Abstract
In this paper, we establish the existence of piece wise (PC)-mild solutions (defined in Section 2) for non local fractional impulsive functional integro-differential equations with finite delay. The proofs are obtained using techniques of fixed point theorems, semi-group theory and generalized Bellman inequality. In this paper, we used the distributed characteristic operators to define a mild solution of the system. We also discussed the controversy related to the solution operator for the fractional order system using weak and strong Caputo derivatives. Examples are given to illustrate the theory.
Highlights
Fractional calculus has gained considerable popularity and importance during the past four decades because fractional derivatives provide an excellent tool for the description of the memory and hereditary properties of various processes
Modeling of the shape memory phenomena with this powerful tool is studied from different perspectives, as well as presenting physical interpretations
Motivated by the above mentioned paper, we study the existence of mild solutions for non local impulsive fractional semi-linear integro-differential equations of the form
Summary
Fractional calculus has gained considerable popularity and importance during the past four decades because fractional derivatives provide an excellent tool for the description of the memory and hereditary properties of various processes. The study of fractional differential equations has gained considerable importance due to their application in various fields including bio-engineering, mechanics, electrical networks, control theory of dynamical systems, viscoelasticity and so on. [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. Motivated by the above mentioned paper, we study the existence of mild solutions for non local impulsive fractional semi-linear integro-differential equations of the form.
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