EXISTENCE AND STABILITY OF SOLUTION IN BANACH SPACE FOR AN IMPULSIVE SYSTEM INVOLVING ATANGANA–BALEANU AND CAPUTO–FABRIZIO DERIVATIVES

Abstract

This paper investigates the necessary conditions relating to the existence and uniqueness of solution to impulsive system fractional differential equation with a nonlinear p-Laplacian operator. Our problem is based on two kinds of fractional order derivatives. That is, Atangana–Baleanu–Caputo (ABC) fractional derivative and the Caputo–Fabrizio derivative. To achieve our main aims, we will first convert the proposed impulse system into an integral equation form. Next, we prove the existence and uniqueness of solutions with the help of Leray–Schauder’s theory and the Banach contraction principle. We analyze the operator for continuity, boundedness, and equicontinuity. Further, we investigate the stability solution to the proposed impulsive system by using stability techniques. In the last part, we demonstrate the results via an illustrative example for the application of the results.