({\mathbb {R}}_+,X)$ there exist $x_0\in X$ and constant $M>0$ such that

$$\begin{aligned} \sup _{t\ge 0}\left\| U(t,0)x_0 +\int ^{t}_{0}U(t,\tau )f(\tau )\mathrm{d}\tau \right\| \le M\Vert f \Vert _{{C_b({\mathbb {R}}_+,X)}}. \end{aligned}$$

(2.7)

Then, if f is T-periodic, there exists ${\hat{x}}\in X$ such that the function

$$\begin{aligned} w(t) = U(t,0){\hat{x}}+\int ^{t}_{0}U(t,\tau )f(\tau )\mathrm {d}\tau ~\mathrm {{for}~\mathrm {all}}~t\ge 0 \end{aligned}$$

is T-periodic and

$$\begin{aligned} \Vert w(t)\Vert \le (M+T)Ke^{\alpha T}\Vert f \Vert _{{C_b({\mathbb {R}}_+,X)}} ~\mathrm {{for}~\mathrm {all}}~t\ge 0, \end{aligned}$$

(2.8)

with constants K and $\alpha$ as in Definition 2.1.

Furthermore, if the evolution family $(U(t,s))_{t\ge s\ge 0}$ satisfies

$$\begin{aligned} \lim _{t\rightarrow \infty }\Vert U(t,0)x\Vert =0\quad \mathrm {{for}~\mathrm {all}}~ x\in X~ \mathrm {{such}~\mathrm {that}}~U(t,0)x~\mathrm {{is}~\mathrm {bounded}~\mathrm {in}}~ {\mathbb {R}}_+, \end{aligned}$$

(2.9)

then such an ${\hat{x}}$ is unique.

Proof

See [20, Theorem 2.3]. $\square$

Theorem 2.6

Suppose that the hypotheses of Lemma 2.5 are satisfied and let the difference operator F satisfy Assumption 2.3. Then, (2.3) has a T-periodic solution $\hat{u}(t)$ satisfying

$$\begin{aligned} \Vert \hat{u}(t)\Vert \le \frac{1}{1-\Vert \Psi \Vert }(M+T)Ke^{\alpha T}\Vert f \Vert _{C_b({\mathbb {R}}_+,X)}~\mathrm{{for}~\mathrm {all}}~t\ge 0. \end{aligned}$$

(2.10)

Furthermore, if the evolution family $U(t,s)_{t\ge s\ge 0}$ satisfies

$$\begin{aligned} \lim _{t\rightarrow \infty }\Vert U(t,0)x\Vert =0 \quad ~\mathrm{{for}~\mathrm {all}}~x\in X~\mathrm{{such}~\mathrm {that}}~ U(t,0)x~\mathrm{{is}~\mathrm {bounded}~\mathrm {in}}~{\mathbb {R}}_+, \end{aligned}$$

(2.11)

then the T-periodic solution ${\hat{u}}(t)$ is unique.

Proof

Let ${\hat{x}}$ be the initial vector obtained in Lemma 2.5 such that

$$\begin{aligned} w(t)=U(t,0)\hat{x}+\int ^t_0U(t,\tau )f(\tau )\mathrm{d}\tau \text { for } t \ge 0 \end{aligned}$$

is T-periodic. For this w(t), we define a mapping $\Omega : [-r,\infty )\rightarrow X$ by

$$\begin{aligned} \Omega (t):={\left\{ \begin{array}{ll} w(t) &{}\text { for } t \ge 0\\ {{\widetilde{w}}}(t) &{}\text { for } -r\le t < 0. \end{array}\right. } \end{aligned}$$

(2.12)

where ${{\widetilde{w}}}(t), -r\le t\le 0,$ is the T-periodic extension of w(t) on the interval $[-r,0).$ Since w(t) is T-periodic, it implies that $\Omega (t)$ is T-periodic. Clearly, $\Omega \in C_b([-r,+\infty ),X)$ and

$$\begin{aligned} \Vert \Omega \Vert _{C_b([-r,+\infty ),X)}\le (M+T)Ke^{\alpha T}\Vert f\Vert _{C_b({\mathbb {R}}_+,X)}. \end{aligned}$$

(2.13)

We next define the operator ${\tilde{\Psi }}: C_b([-r,+\infty ),X) \rightarrow C_b([-r,+\infty ),X)$ by

$$\begin{aligned}{}[{{{\widetilde{\Psi }}}} u](t):= {\left\{ \begin{array}{ll} \Psi (u_t) &{} \text{ for } t \ge 0 \\ \Psi (u_0) &{} \text{ for } -r \le t < 0 \end{array}\right. } \end{aligned}$$

(2.14)

Since $\Vert \Psi \Vert <1$ we have $\Vert \widetilde{\Psi }\Vert \le \Vert \Psi \Vert <1$. Therefore, the operator $I-{{\tilde{\Psi }}}$ is invertible.

We now put

$$\begin{aligned} \hat{u}(t)=\left[ {(I-{\widetilde{\Psi }})}^{-1}\Omega \right] (t). \end{aligned}$$

(2.15)

Since F satisfies Assumption 2.3, we obtain that

$$\begin{aligned} F\hat{u}_t =\hat{u}(t)-\Psi (\hat{u}_t) =(I \hat{u})(t)-\left[ {{\widetilde{\Psi }}} \hat{u}\right] (t) =\left[ \left( I-{\widetilde{\Psi }} \right) \hat{u}\right] (t)\hbox { for all }t\ge 0. \end{aligned}$$

(2.16)

Combining (2.15) and (2.16) we have

$$\begin{aligned} F\hat{u}_t=\left[ \left( I-{\widetilde{\Psi }} \right) \left( I-\widetilde{\Psi }\right) ^{-1}\Omega \right] (t) =\Omega (t) \text { for } t \ge 0. \end{aligned}$$

(2.17)

It implies that

$$\begin{aligned} F{\hat{u}}_t= U(t,0){\hat{x}}+\int ^{t}_{0}U(t,\tau )f(\tau )\mathrm{d}\tau ~\mathrm{{for}~\mathrm {all}}~t\ge 0. \end{aligned}$$

(2.18)

Clearly, letting $t=0$ in the above equation, we arrive at $F{\hat{u}}_0={\hat{x}}$. Therefore, ${\hat{u}}(t)$ is a solution to (2.3).

Next, we prove that $\hat{u}(t)$ is T-periodic. Indeed, by using the Neumann’s series, we have

$$\begin{aligned} \hat{u}(T)=\left[ {(I-{\widetilde{\Psi }})}^{-1}\Omega \right] (T) = \left[ \left( \sum _{n=0}^\infty \widetilde{\Psi }^n\right) \Omega \right] (T) =\sum _{n=0}^\infty \left( \widetilde{\Psi }^n\Omega \right) (T). \end{aligned}$$

Since $\Omega (t)$ is T-periodic, it follows that $({\widetilde{\Psi }}\Omega )(T)=\Psi (\Omega _T)=\Psi (\Omega _0) =({\widetilde{\Psi }}\Omega )(0)$. By induction, one can easily see that $\left( \widetilde{\Psi }^n\Omega \right) (T)= \left( {\widetilde{\Psi }}^n\Omega \right) (0)$ for all $n\in {\mathbb {N}}$. Therefore,

$$\begin{aligned}&\hat{u}(T) =\sum _{n=0}^\infty \left( {\widetilde{\Psi }}^n\Omega \right) (0) = \left( \sum _{n=0}^\infty {\widetilde{\Psi }}^n\Omega \right) (0) = \left( \left( \sum _{n=0}^\infty \widetilde{\Psi }^n\right) \Omega \right) (0) \\&\quad = \left( \left( I-\widetilde{\Psi }\right) ^{-1}\Omega \right) (0) =\hat{u}(0) \end{aligned}$$

It implies that $\hat{u}(t)$ is T-periodic.

