Fractional q-Difference Inclusions in Banach Spaces

Abstract

In this paper, we study a class of Caputo fractional q-difference inclusions in Banach spaces. We obtain some existence results by using the set-valued analysis, the measure of noncompactness, and the fixed point theory (Darbo and Mönch’s fixed point theorems). Finally we give an illustrative example in the last section. We initiate the study of fractional q-difference inclusions on infinite dimensional Banach spaces.

A Filippov's Theorem and Topological Structure of Solution Sets for Fractional q-Difference Inclusions

“…Existence of solutions. From Theorem 5 in [13], the operator T verifies all the requirements of Theorem 2.17, and we deduce that T has at least one fixed point w ∈ F(Θ) which is a solution of (1.1)-(1.2).…”

“…Since {w 0n } bounded sequence, there exists a subsequence of {w 0n } converging to w 0 . As in the proof of Theorem 5 in [13], we can show that {w α } is compact on Θ. We deduce that there exists a subsequence of {w α } converging to w in F(Θ).…”

Positive Solvability for Conjugate Fractional Differential Inclusion of (k, n − k) Type without Continuity and Compactness

“…It is worth mentioning that the literature on the existence and uniqueness of solutions to fractional differential equations is expanding at present, and this problem has drawn the attention of many contributors [19][20][21][22][23][24][25][26][27].…”