Existence and Asymptotic Periodicity of Solutions for Neutral Integro-Differential Evolution Equations with Infinite Delay

1 Introduction

In this paper, we study the existence, regularity and asymptotic periodicity of solutions for neutral functional integro-differential evolution equations with infinite delay of the form

where x(·) is the state variable taking values in a Banach space X and the notation x_t represents the functional defined by x_t: (−∞, 0] → X, x_t(θ) = x(t + θ), belonging to some abstract phase space ℬ which is defined axiomatically. The operator (A, D(A)) is the infinitesimal generator of an analytic semigroup on X, while F(t): D(A) → D(A) and N(t): X → X are two families of bounded linear operators. The nonlinear function f (·, ·) is to be specified later.

Neutral partial integro-differential equations without or with time delay arise in various areas. For example, the following heat conduction system, see [3, 6, 7, 30], has been frequently used to describe the evolution of the temperature in different types of materials with fading memory,

In this system, Ω ⊂ ℝⁿ is open, bounded and has smooth boundary, (t, x) ∈ [0, ∞) × Ω, u(t, x) represents the temperature in x at time t, c is a physical constant and k_i: ℝ → ℝ, i = 1, 2, are functions of the internal energy and the heat flux relaxation respectively, and the internal energy here the heat flux terms are described as functionals of u and Δu. For more models of partial integro-differential equations and related applications we refer to the paper [2] and the monographs [22, 38]. One of the the important way to deal with partial integro-differential equations is to transform them into (neutral) abstract integro-differential evolution equations in Banach space, and then study the abstract systems by applying theory of resolvent operators. Actually, by assuming the solution u is known on (−∞, 0]. we can transform (1.2) into the abstract neutral integro-differential evolution equation of the following form

In [10], it was illustrated that the equation

can be regarded as the abstract formulation of the model proposed and studied by Coleman and Gurtin [8], Gurtin and Pipkin [20], and Miller [31] for the heat flow in a rigid isotropic viscoelastic material in the elastic case, and the model in thermoviscoelasticity considered by Leugering [28].

An important approach to study the above abstract equations is the theory of resolvent operators founded in [17]–[19]. Grimmer et al. proved in [17]–[19] that there is a (unique) resolvent operator (R(t))_t≥0 associated to the linear homogeneous equation

Based on it the authors obtained the representation, existence, and uniqueness of mild solutions of the following inhomogeneous integro-differential evolution equation in Banach space X

via the resolvent operator, where f : ℝ⁺ → X is a continuous function. Actually, the resolvent operator, replacing the role of C₀-semigroup for evolution equations, plays an essential role in solving Eq. (1.3) in weak and strict senses. Based on these important works, in these years much works on various topics, such as existence, regularity and stability of solutions and control problems, for semilinear integro-differential evolution equations has been done by many authors through applying the theory of resolvent operator, see [4, 5, 11, 14–16, 32, 40] and the references therein.

On the other hand, existence of periodic solutions is one of the most interesting and important topics in the qualitative theory of differential equations due to their significance in practical sciences. Many contributions on the existence of periodic solutions for differential equations have been made. Meanwhile, the concept of asymptotically ω-periodic solutions is more general than periodic solutions and, from view of applications, asymptotically periodic systems describe the models more realistically and accurately than periodic ones. The notion of S-asymptotically ω-periodic functions was introduced by Henríquez et al. in [25, 26] (i.e. a bounded continuous function f (·) is called to be S-asymptotically ω-periodic if there exists ω > 0 such that limt→∞‖f(t+ω)−f(t)‖=0) and it is related to and more general than that of asymptotic ω-periodic functions. Moreover, the authors have established there a relationship between S-asymptotically ω-periodic functions and the class of asymptotically ω-periodic functions. In these years this topic is attracting much attention increasing interest of many researchers, some recent works of this area can be found in [1, 9, 12, 24, 29, 35, 37, 39] among others. Particularly, Dos Santos and Henríquez [37] investigated existence of S-asymptotically ω-periodic solutions for neutral integro-differential equations of the following form

via the theory of resolvent operators associated to the linear homogeneous equation

which was founded in [36]. In [36] the authors first established the theory of resolvent operators for (1.4) and then they discussed the existence and uniqueness of solutions for a semilinear neutral integro-differential equation with infinite delay.

