Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator

1 Introduction

In this paper, we are concerned with the existence of solutions for the following neutral evolution equations with nonlocal conditions and delay

where u ( ⋅ ) takes value in a subspace D ( A α ) of Banach space X , which will be defined later, A : D ( A ) ⊂ X → X is a linear operator and − A generates an analytic semigroup T ( t ) ( t ≥ 0 ) , α , h and a be three constants such that 0 < α < 1 and 0 < h , a < + ∞ . For every t ∈ [ 0 , a ] , F , G are given functions satisfying some assumptions, ϕ ∈ C ( [ − h , 0 ] , D ( A α ) ) is a priori given history, while the function g : C ( [ − h , a ] , D ( A α ) ) → C ( [ − r , 0 ] , D ( A α ) ) implicitly defines a complementary history, chosen by the system itself. Following the standard notation (see [1]), if u ∈ C ( [ − h , a ] , D ( A α ) ) and t ∈ [ 0 , a ] , we denote by u t the function u t : [ − h , 0 ] → D ( A α ) , defined by u t ( θ ) c o l o n e u ( t + θ ) for θ ∈ [ − h , 0 ] .

Many complex processes in nature and technology are described by functional differential equations, which are dominant nowadays because the functional components in equations allow one to consider after-effect or prehistory influence. Delay differential equations are one of the important types of functional differential equations, in which the response of the system depends not only on the current state of the system but also on the history of the system. For more details on this topic, see, for example, the books of Hale and Verduyn Lunel [1] and Wu [2], and the papers of Chen et al. [3], Chen [4], Dong and Li [5], Fu [6], Fu and Ezzinibi [7], Li [8], Travis and Webb [9,10], Vrabie [11,12] and Wang et al. [13].

The theory of neutral partial differential equations with a delay has an extensive physical background and realistic mathematical model; hence, it has been considerably developed and the numerous properties of their solutions have been studied, see [6,8,14, 15,16,17, 18,19] and the references therein. The problem concerning neutral partial differential equations with nonlocal conditions and delay is an important area of investigation in recent years. Especially, the existence of solutions of neutral evolution equations with nonlocal conditions and delay have been considered by several authors, see [7,20].

In problem (1.1), when α = 0 in interpolation space X α , Dong et al. [21] studied the neutral partial functional differential equations with nonlocal conditions

With the aid of Hausdorff’s measure of noncompactness and fixed-point theory, the authors established some existence results of mild solutions for the problem (1.2). However, their results cannot be applied to equations with terms involving spatial derivatives.

The existence and regularity problem on the compact interval [ 0 , a ] , in the very simplest case when r = 0 , i.e., when the delay is absent, was studied by Chang and Liu [22]. In this case, C 0 α identifies with X α , F ( t , u , u 0 ) identifies with a function F from [ 0 , a ] × X α → X , and G ( t , u , u 0 ) identifies with a function G from [ 0 , a ] × X α → X , and so the previous paper [22] has considered the problem:

where the operator − A : D ( A ) ⊂ X → X generates an analytic and compact semigroup. The authors of [22] showed existence results of solutions to neutral evolution equations with nonlocal conditions (1.3) in the α -norm under the assumptions that the linear part of equations generates a compact analytic semigroup and the nonlinear part satisfies some Lipschitz conditions with respect to the α -norm. In their work, a key assumption is that the associated semigroup is compact.

In 2013, Fu [6] investigated the existence of mild solutions for the following abstract neutral evolution equation with an infinite delay

where u ( ⋅ ) takes values in a subspace of Banach space X . Under the assumption that the linear part of equations generates a compact analytic semigroup, the author obtained the existence of mild solutions to the problem (1.4) by using fractional power theory and α -norm. The results in [6] can be applied to equations with terms involving spatial derivatives. However, to the best of the authors’ knowledge, in all of the existing articles, such as [6], the neutral evolution equations with nonlocal conditions via fractional operator have been studied under the hypothesis that the corresponding linear partial differential operator generates a compact semigroup, and the existence of mild solutions for neutral evolution equations with nonlocal conditions via fractional operator with noncompact semigroup has not been investigated yet.

Motivated by all the aforementioned aspects, in this article, we apply the theory of fractional power operator, α -norm, Kuratowski measure of noncompactness and corresponding fixed-point theory to obtain the existence and uniqueness of mild solutions for neutral evolution equations with nonlocal conditions and delay via fractional operator (1.1) without the assumptions of compactness on the associated semigroup. Particularly, our results cover the cases where the nonlinear term F takes values in different spaces such as X α and X μ .

