Existence and Controllability Results for Fractional Impulsive Integrodifferential Systems in Banach Spaces

Abstract

We firstly study the existence of PC-mild solutions for impulsive fractional semilinear integrodifferential equations and then present controllability results for fractional impulsive integrodifferential systems in Banach spaces. The method we adopt is based on fixed point theorem, semigroup theory, and generalized Bellman inequality. The results obtained in this paper improve and extend some known results. At last, an example is presented to demonstrate the applications of our main results.

Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces

“…Our work generalizes the work done in Ref. [29] with abstract formulation. According to our knowledge this is an untreated article in the literature.…”

“…Recently, in Ref. [29], the authors studied the existence of mild solutions for impulsive fractional semi-linear integro-differential equations using the Banach contraction principle and Schaefer's fixed point theorem. They have considered the system without delay and without non local condition.…”

Existence of Mild Solutions and Controllability of Fractional Impulsive Integrodifferential Systems with Nonlocal Conditions

“…For instance, the diffusion phenomenon of a small amount of gas in a transparent tube can be given a better description than using the usual local Cauchy problem. On the other hand, controllability of nonlocal problems in Banach spaces has become an active area of investigation; we refer the reader to, for example, the papers [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. The most common method is to transform the controllability problem into a fixedpoint problem of solutions for an appropriate operator in a function space, that is, the existence problem of differential and integrodifferential equations.…”

Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces

“…For the importance of non local conditions in different fields, we refer the reader to [9,11,29,44]. Recently in [33], the authors studied the existence of mild solutions for impulsive fractional semi-linear integro-differential equations using Banach contraction principle and Schaefer's fixed point theorem. They have considered the system without delay and without non local condition.…”