To the best of our knowledge, the regularity and asymptotic periodicity of mild solutions for the neutral partial functional integro-differential equation ((1.1) are untreated topics in the literature up to know. The purpose of this paper is to explore the existence, regularity and S-asymptotic periodicity of solutions for the equation (1.1). The main tool in our discussion is also the theory of resolvent operators for neutral linear integro-differential systems of 1.1) established in [36]. We will represent the global mild solutions of Eq. (1.1) by the resolvent operator R(t) and analyze existence and uniqueness of solutions by using theory of resolvent operator and Banach fixed point theorem. Following this we further study the regularity of mild solutions for Eq. (1.1) under some smoothness conditions. It is seen that the results obtained in this part are better than those in [14, 16, 27] due to the application of the newly theory of resolvent operators. More precisely, other than in [14, 16, 27], we do not impose here any smoothness condition on the neutral term. At last, we study the existence and uniqueness of S-asymptotically ω-periodic solutions assuming the nonlinear function is so. The results are proved by utilizing Schauder’s fixed point theorem and Banach fixed point theorem, respectively, for the different cases that the resolvent operator is compact and non-compact. We point out here that, compared to [29, 37], the conditions for the existence of S-asymptotically ω-periodic solutions are somewhat weaker (see Theorem 4.1 and Theorem 4.2). Clearly, our obtained results in this article extend and develop the existing results in literature mentioned above.

Subsequently the paper is organized as follows. In Section 2, we state briefly the basic theory of analytic semigroup and resolvent operators. In Section 3 we discuss the existence and regularity of global mild solutions of (1.1). The existence and uniqueness of S-asymptotically ω-periodic solutions is then studied in Section 4. Finally, in Section 5, we provide an example to illustrate the applications of the obtained results.

2 Preliminaries

In this section, we present some preliminary facts on theory of analytic semigroup and resolvent operators to be used in this paper.

Let (X, ǁ·ǁ and (Z, ǁ·ǁ) be two Banach spaces, we denote by ℒ(X, Z) the Banach space of bounded linear operators from X into Z endowed with the general operator norm and we abbreviate this notation to ℒ(X) when X = Z. For a linear operator A:D(A)⊆X→X, let Y be the Banach space (D(A), ǁ · ǁ₁) with the graph norm ǁxǁ₁ = ǁAxǁ + ǁxǁ, for x ∈ D(A). Hereafter C([0, T], X) denotes the Banach space consisting of continuous functions from [0, T] to X with the norm

C_b([0, ∞); X) denotes the Banach space consisting of bounded and continuous functions from [0, ∞) to X with ǁ · ǁ_∞ defined by ‖x‖∞=supt≥0‖x(t)‖.

Next we present briefly the basic theory of resolvent operators for the following neutral integro-differential equation associated to Eq. (1.1).

1. The function R(·): [0, ∞) → ℒ(X) is strongly continuous, and there exist M₀ ≥ 1 and ω ∈ ℝ, such that ǁR(t)ǁ ≤ M₀e^ωt for t ≥ 0;

2. R(0) = I;

3. For x ∈ Y, R(·)x ∈ C¹((0, ∞), X) ⋂ C([0, ∞), Y), and

   (2.2)ddt[R(t)x+∫0tN(t−s)R(s)xds]=A[R(t)x+∫0tF(t−s)R(s)xds],

   (2.3)ddt[R(t)x+∫0tR(t−s)N(s)xds]=R(t)Ax+∫0tR(t−s)AF(s)xds,

   for each t ≥ 0.

We assume for the operators in Eq. ((1.1) (or 2.1)) that the following conditions are fulfilled.

1. There are constants M₀ > 0 and θ∈(π2,π) such that

   ρ(A)⊇Λθ={λ∈ℂ:|arg(λ)|<θ} and ‖R(λ,A)‖≤M0|λ|−1,λ∈Λθ.

   Hence in this case the operator A generates an analytic semigroup (T(t))_t≥0 on X (see [13, 34]).

2. The function N: [0, ∞) → ℒ(X) is strongly continuous and N^(λ)x is absolutely convergent for x ∈ X and Re(λ) > 0. There exist β > 0 and an analytical extension of N^(λ) (still denoted by N^(λ)) to Λ_θ such that ‖N^(λ)x‖≤N0|λ|−β‖x‖1 for each λ ∈ Λ_θ and x ∈ D(A). Here N₀ is a constant, the notation f^(λ) represents the Laplace transform of f (t).

3. For all t ≥ 0, F(t) ∈ ℒ(X). F(t): X → D(A), and AF(·)x is strongly measurable on (0, ∞) for any x ∈ X. There exists a b(⋅)∈Lloc1(R+) such that b^(λ) exists for Re λ > 0 and ǁAF(t)xǁ ≤ b(t)ǁxǁ₁ for all t > 0 and x ∈ D(A). Moreover, the operator valued function F^:Λπ2→ℒ(Y) has an analytical extension (still denoted by F^) to Λ_θ such that ‖AF^(λ)x‖≤‖AF^(λ)‖‖x‖1 for all x ∈ D(A), and ‖AF^(λ)‖→0 as |λ| → ∞.