Next, we will estimate $\Vert \hat{u}(t)\Vert$. In fact, we have

$$\begin{aligned} \hat{u}(t) =\left( \left( I-\widetilde{\Psi }\right) ^{-1}\Omega \right) (t) = \left( \left( \sum _{n=0}^\infty {\widetilde{\Psi }}^n\right) \Omega \right) (t). \end{aligned}$$

Since $\Vert \widetilde{\Psi }\Vert <1$ and using estimate (2.13) we obtain that

$$\begin{aligned} \Vert \hat{u}(t)\Vert\le & {} \frac{1}{1-\Vert \widetilde{\Psi } \Vert }\left\| \Omega \right\| _{C_b ([-r,\infty ),X)} \\\le & {} \frac{1}{1-\Vert \widetilde{\Psi } \Vert } (M+T)Ke^{\alpha T}\Vert f \Vert _{C_b ({\mathbb {R}}_+,X)} \hbox { for all }t\ge 0. \end{aligned}$$

From $\Vert \widetilde{\Psi }\Vert \le \Vert \Psi \Vert$, it follows that

$$\begin{aligned} \Vert \hat{u}(t)\Vert < \frac{1}{1-\Vert \Psi \Vert }(M+T)Ke^{\alpha T} \Vert f \Vert _{C_b ({\mathbb {R}}_+,X)}. \end{aligned}$$

(2.19)

We now prove that if the evolution family $(U(t,s))_{t\ge s\ge 0}$ satisfies (2.11), then the T-periodic mild solution is unique. Indeed, let $\hat{u}_1(t)$ and $\hat{u}_2(t)$ be two T-periodic solutions to (2.3). By periodically extending to $[-r,0)$ we can consider $\hat{u}_1(t)$ and $\hat{u}_2(t)$ as T-periodic functions on $[-r,\infty )$. Put now $\Omega _1=(I-{\widetilde{\Psi }})\hat{u}_1$ and $\Omega _2=(I-\widetilde{\Psi })\hat{u}_2$. Then, obviously

$$\begin{aligned} \Omega _1(t)= & {} {\left\{ \begin{array}{ll} v_1(t)=U(t,0)\hat{x}_1+\int ^t_0U(t,\tau )f(\tau )\mathrm{d}\tau &{}\text { for } t \ge 0\\ {{\widetilde{v}}}_1(t) &{}\text { for } -r\le t< 0. \end{array}\right. }\\ \Omega _2(t)= & {} {\left\{ \begin{array}{ll} v_2(t)=U(t,0)\hat{x}_2+\int ^t_0U(t,\tau )f(\tau )\mathrm{d}\tau &{}\text { for } t \ge 0\\ {{\widetilde{v}}}_2(t) &{}\text { for } -r\le t < 0. \end{array}\right. } \end{aligned}$$

where ${{\widetilde{v}}}_1$ and ${{\widetilde{v}}}_2$ are T-periodic extensions to $[-r, 0)$ of $v_1$ and $v_2$, respectively.

Then, putting $\Phi =\Omega _1-\Omega _2$ we have that $\Phi$ is T-periodic and

$$\begin{aligned} \Phi (t)=U(t,0)({\hat{x}}_1-{\hat{x}}_2)~\mathrm{{for}~\mathrm {all}}~t\ge 0. \end{aligned}$$

Since $\Phi (\cdot )$ is bounded on ${\mathbb {R}}_+$, inequality (2.11) then implies that

$$\begin{aligned} \lim _{t\rightarrow \infty }\Vert \Phi (t)\Vert =0. \end{aligned}$$

(2.20)

This fact, together with the periodicity of $\Phi$, implies that $\Phi (t) = 0$ for all $t\ge 0$. This yields $\Omega _1=\Omega _2$ and therefore $\hat{u}_1=(I-{\widetilde{\Psi }})^{-1}\Omega _1 =(I-{\widetilde{\Psi }})^{-1}\Omega _2= \hat{u}_2$. $\square$

3 Periodic solutions to semilinear partial neutral functional differential equations

For a Banach space X with a separable pre-dual Y as in the previous section, we now consider the following partial neutral functional differential equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial Fu_t}{\partial t}=A(t)Fu_t+g(t,u_t), t> 0, \\ u_0 =\phi \in {\mathcal {C}}:=C([-r, 0], X); \end{array}\right. } \end{aligned}$$

(3.1)

where the linear operators A(t), $t\ge 0$, act on X and satisfy the hypotheses of Lemma 2.5, and the nonlinear term $g:[0,\infty )\times {\mathcal {C}}\rightarrow X$ satisfies:

$$\begin{aligned} \begin{aligned} (1)&\ \Vert g(t, 0)\Vert \le \gamma \text{ where } \gamma \text{ is } \text{ a } \text{ nonnegative } \text{ constant, } \\ (2)&\text{ the } \text{ function } g(t,v) \text{ is } T\hbox {-periodic with respect to }t \text{ for } \text{ each } v \in {\mathcal {C}},\\ (3)&\text{ there } \text{ exist } \text{ positive } \text{ constants } \rho \text{ and } L \text{ such } \text{ that }\\ {}&\left\| g(t, v_1)-g(t,v_2)\right\| \le L\Vert v_1-v_2\Vert _{\mathcal {C}} \text{ for } \text{ all } v_1, v_2\in {\mathcal {C}} \text{ with } \Vert v_1\Vert _{\mathcal {C}}, \Vert v_2\Vert _{\mathcal {C}}\le \rho . \end{aligned} \end{aligned}$$

(3.2)

Furthermore, by a mild solution to (3.1), we mean the function u satisfying the following equation

$$\begin{aligned} Fu_t=U(t,0)Fu_0+\int ^{t}_{0}U(t,\tau )g(\tau ,u_{\tau })\mathrm{d}\tau \hbox { for all } t\ge 0. \end{aligned}$$

(3.3)

We refer to Wu [30] for the detailed discussion on mild solutions to partial functional equations and their relation to strong and classical solutions. We then come to our next result on the existence and uniqueness of the periodic mild solution to (3.1).

Theorem 3.1

Let the Banach X and evolution family $(U(t,s))_{t\ge s \ge 0}$ satisfy Assumption 2.2 and let the condition (2.7) in Lemma 2.5 hold true. Suppose that F satisfies Assumption 2.3. Suppose further that the evolution family $U(t,s)_{t\ge s\ge 0}$ satisfies:

$$\begin{aligned} \lim _{t\rightarrow \infty }\Vert U(t,0)x\Vert =0\quad \text { for all } x\in X \text{ such } \text{ that } U(t,0)x \text{ is } \text{ bounded } \text{ in } {\mathbb {R}}_+ \end{aligned}$$

and let g satisfy the conditions in (3.2).

