Abstract
Herein, we investigated the controllability of a semilinear multi-valued differential equation with non-instantaneous impulses of order α∈(1,2), where the linear part is a strongly continuous cosine family without compactness. We did not assume any compactness conditions on either the semi-group, the multi-valued function, or the inverse of the controllability operator, which is different from the previous literature.
Highlights
Due to the multiple applications of fractional differential equations in science, many have authors studied various types of these applications, such as [1,2,3,4].The motivation for considering nonlocal Cauchy problems is the physical problems
In [5,6,7,8], there are many results concerning the existence of solutions of differential equations or inclusions with non-instantaneous impulses of fractional order γ ∈ (0, 1), while in [9,10,11], the authors considered second-order non-instantaneous impulsive differential equations
Many authors have investigated the existence of solutions for differential equations or inclusions of order γ ∈ (1, 2); for example, Li et al [12] considered an abstract Cauchy problem, He et al [13] treated with nonlocal fractional evolution inclusions, and Wang et al [14] generalized the work done by He et al [13] to a case when there are non-instantaneous impulses
Summary
Due to the multiple applications of fractional differential equations in science, many have authors studied various types of these applications, such as [1,2,3,4]. Many authors have investigated the existence of solutions for differential equations or inclusions of order γ ∈ (1, 2); for example, Li et al [12] considered an abstract Cauchy problem, He et al [13] treated with nonlocal fractional evolution inclusions, and Wang et al [14] generalized the work done by He et al [13] to a case when there are non-instantaneous impulses. Motivated by the works cited above, we prove, in this paper, without assuming that the semi-group {C(θ) : θ ∈ R} is compact or the multi-valued function F is Libschitz in the second variable or satisfies any condition involving a measure of non-compactness, and by using a fixed point theorem for weakly sequentially closed graph operators, the controllability of problem (1).
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