Existence, Uniqueness and Stability Analysis of Switched Coupled Systems of Fractional Differential Equations

Abstract

The analysis of existence, uniqueness and stability of nonlinear fractional differential equations is an important branch of qualitative theory of differential equations. The main aim of this thesis is to analyze the existence, uniqueness and to investigate various breeds of Ulam–Hyers stability results related to the solutions of nonlinear implicit coupled systems of fractional differential equations with the service of fixed point theorems, over different Banach spaces. Additionally, we investigate the said analysis for some nonlinear fractional differential equations consisting of different kinds of fractional order derivatives. In the beginning, we deal with the existence, uniqueness and Ulam–Hyers stability of a nonlin ear implicit coupled system of fractional differential equations involving p-Laplacian operator with singularity. First we present a standard framework to stem a formula of solutions to our suggested model and then implement the concepts of Ulam–Hyers stability. Secondly, we consider a non linear coupled system of fractional differential equations involving Hilfer-Hadamard and ψ-Hilfer fractional order derivatives. Few sufficient conditions are constructed to observe the existence, uniqueness and Ulam–Hyers stability of our proposed models, with the use of Banach contraction principle, Krasnoselskii’s fixed point theorem, over Banach space. We also investigate a nonlocal neutral fractional differential equation system involving different types of fractional order deriva tioves for the existence, uniqueness and Ulam–Hyres–Mittag–Lefler stability with the help of Ba nach contraction principle, Schaefer’s and Schauder’s fixed point approaches. We also investigate an implicit coupled system of fractional differential equations consisting of Katugampola–Caputo fractional order derivative for the existence, uniqueness and Ulam–Hyers stability. Furthermore, we provide examples to exhibit the preeminent results. Besides, we study the above properties to the solutions of nonlocal impulsive ψ-Hilfer neutral stochastic fractional differential equations and a system of fractional differential equations consisting of conformable fractional order derivative.