Then, for sufficiently small L, equation (3.1) has a unique T-periodic mild solution $\hat{u}$ on a small ball of $C_b([-r,\infty ),X)$ satisfying

$$\begin{aligned} \Vert \hat{u}\Vert _{C_b([-r,\infty ),X)} \le \dfrac{1}{1-\Vert \Psi \Vert } (M+T)Ke^{\alpha T}(L\rho +\gamma ). \end{aligned}$$

Proof

Consider the following closed set ${\mathcal {B}}^T_{\rho }\subset {C_b ([-r,\infty ),X)}$ defined by

$$\begin{aligned} {\mathcal {B}}^T_{\rho } :=\left\{ v \in C_b([-r,\infty ),X) : \ v \text{ is } T\hbox {-periodic and }\quad \Vert v\Vert _{C_b([-r,\infty ), X)}\le \rho \right\} . \end{aligned}$$

(3.4)

Given $v\in C_b([-r,\infty ),X)$, we consider the equation for u:

$$\begin{aligned} Fu_t=U(t,0)Fu_0+\int ^{t}_{0}U(t,\tau )g(\tau ,v_{\tau })\mathrm{d}\tau \hbox { for all } t\ge 0. \end{aligned}$$

(3.5)

Then, we define the transformation $\Phi$ as follows: For $v\in {{\mathcal {B}}}^T_{\rho }$ we set

$$\begin{aligned} \Phi (v)(t):= {\left\{ \begin{array}{ll} u(t) \hbox { for } t\ge 0 \\ {\tilde{u}}(t) \hbox { for } - r \le t < 0. \end{array}\right. } \end{aligned}$$

(3.6)

where $u(t) \in C_b({\mathbb {R}}_+,X)$ is unique T-periodic solution to (3.5) satisfying

$$\begin{aligned} \Vert u(t)\Vert \le \frac{1}{1-\Vert \Psi \Vert }(M+T)Ke^{\alpha T}\sup _{\tau \ge 0}\Vert g(\tau ,v_{\tau })\Vert \hbox { for all }t\ge 0, \end{aligned}$$

(3.7)

and ${\tilde{u}}(t)$ is the T-periodic extension of u(t) on the interval $[-r,0)$. Note that the existence and uniqueness of such a u satisfying (3.7) is guaranteed by Theorem 2.6.

Next, we prove that for sufficiently small L and $\gamma$, the transformation $\Phi$ acts from ${{\mathcal {B}}}^T_{\rho }$ into itself. Indeed, taking any $v\in B^T_{\rho }$, by the properties of g given in (3.2) we have

$$\begin{aligned} \begin{aligned} \Vert g(t,v_t)\Vert&\le \Vert g(t,v_t)-g(t,0)\Vert +\Vert g(t,0)\Vert \le L\Vert v_t\Vert _{\mathcal {C}}+\gamma \\ {}&\quad \le L\Vert v\Vert _{C_b([-r,\infty ),X)} +\gamma \le L\rho +\gamma \end{aligned} \end{aligned}$$

(3.8)

for all $t\ge 0$. Therefore,

$$\begin{aligned} \Vert u(t)\Vert \le \frac{1}{1-\Vert \Psi \Vert }(M+T)Ke^{\alpha T}(L\rho +\gamma ) \hbox { for all } t\ge 0. \end{aligned}$$

(3.9)

Thus, for $v\in {\mathcal {B}}^T_{\rho }$, by above definition of $\Phi$ and from (3.9) we have

$$\begin{aligned} \Vert \Phi (v)\Vert _{C_b([-r,\infty ),X)}\le \frac{1}{1-\Vert \Psi \Vert } (M+T)Ke^{\alpha T}(L\rho +\gamma ). \end{aligned}$$

(3.10)

Therefore, we obtain that for sufficiently small L and $\gamma$, the map $\Phi$ acts from ${\mathcal {B}}^T_{\rho }$ into itself.

Next, we prove that for sufficiently small L and $\gamma$, the map $\Phi : {{\mathcal {B}}}^T_{\rho }\rightarrow {{\mathcal {B}}}^T_{\rho }$ is a contraction. To do this, let $v_1, v_2 \in {\mathcal {B}}^T_{\rho }$, put $w_1:=\Phi (v_1)$, $w_2:=\Phi (v_2)$ and set $w=w_1-w_2$. Then, by definition of $\Phi$, we have that w is the unique T-periodic solution to the equation

$$\begin{aligned} Fw_{t}=U(t,0)Fw_0+\int ^t_0U(t,\tau )(g(\tau ,(v_1)_\tau ) -g(\tau ,(v_2)_\tau ))\mathrm {d}\tau \quad \text{ for } t \ge 0. \end{aligned}$$

Theorem 2.6 now implies that

$$\begin{aligned}&\Vert \Phi (v_1)-\Phi (v_2)\Vert _{C_b ([-r,\infty ),X)}\\&\quad =\Vert w_1-w_2\Vert _{C_b ([-r,\infty ),X)}=\Vert w\Vert _{C_b ([-r,\infty ),X)}\\&\quad \le \dfrac{1}{1-\Vert \Psi \Vert }(M+T)Ke^{\alpha T}\sup \limits _{\tau \ge 0}\Vert g(\tau ,(v_1)_\tau )-g(\tau ,(v_2)_\tau )\Vert . \end{aligned}$$

Hence, from (3.2), it follows that

$$\begin{aligned} \Vert \Phi (v_1)-\Phi (v_2)\Vert _{C_b ([-r,\infty ),X)} \le \dfrac{(M+T)Ke^{\alpha T}L}{1-\Vert \Psi \Vert } \Vert v_1- v_2\Vert _{C_b ([-r,\infty ),X)}. \end{aligned}$$

We thus obtain that for sufficiently small L and $\gamma$, the operator $\Phi : {{\mathcal {B}}}^T_{\rho } \rightarrow {\mathcal B}^T_{\rho }$ is a contraction. Therefore, there exists a unique fixed point $\hat{u}$ in ${\mathcal {B}}^T_{\rho }$ of $\Phi$, and by definition of $\Phi$, this function $\hat{u}$ is the unique T-periodic mild solution to (3.1) and

$$\begin{aligned} \Vert \hat{u}\Vert _{C_b([-r,\infty ),X)} \le \dfrac{1}{1-\Vert \Psi \Vert } (M+T)Ke^{\alpha T}(L\rho +\gamma ). \end{aligned}$$

$\square$

4 Periodic solutions in the case of exponential dichotomy

In this section, we study the case that the evolution family $(U(t,s))_{t\ge s\ge 0}$ has an exponential dichotomy. In this case, the existence of bounded solutions to (2.3) can be conveniently proved. Therefore, the existence and uniqueness of periodic solutions to (2.3) and hence to (3.3) easily follow. Moreover, using the Gronwall-type inequalities, we will show the conditional stability of such periodic solutions. We first recall the notions of exponential dichotomy and stability of an evolution family.

Definition 4.1

Let ${\mathcal {U}}:=(U(t,s))_{t\ge s\ge 0}$ be an evolution family on Banach space X.

