Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations

Abstract

We study the existence and uniqueness of a weighted pseudo-almost periodic (mild) solution to the semilinear fractional equation ${\partial }_t^{\alpha }u=Au+{\partial }_t^{\alpha -1}f(\cdot ,u)$, $1<\alpha <2$, where $A$ is a linear operator of sectorial negative type. This article also deals with the existence of these types of solutions to abstract partial evolution equations.

Introduction

Pseudo-almost periodic functions have many applications in several problems, for example in theory of functional differential equations, integral equations and partial differential equations. The concept of pseudo-almost periodicity, which is the central issue in this work, was introduced by Zhang [1], [2], [3], [4], [5] in the early nineties. Since then, such a notion became of great interest to several mathematicians (see [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]). The pseudo-almost periodicity is a natural generalization of the classical almost periodicity in the sense of Bochner. In [19], [20], [21], [22], a new generalization of the concept of almost periodicity was introduced. Such a new concept is called weighted pseudo-almost periodicity. To construct those weighted pseudo-almost periodic functions, the main idea consists of enlarging the so-called ergodic component.

Firstly, we study in this work some sufficient conditions for the existence and uniqueness of a weighted pseudo-almost periodic mild solution to the following semilinear fractional differential equation

$$D_t^{\alpha }u\left(t\right)=Au\left(t\right)+D_t^{\alpha -1}f(t,u\left(t\right))\text{,}t\in \mathbb{R}\text{,}$$

where $1<\alpha <2$, $A:D\left(A\right)\subset X\to X$ is a linear densely defined operator of sectorial type on a complex Banach space $X$ and $f:\mathbb{R}\times X\to X$ is a weighted pseudo-almost periodic function (see Definition 2.2) satisfying suitable conditions in $x$. The fractional derivative is understood in the Riemann–Liouville sense. The origin of fractional calculus goes back to Newton and Leibnitz in the seventieth century. We observe that fractional order can be complex in viewpoint of pure mathematics and there is much interest in developing the theoretical analysis and numerical methods to fractional equations, because they have recently proved to be valuable in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, electromagnetism, biology and hydrogeology. For example space-fractional diffusion equations have been used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium [23], [24] or to model activator–inhibitor dynamics with anomalous diffusion [25]. For details, including some applications and recent results, see the monographs of Ahn and MacVinisch [26], Gorenflo and Mainardi [27], Hilfer [28], Kilbas et al. [29], Kiryakova [30], Miller and Ross [31], Ross [32], Podlubny [33] and Samko et al. [34], and the papers of Agarwal et al. [35], [36], [37], [38], Benchohra et al. [39], Diethelm et al. [40], [41], El-Borai [42], [43], [44], El-Sayed [45], [46], [47], Chen et al. [48], Gaul et al. [49], Hu and Wang [50], Mophou and N’Guérékata [51], [52], [53], Nieto et al. [54], [55], N’Guérékata [56], Lakshmikantham [57], Lakshmikantham et al. [58], [59], [60], Mainardi [61] and the references therein. Type (1.1) equations are attracting increasing interest (cf [26], [62], [63], [64]). The study of pseudo-almost periodic mild solution of (1.1) was initiated recently by the authors in [65] (see also [66], [67]). To the best of the authors’ knowledge, no results yet exist for weighted pseudo-almost periodic mild solutions of (1.1). The investigation of the theory of fractional differential equations has only been started quite recently and therefore seem to deserve an independent study of their theory. This is a significant contribution from the perspective of developing the theory of almost periodicity and ergodicity for fractional equations and considering the fact that only a few articles investigated the existence and properties of solutions of functional integral or differential equations of fractional order on an unbounded interval (see [68], [69], [70]). It is worth noting that our assumptions are very natural and we have tested them in the practical context. In particular to build intuition and throw some light on the power of our results, we examine sufficient conditions for the existence of weighted pseudo-almost periodic solutions to a fractional relaxation–oscillation equation (see Example 2). On the other hand, we use the machinery developed in Section 3.1 (see Proposition 3.1 and Theorem 3.2) to analyze in a detailed manner sufficient conditions for the existence of a weighted pseudo-almost periodic solutions to the class of abstract partial evolution equations

$$\frac{d}{dt}[u\left(t\right)+f(t,Bu\left(t\right))]=Au\left(t\right)+g(t,Cu\left(t\right))\text{,}t\in \mathbb{R}\text{,}$$

where $A:D\left(A\right)\subset X\to X$ is a sectorial linear operator on $X$ whose corresponding analytic semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ is hyperbolic, $B$ and $C$ are arbitrary densely defined closed linear operators on $X$ and $f$ and $g$ are weighted pseudo-almost periodic in $t\in \mathbb{R}$ uniformly in the second variable (see Section 3.3). Note that Eq. (1.2) in the case when $A$ is a sectorial operator corresponds to several interesting situations encountered in the literature (see Diagana’s paper [15]). We point out that the main result of Section 3.3 (Theorem 3.6) generalizes Theorem 3.5 in [21].

We now turn to a summary of this work. Section 2 provides the definitions and preliminary results to be used in theorems stated and proved in the article. In particular to facilitate access to the individual topics, we review in Section 2.1 some of the standard properties of sectorial operators. Weighted pseudo-almost periodic functions, which is the Leitmotiv of this work, is discussed in detail in Section 2.2. In Section 2.3, we give a compactness criterion in $C_h\left(Z\right)$ (see Lemma 2.6). Section 2.4 deals with intermediate spaces between $X$ and $D\left(A\right)$, which are of fundamental importance in a part of this article. Section 3 is divided into three parts. In the first part, Section 3.1, we obtain very general results on the existence of weighted pseudo-almost periodic (w.p.a.p. in short) mild solutions for semilinear fractional differential equations. The second part is concerned with semilinear differential equations (see Section 3.2). Using the formalism presented in Section 2.4, we deal in the third part, Section 3.3, with the existence of w.p.a.p. solutions for partial evolution equations. From a more calculation point of view, throughout this article, we give a few applications.