Impulsive fractional order integrodifferential equation via fractional operators

Abstract

In this paper, we establish the existence and uniqueness of the solution to fractional equations abstract integrodifferential equation with impulsive as d η w ( z ) d z η + ζ w ( z ) = f ( z , w ( z ) ) + K ( w ) ( z ) , z > z 0 , z ≠ z k , 0 < η ≤ 1 , Δ w | z = z k = I k ( w ( z k - ) ) , k = 1 , . . . , m , w ( z 0 ) + φ ( w ) = w 0 ∈ X , where K ( w ) ( z ) = ∫ z 0 z q ( z - s ) g ( s , w ( s ) ) d s , and we used the fixed point theorems due to Banach space. The main conclusion is obtained by using fractional calculus, operator semigroup and fixed point theorem. Finally, in support, an example is presented to validate the obtained results.