• (1) The evolution family ${\mathcal {U}}$ is said to have an exponential dichotomy on $[0,\infty )$ if there exist bounded linear projections $P(t),\,t\ge 0$, on X and positive constants $N,\, \nu$ such that

  • (a) $U(t,s)P(s)=P(t)U(t,s),\quad t\ge s\ge 0$,

  • (b) the restriction $U(t,s)_|:\text {Ker}P(s)\rightarrow \text {Ker}P(t),\,t\ge s\ge 0$, is an isomorphism, and we denote its inverse by $U(s,t)_|:=(U(t,s)_|)^{-1}$, $0\le s\le t$,

  • (c) $\Vert U(t,s)x\Vert \le Ne^{-\nu (t-s)}\Vert x\Vert$ for $x\in P(s)X,\,t\ge s\ge 0$,

  • (d) $\Vert U(s,t)_|x\Vert \le Ne^{-\nu (t-s)}\Vert x\Vert$ for $x\in \text {Ker}P(t),\,t\ge s\ge 0$.

  The projections $P(t),\,t\ge 0$, and the constants $N,\,\nu$ are called the dichotomy projections and dichotomy constants, respectively.

• (2) The evolution family ${\mathcal {U}}$ is called exponentially stable if it has an exponential dichotomy with the dichotomy projections $P(t)=Id$ for all $t\ge 0$. In other words, ${\mathcal {U}}$ is exponentially stable if there exist positive constants N and $\nu$ such that

  $$\begin{aligned} \Vert U(t,s)\Vert \le Ne^{-\nu (t-s)}\hbox { for all }t\ge s\ge 0. \end{aligned}$$

  (4.1)

We remark that properties (a)–(d) of dichotomy projections P(t) already imply that

• i) $H:=\sup _{t\ge 0}\Vert P(t)\Vert <\infty$,

• ii) $t\mapsto P(t)$ is strongly continuous

(see [25, Lemm. 4.2]). We refer the reader to [15] for characterizations of exponential dichotomies of evolution families in general admissible spaces.

In case $(U(t,s))_{t\ge s\ge 0}$ has an exponential dichotomy with dichotomy projections $(P(t))_{t\ge 0}$ and constants $N, \nu \ >0$, we can then define the Green’s function on a half-line as follows:

$$\begin{aligned} {\mathcal {G}}(t,\tau ) :={\left\{ \begin{array}{ll} P(t)U(t,\tau ) &{}\hbox { for } t> \tau \ge 0,\\ -U(t,\tau )_\mid (I-P(\tau )) &{}\hbox { for } 0\le t< \tau . \end{array}\right. } \end{aligned}$$

(4.2)

Then ${\mathcal {G}}(t,\tau )$ satisfies the estimate

$$\begin{aligned} \Vert {\mathcal {G}}(t,\tau )\Vert \le (1+H)Ne^{-\nu |t-\tau |}\hbox { for }t\not =\tau ,\ t,\tau \ge 0. \end{aligned}$$

(4.3)

The following lemma gives the form of bounded solutions to (2.3) and (3.3).

Lemma 4.2

Let the evolution family $(U(t,s))_{t\ge s\ge 0}$ have an exponential dichotomy with the corresponding dichotomy projections $(P(t))_{t\ge 0}$ and dichotomy constants $N,\nu >0$. Let $f\in C_b({\mathbb {R}}_+,X)$, and g satisfy conditions given in (3.2). Then, the following assertions hold true.

• (a) Let $w\in C_b({\mathbb {R}}_+,X)$ be given by

  $$\begin{aligned} w(t)=U(t,0)w(0) +\int ^{t}_{0}U(t,\tau )f(\tau )d\tau . \end{aligned}$$

  (4.4)

  Then w satisfies

  $$\begin{aligned} \qquad w(t)=U(t,0)\zeta +\int _{0}^\infty {\mathcal {G}}(t,\tau )f(\tau )\mathrm{d}\tau \hbox { for some }\zeta \in X_0:=P(0)X,\ t\ge 0, \end{aligned}$$

  (4.5)

  where ${\mathcal {G}}(t,\tau )$ is the Green’s function defined by (4.2).

• (b) Let $u\in C_b([-r,\infty ),X)$ be a solution to (3.3) with $\sup _{t\ge -r}\Vert u(t)\Vert \le \rho$ for a fixed $\rho > 0$. Then, u satisfies

  $$\begin{aligned} {\left\{ \begin{array}{ll} Fu_t=U(t,0)\eta +\int _{0}^\infty {\mathcal {G}}(t,\tau )g(\tau ,u_{\tau }))\mathrm{d}\tau \hbox { for }t\ge 0,\\ \; \ u_0=\phi \in {\mathcal {C}} \end{array}\right. } \end{aligned}$$

  (4.6)

  for some $\eta =P(0)F\phi \in X_0,$ where ${\mathcal {G}}$ and $X_0$ are determined as in (a).

Proof

See [20, Lemma 4.2]. $\square$

Remark 4.3

By straightforward computations, we can prove that the converses of statements (a) and (b) are also true, i.e., we can prove the following:

• (1) A solution to (4.5) also satisfies (4.4) for $t\ge 0$.

• (2) A solution to (4.6) satisfies (3.3) for $t\ge 0$.

We next prove the existence of bounded solutions to (2.3) and (3.3) (i.e., bounded mild solutions to (2.1) and (3.1)) and hence that of periodic solutions in the following theorem.

Theorem 4.4

Let Assumptions 2.2 and 2.3 be satisfied and let the evolution family $(U(t,s))_{t\ge s\ge 0}$ have an exponential dichotomy with the dichotomy projections P(t), $t\ge 0$, and constants $N,\nu .$ Let $f\in C_b ({\mathbb {R}}_+,X)$ be T-periodic and suppose that g satisfies the conditions in (3.2) with given positive constants $\rho$, L, $\gamma$. Then, the following assertions hold true.

• (a) Equation (2.3) has a unique T-periodic solution $\hat{u}(t)$ in $C_b ({\mathbb {R}}_+,X)$.

• (b) If $L, \gamma$ is sufficiently small, then (3.3) has a unique T-periodic solution $\hat{u}(t)$ in $C_b([-r,\infty ),X)$.

Proof

• (a) For a given $f \in C_b ({\mathbb {R}}_+,X)$, take $\zeta =0\in X_0$ in (4.5). Then, for $w(t)=\int _{0}^\infty {\mathcal {G}}(t,\tau )f(\tau )d\tau$, by Lemma 4.2 and Remark 4.3 we have that $w(t)=U(t,0)w(0) +\int ^{t}_{0}U(t,\tau )f(\tau )\mathrm{d}\tau$, and w belongs to $C_b ({\mathbb {R}}_+,X)$. Furthermore, using inequality (4.3), we have

  $$\begin{aligned} \begin{aligned} \qquad \Vert w(t)\Vert&\le (1+H)N\int _0^\infty e^{-\nu |t-\tau |}\Vert f(\tau )\Vert \mathrm {d}\tau \\ {}&\le (1{+}H)N\left[ \int _0^t e^{-\nu |t-\tau |}\mathrm {d}\tau +\int _t^\infty e^{-\nu |\tau -t|}\mathrm {d}\tau \right] \sup _{\tau \ge 0}\Vert f(\tau )\Vert \nonumber \text{ for } \text{ all } t{\ge } 0. \end{aligned} \end{aligned}$$

  Thus,

  $$\begin{aligned} \sup _{t\ge 0}\Vert w(t)\Vert \le \frac{2(1+H)N}{\nu }\Vert f\Vert _{C_b({\mathbb {R}}_+,X)}. \end{aligned}$$

  Moreover, for $x\in X$ such that $\sup _{t\ge 0}\Vert U(t,0)x\Vert <\infty$, the exponential dichotomy of $(U(t,s))_{t\ge s}$ yields that $x\in P(0)X$. Therefore, $\Vert U(t,0)x\Vert \le Ne^{-\nu t}\Vert x\Vert \longrightarrow 0$ as $t\rightarrow \infty$.

