Stability of nonlinear impulsive higher order differential – fractional integral delay equations with nonlocal initial conditions

Abstract

The aim of this paper is to investigate some types of stability such as generalized Hyers-Ulam- Rassias stability(G-H-U-R-stabile) and the relation with Hyers-Ulam(H-U-stabile) stable and Hyers-Ulam-Rassias stable (H-U-R-stabile) and generalized Hyers-Ulam stable (G-H-U- stable) to obtain which one guarantee to satisfy stability of equations included a nonlinear function some of them contains a delay time of solution and the other contain a vector of different order of derivatives for the solution to n-time and vector of fractional order of integrals with different fractional orders and that was the for using a claculse of fractional calculus to satisfies the issue of this techniques. Moreover, the nonlocal initial values for the proposal equation of nonlinear impulsive higher order differential – fractional integral delay time equations which are adding more interesting for nonlinear analytic object of nonlinear higher order integro – fractional order impulsive classes, and the impulsive difference of the equation has some necessary conditions to prove the results of solution to be stable with certain type has related with other types. The necessary and sufficient conditions which assumed on this nonlinear higher order integro-differential impulsive equation have been achieved the stability with interesting certain estimates obtain through the proving technique. Also the uniqueness of solution has been studied with same conditions was presented for stability and used for that issue a contraction fixed point theorem.