On a Periodic Boundary Value Problem for a Fractional–Order Semilinear Functional Differential Inclusions in a Banach Space

Abstract

We consider the periodic boundary value problem (PBVP) for a semilinear fractional-order delayed functional differential inclusion in a Banach space. We introduce and study a multivalued integral operator whose fixed points coincide with mild solutions of our problem. On that base, we prove the main existence result (Theorem ). We present an example dealing with existence of a trajectory for a time-fractional diffusion type feedback control system with a delay satisfying periodic boundary value condition.

On a Periodic Boundary Value Problem for Fractional Quasilinear Differential Equations with a Self-Adjoint Positive Operator in Hilbert Spaces

“…Fractional calculus and the theory of fractional differential equations have gained significant popularity and importance over the past three decades, mainly due to demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering (see monographs [1,2], papers [3][4][5][6][7][8][9]). This branch of mathematics really provides many useful tools for various problems related to special functions of mathematical physics, as well as their extensions and generalizations for one or more variables (see e.g., paper [10]).…”

On the Existence of a Unique Solution for a Class of Fractional Differential Inclusions in a Hilbert Space

“…(see, for example, [32,33]). Let us recall (see, for example, [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]) that a mild solution to problems (15) and 16is a function x ∈ C([0, T], H) of the form…”

“…Moreover, it is known (see [5,8,[10][11][12][13][14]) that the family (34) has compact convex values and is condensing with respect to the MNC ν in C([0, T]; H) (see Section 2). Since the multioperators λF satisfy conditions (F1)-(F5) independently on λ, by applying Theorem 1, we conclude that there exists a constant C(T) such that all solutions to problems (35) and (36) satisfy the a priori estimate…”

“…The increasing interest in this subject is explained by its numerous applications in various branches of applied mathematics, physics, engineering, biology, economics, and other sciences (see, e.g., the monographs of [1][2][3]). A number of various approaches to the solving of boundary value problems for differential equations and inclusions in the case of fractional order q ∈ (0, 1) have been widely described in the literature (see [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and references therein).…”

Remotely Almost Periodic Solutions of Ordinary Differential Equations

“…It is well known that Sarason defined the notion of a scalar-valued remotely almost periodic function in [11]. e class of vector-valued remotely almost periodic functions defined on R n was introduced by Yang and Zhang in [12], where the authors have provided several applications in the study of existence and uniqueness of remotely almost periodic solutions for parabolic boundary value problems (for some results about parabolic boundary value problems, one may refer to [13][14][15] and references cited therein). In Propositions 2.4-2.6 in [16], the authors have examined the existence and uniqueness of remotely almost periodic solutions of multidimensional heat equations, while the main results of Section 3 are concerned with the existence and uniqueness of remotely almost periodic type solutions of the certain types of parabolic boundary value problems (see also [17,18], where the authors have investigated almost periodic type solutions and slowly oscillating type solutions for various classes of parabolic Cauchy inverse problems).…”