  Thus, applying Theorem 2.6, we obtain that for the T-periodic function $f\in C_b ({\mathbb {R}}_+,X)$ there exists a unique T-periodic solution $\hat{u}$ of (2.3) satisfying

  $$\begin{aligned} \Vert \hat{u}\Vert _{C_b({\mathbb {R}}_+,X)}\le \frac{1}{1-\Vert \Psi \Vert }\left( \frac{2(1+H)N}{\nu }+T\right) Ke^{\alpha T}\Vert f \Vert _{C_b({\mathbb {R}}_+,X)}. \end{aligned}$$

  (4.7)

• (b) By Assertion (a), for each T-periodic input function f, the linear problem (2.1) has a unique T-periodic solution $\hat{u}\in C_b({\mathbb {R}}_+,X)$ satisfying inequality (4.7). Therefore, Assertion (b) then follows from Theorem 3.1.

$\square$

We now prove the conditional stability of periodic solutions to (3.3). To do this, for $x\in X, {\hat{\phi }}\in {\mathcal {C}},$$\hat{v}\in C_b([-r,\infty ),X)$ and a real number $a>0$, we denote by

$$\begin{aligned} \begin{aligned} B_a(x)&:=\{y\in X:\Vert x-y\Vert \le a \},\\{\mathbb {B}}_a({\hat{\phi }})&:=\{\phi \in {\mathcal {C}}:\Vert \phi -{\hat{\phi }}\Vert _{\mathcal {C}}\le a\},\\{\mathcal {B}}_a(\hat{v})&:=\{v\in C_b([-r,\infty ),X):\Vert v-\hat{v}\Vert _{C_b}\le a\}. \end{aligned} \end{aligned}$$

Let ${\mathcal {B}}_\rho (0)$ be the ball containing $\hat{u}$ as in Assertion (b) of Theorem 4.4. Suppose further that there exists a positive constant $L_1$ such that:

$$\begin{aligned} \left\| g(t,\phi _1)-g(t,\phi _2)\right\| \le L_1\Vert \phi _1-\phi _2\Vert _{{\mathcal {C}}} \text{ for } \text{ all } \phi _1, \phi _2\in {\mathbb {B}}_{2\rho }(0)\quad \text{ and } \text{ all } t\ge 0. \end{aligned}$$

(4.8)

Theorem 4.5

Let the hypotheses of Theorem 4.4 be satisfied. Let $\hat{u}$ be the T-periodic solution of (3.3) obtained in Assertion (b) of Theorem 4.4. Let g satisfy the conditions given in (3.2) and (4.8). Then, if $L_1$ is small enough, to each $\phi \in {\mathcal {C}}$ with $\Vert F\phi -F\hat{u}_0\Vert \le \rho /2$ and $P(0)F\phi \in B_{\frac{\rho }{2N}}(P(0)F\hat{u}_0)\cap P(0)X$ there corresponds one and only one solution $u(\cdot )$ of (3.3) on $[-r,\infty )$ satisfying the conditions $u_0=\phi$ and $u \in {\mathcal {B}}_\rho (\hat{u})$. Moreover, the following estimate is valid for u(t) and $\hat{u}(t)$:

$$\begin{aligned} \Vert u_t- \hat{u}_t\Vert _{\mathcal {C}} \le Ce^{-\mu t}\Vert P(0)Fu_0-P(0)F{\hat{u}}_0\Vert , \end{aligned}$$

(4.9)

for some positive constants C and $\mu$ independent of u, $\hat{u}$ and $\rho$.

Proof

Putting $w=u-\hat{u}$ with u in ${\mathcal {B}}_\rho (\hat{u})$ then w in ${\mathcal {B}}_\rho (0);$ and $\tilde{g}(t,w_t)=g(t,w_t+\hat{u}_t)-g(t,\hat{u}_t)$, we obtain that $\tilde{g}(t,0) = 0$ and $\Vert {{\tilde{g}}}(t,w_t)\Vert \le L_1\Vert w_t\Vert _{\mathcal {C}}.$

We define the operator ${\tilde{\Psi }}: C([-r,+\infty ),X) \rightarrow C([-r,+\infty ),X)$ by

$$\begin{aligned}{}[{{\widetilde{\Psi }}} v](t)={\left\{ \begin{array}{ll} \Psi (v_t)&{} \text{ for } t \ge 0\\ \Psi (v_0)&{} \text{ for } -r \le t \le 0 \end{array}\right. } \end{aligned}$$

Since $\Vert \Psi \Vert <1$ we have $\Vert {\tilde{\Psi }}\Vert \le \Vert \Psi \Vert <1$. Therefore, the operator $(I-{{\tilde{\Psi }}})$ is invertible.

Setting $\xi =P(0)F\phi -P(0)F{\hat{u}}_0$ then we have $\Vert \xi \Vert =\Vert P(0)F\phi -P(0)F\hat{u}_0\Vert \le \frac{\rho }{2N}.$

We define a mapping ${{\widetilde{F}}}_\xi : C_b([-r,\infty ),X) \rightarrow C_b([-r,\infty ),X)$ by

$$\begin{aligned} ({{\widetilde{F}}}_\xi w)(t)= {\left\{ \begin{array}{ll} U(t, 0)\xi +\displaystyle \int _0^{\infty } {\mathcal {G}}(t,\tau )\tilde{g}(\tau ,w_{\tau })\mathrm{d}\tau &{} \text { for } t \ge 0\\ U(-t, 0)\xi +\displaystyle \int _0 ^{\infty } {\mathcal {G}}(-t ,\tau )\tilde{g}(\tau ,w_{\tau })\mathrm{d}\tau &{} \text { for } -r\le t \le 0. \end{array}\right. } \end{aligned}$$

(4.10)

For $t \ge 0$, we then estimate

$$\begin{aligned} \begin{aligned} \Vert ({{\widetilde{F}}}_\xi w)(t)\Vert&\le \Vert U(t, 0)\xi \Vert +\displaystyle \int _0^{\infty } \Vert {\mathcal {G}}(t,\tau ){{\tilde{g}}}(\tau ,w_{\tau })\Vert \mathrm{d}\tau \\&\le Ne^{-\nu t}\Vert \xi \Vert +(1+H)NL_1\displaystyle \int _0^\infty e^{-\nu |t-\tau |}\Vert w_{\tau }\Vert _{\mathcal {C}}\mathrm{d}\tau \\&\le Ne^{-\nu t}\Vert \xi \Vert +(1+H)NL_1\sup _{s\ge -r}\Vert w(s)\Vert \displaystyle \int _0^\infty e^{-\nu |t-\tau |}\mathrm{d}\tau \\&\le Ne^{-\nu t}\frac{\rho }{2N}+\frac{2(1+H)NL_1\rho }{\nu }\\&\le \frac{\rho }{2}+\frac{2(1+H)NL_1\rho }{\nu }. \end{aligned} \end{aligned}$$