4. There exist a subspace D⊆D(A) dense in Y and positive constants C_i, i = 1, 2, such that A(D)⊆D(A), F^(λ)(D)⊆D, N^(λ)(D)⊆D(A), and ‖AF^(λ)x‖1≤C1‖x‖1, ‖N^(λ)x‖1≤C2|λ|−β‖x‖1 for every x ∈ D and λ ∈ Λ_θ.

Observe that the above assumptions guarantee that AF(·) verifies the conditions of the family B(·) in [36]. Hence, according to [36], we infer readily that, under the hypotheses (V₁)−(V₄) there is a resolvent operator R(t) for the linear part of (1.1) (i.e. (2.1)), which is given by

where G(λ)=(λI+λN^(λ)−A−AF^(λ))−1 satisfying ǁG(λ)ǁ ≤ M₀|λ⁻¹| for λ ∈ Λ_{r, ϑ} and some M₀ > 0. We put Ω(G)={λ∈∑F∩∑N:λI+λN^(λ)−A−AF^(λ) is invertible and G(λ)∈ℒ(X)}, σ(G) = ℂ \ Ω(G) and W_G = sup{Re(λ) : λ ∈ σ(G)}. Here the notations ∑F and ∑N stand for the domain of maximal extension of the operators F^(⋅) and N^(⋅) on ℂ, respectively. The region Λ_r,ϑ = {λ ∈ ℂ \ {0} : |λ| > r, |arg(λ)| < ϑ} and Γr,ϑ=∪i=13Γr,ϑi the curves Γr,ϑi,i=1, 2, 3, are given respectively by

for r>0,ϑ∈(π2,θ) being fixed numbers. These curves are oriented so that Im(λ) is increasing.

The following theorem founded in [37] will be used in this paper.

In this paper, we will use an axiomatic definition of the phase space ℬ introduced by Hale and Kato in [21] and follow the terminology used in [23]. Thus, ℬ is a linear space of functions mapping (−∞, 0] into X endowed with a seminorm ǁ · ǁ_ℬ. We assume that ℬ satisfies the following axioms:

1. If x: (–∞,σ + a) → X, a > 0 is continuous on [σ, σ + a) and x_σ ∈ ℬ, then for every t ∈ [σ, σ + a) the following conditions hold:

   1. x_t is in ℬ.

   2. ǁx(t)ǁ ≤ Hǁx_tǁ_ℬ.

   3. ǁx_tǁ_ℬ ≤ K(t – σ) sup{ǁx(s)ǁ : σ ≤ s ≤ t} + M(t — σ)ǁx_σǁ_ℬ.

Here H ≥ 0 is a constant, K, M: [0, +∞) → [0, +∞), K(·) is continuous and M(·) is locally bounded, and H, K(·), M(·) are independent of x(·).

1. For the function x(·) in (A), the function t → x_t is continuous from [σ, σ + a) into ℬ.

2. The space ℬ is complete.

For our later discussion we also need some additional properties of the space ℬ. Let C₀₀ be the space of continuous functions from (−∞, 0] into X with compact support. Then from [23: Proposition 2.1], C00⊆ℬ. We assume that the phase space ℬ satisfies additionally the following axioms:

1. If {ϕⁿ} is a Cauchy sequence in ℬ and converges compactly to ϕ on (−∞,0], then ϕ ∈ ℬ and limn→+∞‖ϕn−ϕ‖ℬ=0.

2. If a uniformly bounded sequence {ϕⁿ(θ)} in C₀₀ converges to a function ϕ(θ) compactly on (–∞, 0], then ϕ ∈ ℬ and limn→+∞‖ϕn−ϕ‖ℬ=0.

Thus we have that

In addition, from [33], one has that

Finally we introduce the concept of fading memory spaces. To do this, let ℬ₀ = {ϕ ∈ ℬ : ϕ(0) = 0} which is a closed subspace of the space ℬ, and let the operator S₀ (t) on ℬ₀ be defined by

for ϕ ∈ ℬ₀. Clearly, the operator family S₀(t), t ≥ 0, is a C₀-semigroup on ℬ₀.

3 Existence and regularity of solutions

The purpose of this section is to study the existence, uniqueness and regularity of mild solutions for the equation (1.1) on (–∞, ∞).

4 Asymptotic periodicity of solutions

The goal of this section is to establish the existence and uniqueness of S-asymptotically ω-periodic mild solutions for the equation (1.1). Initially we introduce some concepts and properties related to S-asymptotically ω-periodic functions.

We denote by SAP_ω(X) the set of all S-asymptotically ω-periodic functions from [0, ∞) to X. Note that SAP_ω(X) is a Banach space with the sup-norm ǁ · ǁ_∞.

The following two results, established respectively in [25] and [26], will play an essential role in proving the S-asymptotically ω-periodicity of solutions for Eq. (1.1).

Next we give the definition of S-asymptotically ω-periodic mild solutions for Eq. (1.1).

Now, we establish the first result of this part.