On the nonlocal noninstataneous Impulsive Integro-differential equations in the frame of a fractional Caputo type derivative

Document Type : Regular research papers

Authors

1 Department of Mathematics, Gaston Berger University, Sénégal

2 Department of Mathematics, Gaston Berger University

3 Institute of Mathematics and Physical Sciences, URMPM P.O. Box 613, Porto Novo, Benin

4 Faculty of Sciences Semlalia, Department of Mathematics, Cadi Ayyad University, Marrakesh, Morocco.

Abstract

Study of evolution equations subjected to impulsive action, which starts abruptly and stays active on a finite time interval, has been a subject of interest in the last few years due to its applicability in practical problems. The pioneering work of such a model, known as non-instantaneous impulsive differential equation, is reported in the work of Hernandez and O’Regan \cite{Hernan}. The concept of resolvent operator is precisely relevant to dealing
integro-differential equations in Banach spaces. This strategy
has been utilized to a substantial type of nonlinear differential equations in Banach spaces.
For more points of interest on this concept, we refer to Grimmer\cite{R.GRIMMER}.
In this work, we investigate the existence of mild solutions for semilinear integro-differential systems in Banach spaces. Using M\"onch's fixed point theorem, the theory of Grimmer's resolvent operator and the measures of non compactness, we prove the main results of this work. At the end, an example is given to further illustrate the conclusions drawn from the theoretical study.

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