Similarly, for $-r\le t <0$, we then

$$\begin{aligned} \begin{aligned} \Vert ({{\widetilde{F}}}_\xi w)(t)\Vert&\le \Vert U(-t, 0)\xi \Vert +\displaystyle \int _0^{\infty } \Vert {\mathcal {G}}(-t,\tau ){{\tilde{g}}}(\tau ,w_{\tau })\Vert \mathrm{d}\tau \\&\le Ne^{\nu t}\frac{\rho }{2N}+(1+H)NL_1\displaystyle \int _0^\infty e^{-\nu |-t-\tau |}\Vert w_{\tau }\Vert _{\mathcal {C}}\mathrm{d}\tau \\&\le \frac{\rho }{2}+\frac{2(1+H)NL_1\rho }{\nu }. \end{aligned} \end{aligned}$$

Therefore

$$\begin{aligned} \sup _{t\ge -r}\Vert ({{\widetilde{F}}}_\xi w)(t)\Vert \le \frac{\rho }{2}+ \frac{2(1+H)NL_1\rho }{\nu }. \end{aligned}$$

Moreover, for $x(\cdot ), y(\cdot )$ in ${\mathcal {B}}_\rho (0),$ we have

$$\begin{aligned} \begin{aligned} \Vert {{\tilde{g}}}(t,x_t)-{{\tilde{g}}}(t,y_t)\Vert&=\Vert g(t,x_t+{\hat{u}}_t)- g(t,{\hat{u}}_t)- g(t,y_t+{\hat{u}}_t)+ g(t,{\hat{u}}_t)\Vert \\&=\Vert g(t,x_t+{\hat{u}}_t)- g(t,y_t+{\hat{u}}_t)\Vert \\&\le L_1\Vert x_t-y_t\Vert _{\mathcal {C}}. \end{aligned} \end{aligned}$$

Therefore

$$\begin{aligned} \begin{aligned} \Vert ({{\widetilde{F}}}_\xi x)(t)-({{\widetilde{F}}}_\xi y)(t)\Vert&\le \displaystyle \int _0^{\infty }\Vert {\mathcal {G}}(t,\tau )\Vert \Vert {{\tilde{g}}}(\tau ,x_{\tau })-{{\tilde{g}}}(\tau ,x_{\tau })\Vert \mathrm{d}\tau \\&\le \frac{2(1+H)NL_1}{\nu } \sup _{s\ge -r}\Vert x(s)-y(s)\Vert \end{aligned} \end{aligned}$$

for all $t\ge -r$. Hence,

$$\begin{aligned} \sup _{t\ge -r} \Vert ({{\widetilde{F}}}_\xi x)(t)-({{\widetilde{F}}}_\xi y)(t)\Vert \le \frac{2(1+H)NL_1}{\nu } \sup _{t\ge -r}\Vert x(t)-y(t)\Vert . \end{aligned}$$

We now put $H:=(I-\widetilde{\Psi })^{-1} {{\widetilde{F}}}_{\xi }$. Using the Neumann’s series, we prove that H acts from ${\mathcal {B}}_\rho (0)$ into ${\mathcal {B}}_\rho (0)$ and is a contraction. In fact, for $t> -r$ we have

$$\begin{aligned} \begin{aligned} \Vert (Hw)(t)\Vert&=\left\| \left( (I-\widetilde{\Psi })^{-1} \widetilde{F}_{\xi }w\right) (t) \right\| =\left\| \left( \left( \sum ^{\infty }_{n=0}\widetilde{\Psi }^{n} \right) \widetilde{F}_{\xi }w\right) (t)\right\| \\&= \left\| \sum ^{\infty }_{n=0}\left( \widetilde{\Psi }^{n} {{\widetilde{F}}}_{\xi }w\right) (t)\right\| \\&\le \sum ^{\infty }_{n=0} \left\| \left( \widetilde{\Psi }^{n} {{\widetilde{F}}}_{\xi }w\right) (t)\right\| \le \sum ^{\infty }_{n=0} \Vert \widetilde{\Psi }\Vert ^{n}\left\| \left( {{\widetilde{F}}}_{\xi }w\right) (t)\right\| \\&\le \frac{1}{1-\Vert \Psi \Vert } \left( \frac{\rho }{2}+\frac{2(1+H)NL_1\rho }{\nu }\right) . \end{aligned} \end{aligned}$$

Thus, we obtain that if $L_1$ is small enough then the transformation H acts from ${\mathcal {B}}_\rho (0)$ into itself. And we estimate

$$\begin{aligned} \begin{aligned} \Vert (Hx)(t)-(Hy)(t)\Vert&=\left\| \left( (I-\widetilde{\Psi })^{-1} {{\widetilde{F}}}_{\xi }x\right) (t) -\left( (I-\widetilde{\Psi })^{-1} {{\widetilde{F}}}_{\xi }y\right) (t) \right\| \\&=\left\| \left( \left( \sum ^{\infty }_{n=0}\widetilde{\Psi }^{n} \right) {{\widetilde{F}}}_{\xi }x\right) (t) -\left( \left( \sum ^{\infty }_{n=0}\widetilde{\Psi }^{n} \right) {{\widetilde{F}}}_{\xi }y\right) (t) \right\| \\&= \left\| \sum ^{\infty }_{n=0} \left( \widetilde{\Psi }^{n} ({{\widetilde{F}}}_{\xi }x-{{\widetilde{F}}}_{\xi }y)\right) (t)\right\| \\&\le \sum ^{\infty }_{n=0} \left\| \left( \widetilde{\Psi }^{n} {{\widetilde{F}}}_{\xi }w\right) (t)\right\| \\&\le \sum ^{\infty }_{n=0}\Vert \widetilde{\Psi }\Vert ^{n} \left\| \left( {{\widetilde{F}}}_{\xi }x-{{\widetilde{F}}}_{\xi }y\right) (t)\right\| \\&\le \frac{1}{1-\Vert \Psi \Vert } \left\| \left( {{\widetilde{F}}}_{\xi }x-{{\widetilde{F}}}_{\xi }y\right) (t)\right\| \\&\le \frac{1}{1-\Vert \Psi \Vert } \frac{2(1+H)NL_1}{\nu } \sup _{t\ge -r}\Vert x(t)-y(t)\Vert . \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \sup _{t\ge -r}\Vert (Hx)(t)-(Hy)(t)\Vert \le \frac{2(1+H)NL_1}{\nu (1-\Vert \Psi \Vert )} \sup _{t\ge -r}\Vert x(t)-y(t)\Vert . \end{aligned}$$

Hence, if $L_1$ is small enough then the transformation H is a contraction acts from ${\mathcal {B}}_\rho (0)$ in to it self.

