QUALITATIVE ANALYSIS OF IMPLICIT DELAY MITTAG-LEFFLER-TYPE FRACTIONAL DIFFERENTIAL EQUATIONS

Abstract

This research work is devoted to endeavor some results for a delay implicit impulsive type problem under Atangana-Baleanu fractional derivative. The concerned derivative utilizes a nonlocal and non-singular kernel. We build some hypotheses to prove our results. We use Banach and Krasnoselskii fixed point theorems to derive the required results. We consider the following problem involving nonlocal and non-singular fractional derivative with delay term: {D-ABC(upsilon)(sic)(zeta) = Phi(zeta)(sic)(zeta) + Omega(zeta, (sic)(zeta - tau), D-ABC(upsilon)(sic)(zeta)), zeta is an element of Delta = [0, theta], 0 < v <= 1, (sic)(0) = (sic)0 +integral(0)(theta - psi)(upsilon-1)/Gamma(upsilon)Lambda((sic)(psi))d psi, (1) here 0 < upsilon, tau <= 1, represent the order of the derivative lambda, Phi : Delta -> R is bounded linear operator and Omega : Delta x R -> R shows a nonlinear continuous function. Stability theory of Ulam-Hyers is used to established the stability results. We provide some examples to demonstrate our theoretical findings.