Topological Structure of the Solution Sets for Impulsive Fractional Neutral Differential Inclusions with Delay and Generated by a Non-Compact Demi Group

Abstract

In this paper, we give an affirmative answer to a question about the sufficient conditions which ensure that the set of mild solutions for a fractional impulsive neutral differential inclusion with state-dependent delay, generated by a non-compact semi-group, are not empty compact and an Rδ-set. This means that the solution set may not be a singleton, but it has the same homology group as a one-point space from the point of view of algebraic topology. In fact, we demonstrate that the solution set is an intersection of a decreasing sequence of non-empty compact and contractible sets. Up to now, proving that the solution set for fractional impulsive neutral semilinear differential inclusions in the presence of impulses and delay and generated by a non-compact semigroup is an Rδ-set has not been considered in the literature. Since fractional differential equations have many applications in various fields such as physics and engineering, the aim of our work is important. Two illustrative examples are given to clarify the wide applicability of our results.

Topological Properties of Solution Sets for τ-Fractional Non-Instantaneous Impulsive Semi-Linear Differential Inclusions with Infinite Delay

“…Our work generalizes what was conducted by Wang et al [31], in which Problem (1) was considered without delay (ρ(w, x w ) = 0) and τ(t) = t, ∀t ∈ J, and by Alsheekhhussain et al [30], in which a similar type for Problem (1) was considered in special cases where τ(t) = t, ∀t ∈ J, ρ(w, x w ) = w with finite delay. Moreover, this work generalizes Theorem 4.1 in [44] when the right-hand side is a multi-valued function in the presence of both non-instantaneous impulses and infinite delay.…”

“…Therefore, it is useful and interesting to investigate the topological structure of this set. Many researchers have performed this for different types of differential inclusions, proving that it is an R δ -set and homotopically equivalent to a point (see, for instance, [25,26,28,[30][31][32][33][36][37][38][39][40][41]). None of these works addressed the topological properties of the mild solution set for non-instantaneous impulsive semi-linear differential inclusions involving a τ-Caputo fractional derivative with infinite delay in infinite-dimensional Banach spaces.…”

“…Therefore, this topic is very important and is reasonable and practical to study. Among these studies we mention the following: DeBlasi [23], Papageorgiou [24] and Zhou et al [25] considered differential inclusions; Gabor et al [26], Djebali et al [27], Zhang et al [28] and Ma et al [29] studied IDIs; Alsheekhhussain et al [30] and Wang et al [31] looked at semi-linear FDIs with non-instantaneous impulses; and Zhou et al [32,33] considered fractional stochastic differential inclusions.…”

“…It is worth noting that Alsheekhhussain et al [30] recently demonstrated that the mild solution set for a similar type for Problem (1) is non-empty and an R δ -set in the special cases where τ(t) = t; t ∈ J, ρ(w, x w ) = w and the delay is finite.…”

“…-This work is novel and interesting because the linear part is an operator that generates a non-compact semi-group, the non-linear part is a multi-valued function, and the studied problem contains the τ-Caputo derivative with non-instantaneous impulses and infinite delay. -Our technique helps any researcher interested in extending the results in [23][24][25][26][27][28][29][30]32,33] to cases where the right-hand side is a multi-valued function in the presence of both non-instantaneous impulses and infinite delay, while the left-hand side contains the τ-Caputo derivative.…”