Thus, there exists a unique $w(\cdot )\in {\mathcal {B}}_\rho (0)$ such that $Hw=w.$

This yields that $w(t), t\ge -r$ is the unique solution to

$$\begin{aligned} Fw_t=U(t,0)\xi +\int _0^\infty {\mathcal {G}}(t,\tau ){{\tilde{g}}}(\tau ,w_\tau )\mathrm{d}\tau . \end{aligned}$$

(4.11)

By Lemma 4.2 and Remark 4.3, we have that w is the unique solution in ${\mathcal {B}}_\rho (0)$ of the equation

$$\begin{aligned} Fw_t=U(t,0)\left( F\phi -F\hat{u}_0\right) +\int _0^t U(t,\tau ){{\tilde{g}}}(\tau ,w_\tau )\mathrm{d}\tau . \end{aligned}$$

(4.12)

Returning to the solution $u(\cdot )$ of (3.3) by replacing w by $u-\hat{u}$ we have that there exists a unique solution $u\in {\mathcal {B}}_\rho (\hat{u})$ of (3.3) with $u_0=\phi$.

Finally, we prove the estimate (4.9). To do this, we put $\xi :=P(0)Fu_0-P(0)F\hat{u}_0$, $w:=u-\hat{u}$, and $\tilde{g}(t,w_t):=g(t,w_t+\hat{u}_t)-g(t,\hat{u}_t)$. Then, we can use the formula (4.11) to derive

$$\begin{aligned} Fw_t=U(t,0)\xi +\int _0^\infty {\mathcal {G}}(t,\tau ){{\tilde{g}}}(\tau ,w_\tau )\mathrm{d}\tau . \end{aligned}$$

Using the facts that $\Vert \xi \Vert \le \frac{\rho }{2N}$ and $\Vert Fu_0-F\hat{u}_0\Vert \le \frac{\rho }{2}$, it follows that

$$\begin{aligned} \begin{aligned} \Vert w(t)\Vert&=\Vert (Hw)(t)\Vert =\left\| \left( (I-\widetilde{\Psi })^{-1} {{\widetilde{F}}}_{\xi }w\right) (t) \right\| =\left\| \left( \left( \sum ^{\infty }_{n=0}\widetilde{\Psi }^{n} \right) {{\widetilde{F}}}_{\xi }w\right) (t)\right\| \\&\le \sum ^{\infty }_{n=0} \Vert \widetilde{\Psi }\Vert ^{n}\left\| \left( {{\widetilde{F}}}_{\xi }w\right) (t)\right\| \le \frac{1}{1-\Vert \Psi \Vert }\Vert ({{\widetilde{F}}}_\xi w)(t)\Vert . \end{aligned} \end{aligned}$$

For all $t\ge 0$ we have

$$\begin{aligned} \Vert w(t)\Vert \le \frac{1}{1-\Vert \Psi \Vert }\left[ Ne^{-\nu t}\Vert \xi \Vert +(1+H)NL_1\displaystyle \int _0^\infty e^{-\nu |t-\tau |}\Vert w_{\tau }\Vert _{\mathcal {C}}\mathrm{d}\tau \right] . \end{aligned}$$

Setting $A:=\frac{N}{1-\Vert \Psi \Vert }, B:=\frac{(1+H)NL_1}{1-\Vert \Psi \Vert },$ we obtain that

$$\begin{aligned} \Vert w(t)\Vert \le Ae^{-\nu t}\Vert \xi \Vert +B\displaystyle \int _0^\infty e^{-\nu |t-\tau |}\Vert w_{\tau }\Vert _{\mathcal {C}} \mathrm{d}\tau \hbox { for all }t\ge 0. \end{aligned}$$

Since $t+\theta \in [-r+t,t]$ for fixed $t\in [0,\infty )$ and $\theta \in [-r,0]$, it follows that

$$\begin{aligned} \Vert w_t\Vert _{\mathcal {C}} \le Ae^{\nu r}\Vert \xi \Vert e^{-\nu t}+Be^{\nu r}\displaystyle \int _0^\infty e^{-\nu |t-\tau |}\Vert w_{\tau }\Vert _{\mathcal {C}}\mathrm{d}\tau \hbox { for all }t\ge 0. \end{aligned}$$

Put $\upsilon (t):=\Vert w_t\Vert _{\mathcal {C}}$, $t\ge 0$. Then $\sup _{t\geqslant 0}\upsilon (t)<\infty$ and

$$\begin{aligned} \Vert \upsilon (t)\Vert \le Ae^{\nu r}\Vert \xi \Vert e^{-\nu t}+Be^{\nu r}\displaystyle \int _0^\infty e^{-\nu |t-\tau |}\Vert \upsilon (\tau )\Vert \mathrm{d}\tau \hbox { for all } t\ge 0. \end{aligned}$$

Applying now a Gronwall-type inequality [8, Corollary III.2.3] we obtain that

$$\begin{aligned}&\Vert \upsilon (t)\Vert \le C\Vert \xi \Vert e^{-\mu t}\hbox { for all } t\ge 0 \hbox { where }\mu :=\sqrt{\nu ^2-2\nu \beta },\ C\\&\quad :=\frac{2\alpha \nu }{\nu +\sqrt{\nu ^2-2\nu \beta }},\ \alpha :=Ae^{\nu r},\ \beta :=Be^{\nu r}. \end{aligned}$$

Returning to the solution u of (3.3) by replacing w by $u-\hat{u}$ and $\xi =P(0)Fu_0-P(0)F{\hat{u}}_0$ we have

$$\begin{aligned} \Vert u_t-\hat{u}_t\Vert _{\mathcal {C}} \le Ce^{-\mu t}\Vert P(0)Fu_0-P(0)F{\hat{u}}_0\Vert , \end{aligned}$$

finishing the proof of the theorem. $\square$

Remark 4.6

The assertion of the above theorem shows us the conditional stability of the periodic solution $\hat{u}$ in the sense that for any other solution u such that $\Vert F\phi -F\hat{u}_0\Vert \le \rho /2$ and $P(0)Fu_0\in B_{\frac{\rho }{2N}}(P(0)F\hat{u}_0)\cap P(0)X$ and u being in a small ball ${\mathcal {B}}_\rho (\hat{u})$ we have $\left\| u(t)- \hat{u}(t)\right\| \rightarrow 0$ exponentially as $t\rightarrow \infty$ (see inequality (4.9)).

For an exponentially stable evolution family (see Definition 4.1 (2)), we have the following corollary which is a direct consequence of Theorem 4.5.

Corollary 4.7

Let the assumptions of the Theorem 4.4 hold, and let $\hat{u}$ be the periodic solution of (3.3) obtained in assertion (b) of Theorem 4.4. Let further the evolution family $(U(t,s))_{t\ge s\ge 0}$ be exponentially stable. Then, the periodic solution $\hat{u}$ is exponentially stable in the sense that for any other solution $u \in C_b([-r,\infty ), X)$ of (3.3) such that $\Vert Fu_0-F\hat{u}_0\Vert$ is small enough we have

$$\begin{aligned} \Vert u_t-\hat{u}_t\Vert _{\mathcal {C}}\le C{e^{-\mu t}}\Vert Fu_0-F{\hat{u}}_0\Vert \end{aligned}$$

(4.13)

for some positive constants C and $\mu$ independent of u and $\hat{u}$.

Proof

We just apply Theorem 4.5 for $P(t)=Id$ to obtain the assertion of the theorem.

$\square$

We finally illustrate our results by the following example.

