On antiperiodic boundary value problem for semilinear fractional differential inclusion with deviating argument in Banach space

Abstract

We consider a boundary value problem for a semi-linear differential inclusion of Caputo fractional derivative and a deviating coefficient in a Banach space. We assume that the linear part of the inclusion generates a bounded 0-semigroup. A nonlinear part of the inclusion is a multi-valued mapping depending on the time and the prehistory of the function before a current time. The boundary condition is functional and anti-periodic in the sense that one function is equals to another with an opposite sign. To solve the considered problem, we employ the theory of fractional mathematical analysis, the properties of Mittag-Leffler as well as the theory of topological power for multi-valued condensing maps. The idea is as follws: the original problem is reduced to the existence of fixed points of a corresponding resolving multi-valued integral operator in the space of continuous functions. To prove the existence of the fixed points of the resolving multi-operator we employ a generalized theorem of Sadovskii type on a fixed point. This is why we show that the resolving integral multi-operator is condensing with respect to a vector measure of non-compactness in the space of continuous functions and maps a closed ball in this space into itself.

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…”

“…Now, suppose that x ∈ C([0, T]; H) is any mild solution to problems (15) and (16). Take a selection f ∈ P F (x) satisfying (18). Then, condition (F3) implies that…”

“…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).…”

Impulsive Fractional Differential Inclusions and Almost Periodic Waves

“…Due to their great relevance to reality and their numerous implementations, the almost periodicity is considered a very important qualitative property of solutions. However, the relevant results for fractional inclusions are few [11][12][13]. The main goal of the present paper is to contribute to the development of this area.…”

“…However, the concept of almost periodic solutions (waves) for fractional-order differential inclusions has just begun to be studied [11][12][13]. Since the investigations into the existence of non-periodic solutions and their properties are of great significance for fractional-order inclusions, the theory of almost periodic waves should be further developed.…”