The rest of this paper is organized as follows. In Section 2, we recall some preliminary results on the fractional powers of unbounded linear operators generating analytic semigroups, and the results of Kuratowski’s measure of noncompactness, which will be used in the proof of our main results. Section 3 states and proves the existence and uniqueness of mild solutions for the problems (1.1) in the α -norm by utilizing the fixed-point theorem and Kuratowski’s measure of noncompactness. An example of the neutral partial differential system is also given in Section 4 to illustrate the feasibility of our abstract results.

2 Preliminaries

In this section, we introduce some notations, definitions and preliminary facts that are used throughout this paper.

Assume that X is a real Banach space with norm ‖ ⋅ ‖ , A : D ( A ) ⊂ X → X is a densely defined closed linear operator and − A generates an analytic semigroup T ( t ) ( t ≥ 0 ) . Then there exists a constant M ≥ 1 such that ‖ T ( t ) ‖ ≤ M for t ≥ 0 . Without loss of generality, we suppose that 0 ∈ ρ ( A ) ; otherwise instead of A , we take A − λ I , where λ is chosen such that 0 ∈ ρ ( A − λ I ) where ρ ( A ) is the resolvent set of A . Then it is possible to define the fractional power A α for 0 < α < 1 , as a closed linear operator on its domain D ( A α ) . Furthermore, the subspace D ( A α ) is dense in X and

defines a norm on D ( A α ) . Hereafter, we denote by X α the Banach space D ( A α ) normed with ‖ ⋅ ‖ α . In addition, we have the following properties.

Set

It is easy to see that C 0 α and C a α are Banach spaces with norms

respectively. For any R > 0 , let D R = { u ∈ X α : ‖ u ‖ α ≤ R } , D R ( C 0 α ) = { u ∈ C 0 α : ‖ u ‖ C 0 α ≤ R } , D R ( C a α ) = { u ∈ C a α : ‖ u ‖ C a α ≤ R } .

Moreover, we introduce the Kuratowski measure of noncompactness μ ( ⋅ ) defined by

for a bounded set D in the Banach space X . In this article, we denote by μ ( ⋅ ) , μ α ( ⋅ ) and μ C a α ( ⋅ ) the Kuratowski measure of noncompactness on the bounded set of X , X α and C a α , respectively. For any D ⊂ C a α , s ∈ [ − h , a ] and t ∈ [ 0 , a ] , set

3 Existence results

This section is devoted to investigate the existence of mild solutions for problem (1.1) in the subspace X α of Banach space X . By comparison with the abstract Cauchy problem

whose properties are well known [23], we can obtain the definition of mild solution for the problem (1.1) in X α .

In what follows, we make the following hypotheses on the data of problem (1.1).

1. The function F : [ 0 , a ] × X α × C 0 α → X is Carathéodory continuous, and there exist constants q ∈ [ 0 , 1 − α ) , γ > 0 and function φ R ∈ L 1 q ( [ 0 , a ] , R + ) such that for some positive constant R ,

   ‖ F ( t , u , v ) ‖ ≤ φ R ( t ) , liminf R → + ∞ ‖ φ R ‖ L 1 q ( [ 0 , a ] ) R ≔ γ < + ∞ ,

   for any u ∈ D R , v ∈ D R ( C 0 α ) and a.e. t ∈ [ 0 , a ] .

1. There exists a constant β ∈ ( 0 , 1 ) with α + β ≤ 1 , such that the function G : [ 0 , a ] × X α × C 0 α → X α + β satisfies Lipschitz condition, i.e., there exists a constant L G > 0 such that

   ‖ A β G ( t , u 2 , v 2 ) − A β G ( t , u 1 , v 1 ) ‖ α ≤ L G ( ∣ t 2 − t 1 ∣ + ‖ u 2 − u 1 ‖ α + ‖ v 2 − v 1 ‖ C 0 α ) ,

   where t 1 , t 2 ∈ [ 0 , a ] , u 1 , u 2 ∈ X α and v 1 , v 2 ∈ C 0 α .

1. There exists a positive constant L g such that

   ‖ g ( u ) − g ( v ) ‖ C 0 α ≤ L g ‖ u − v ‖ C a α , u , v ∈ C a α .

2. ℋ + M α 1 − q 1 − α − q 1 − q a 1 − α − q γ < 1 , where ℋ = M ( L g + 2 N β L G ) + 2 N β L G + 2 M 1 − β L G a β β .

3. There exists a function η ∈ L 1 ( [ 0 , a ] , R + ) with ‖ η ‖ L 1 < 1 − ℋ 8 such that for any bounded and countable set D ⊂ C a α ,

   μ α ( T ( s ) F ( t , D ( t ) , D t ) ) ≤ η ( t ) ( μ α ( D ( t ) ) + sup − h ≤ τ ≤ 0 μ α ( D ( t + τ ) ) ) , a.e. t , s ∈ [ 0 , a ] .