4.1 An example

We consider the problem

(4.14)

here, $k,\eta , L\in {\mathbb {R}}$ and $|k|< 1,\ \eta \ne n^2$ for all $n\in {\mathbb {N}}$; the function $a(t)\in L_{1,loc}({\mathbb {R}}_+)$ is 1-periodic and satisfies the condition $0<\gamma _0\le a(t)\le \gamma _1$ for fixed $\gamma _0,\gamma _1;$ the function $h:[0,\pi ]\times {\mathbb {R}}_+\rightarrow {\mathbb {R}}_+$ is continuous on $[0,\pi ]\times {\mathbb {R}}_+$ and 1-periodic with respect to t.

We next put $X:=L_2[0,\pi ],\ {\mathcal {C}}:=C([-1,0],X)$, and let $A: X\supset D(A)\rightarrow X$ be defined by $Ay=y''+\eta y$, with the domain

$$\begin{aligned} D(A)=\{y\in X: y \text{ and } y' \text{ are } \text{ absolutely } \text{ continuous, } y''\in X, y(0)=y(\pi )=0\}. \end{aligned}$$

It can be seen (see [9]) that A is the generator of an analytic semigroup $({\mathbb {T}}(t))_{t\ge 0.}$

Since $\sigma (A)=\{-n^2+\eta : n=1,2,3,...\}$, applying the spectral mapping theorem for analytic semigroups we obtain

$$\begin{aligned} \sigma ({\mathbb {T}}(t))=e^{t\sigma (A)}=\{e^{t(-n^2+\eta )}: n=1,2,3,\ldots \} \end{aligned}$$

(4.15)

and hence $\sigma ({\mathbb {T}}(t))\cap {\Gamma } = \emptyset$ for all $t>0,$ where ${\Gamma }:=\{\lambda \in {\mathbb {C}} : |\lambda |=1\}$.

Putting now $A(t):=a(t)A$ we have that A(t) is 1-periodic, and the family $(A(t))_{t\ge 0}$ generates an 1-periodic (in the sense of Assumption 2.2) evolution family $U(t,s )_{t\ge s\ge 0}$ which is defined by the formula $U(t,s)={\mathbb {T}}(\int _s^ta(\tau )\mathrm{d}\tau )$ for all $t\ge s\ge 0.$

By (4.15) we have that the analytic semigroup $({\mathbb {T}}(t))_{t\ge 0}$ is hyperbolic (or has an exponential dichotomy) with the projection P satisfying

• (i) $\Vert {\mathbb {T}}(t)x\Vert \le Ne^{-\beta t}\Vert x\Vert$ for $x\in PX,\,t\ge 0$

• (ii) $\Vert {\mathbb {T}}(-t)_|x\Vert =\Vert ({\mathbb {T}}(t)_|)^{-1}x\Vert \le Ne^{-\beta t}\Vert x\Vert$ for $x\in \text {Ker}P,\,t\ge 0,$ where the invertible operator ${\mathbb {T}}(t)_|$ is the restriction of ${\mathbb {T}}(t)$ to KerP, and N, $\beta$ are positive constants.

Using the hyperbolicity of $({\mathbb {T}}(t))_{t\ge 0}$, it is straightforward to check that the evolution family $U(t,s )_{t\ge s\ge 0}$ has an exponential dichotomy with the projection $P(t)=P$ for all $t\ge 0$ and the dichotomy constants N and $\nu :=\beta \gamma _0$ by the following estimates:

$$\begin{aligned} \begin{aligned} \Vert U(t,s)x\Vert&\le Ne^{-\nu (t-s)}\Vert x\Vert \text{ for } x\in PX,\,t\ge s\ge 0, \\ \Vert U(s,t)_|x\Vert&\le Ne^{-\nu (t-s)}\Vert x\Vert \text { for } x\in \text {Ker}P,\,t\ge s\ge 0. \end{aligned} \end{aligned}$$

Then we define the delay operators $g: {\mathbb {R}}_+\times {\mathcal {C}}\rightarrow X$ by

$$\begin{aligned} g(t,\phi ):=L(\delta _{-1}\phi )\Vert \phi \Vert + h(\cdot ,t)) \end{aligned}$$

for $\phi \in {\mathcal {C}}:=C([-1,0],X)$ where $\delta _{-1}$ is the Dirac delta function concentrated at $-1.$

We define the difference operators F as

$$\begin{aligned} F:C([-1,0],X) \rightarrow X, \quad F(y):=y(0)-ky(-1). \end{aligned}$$

Then, (4.14) can be rewritten as

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{d}{dt}Fu_t(\cdot )=A(t)Fu_t(\cdot )+g(t,u_t(\cdot ,\theta )),\\ u_0(\cdot ,\theta )=\phi (\cdot ,\theta )\in {\mathcal {C}}, \phi \in {\mathbb {B}}_a. \end{array}\right. } \end{aligned}$$

Clearly, the difference operator F has the form $F=\delta _0-\Psi$ for $\Psi :=k\delta _{-1}$ and hence $\Vert \Psi \Vert \le k<1$, $\delta _0$ being the Dirac function concentrated at 0. Since $h(\cdot ,t)$ is $1-$periodic, it follows that $g(t,\phi )$ is 1-periodic with respect t for each function $\phi \in {\mathbb {B}}_a$.

Moreover,

$$\begin{aligned}&\left\| g(t,0)\right\| =L\Vert h(\cdot ,t)\Vert \le L\gamma \hbox { for }\gamma \\&\quad :=\sup \nolimits _{t\in [0,\infty )}\left( \int _0^\pi |h(x,t)|^2dx\right) ^{1/2},\hbox { and we have}\\&\left\| g(t,u_t(\cdot ,\theta ))-g(t,v_t(\cdot ,\theta ))\right\| \\&\quad =L\left\| \left( u_t(\cdot ,\theta )\Vert u(\cdot ,t)\Vert -v_t(\cdot ,\theta ) \Vert v(\cdot ,t)\Vert \right) \right\| \\&\quad =L\left\| \mathbf {(}u_t(\cdot ,\theta )\Vert u(\cdot ,t)\Vert -v_t(\cdot ,\theta ) \Vert u(\cdot ,t)\Vert \mathbf {)}\right. \\&\qquad \left. +\mathbf {(}v_t(\cdot ,\theta )\Vert u(\cdot ,t)\Vert -v_t(\cdot ,\theta )\Vert v (\cdot ,t)\Vert \mathbf {)}\right\| \\&\quad \le L\left( \Vert u(\cdot ,t)\Vert \left\| (u_t(\cdot ,\theta )-v_t(\cdot ,\theta ))\right\| +\Vert v_t(\cdot ,\theta )\Vert \left\| (\Vert u(\cdot ,t)\Vert -\Vert v(\cdot ,t)\Vert )\right\| \right) \\&\quad \le 2aL\left\| u_t-v_t\right\| _{{\mathcal {C}}} \quad \hbox { for all } u_t, v_t \in {\mathbb {B}}_a, t\in {\mathbb {R}}_+. \end{aligned}$$

Therefore, g satisfies the hypotheses of Theorems 4.4 with $\rho =a$, $L:=\rho +\frac{\gamma }{2\rho }.$ By Theorem 4.4, we obtain that, for sufficiently small $\rho , L$ and $\gamma$, Equation (4.14) has one and only one 1-periodic mild solution $\hat{u}\in {\mathcal {B}}_{\rho }(0),$ and this solution ${\hat{u}}$ is conditionally stable in the sense of Remark 4.6.