The Existence And Hyers-Ulam Stability For Two Classes Of Nonlinear Caputo Fractional Differential Equations

Fractional calculus,especially fractional differential equation is an advantageous mathematical model used to describe properties of various processes and applications in many fields.Generally,fractional differential equations are derived from the study of aerodynamics,fluid flow,electrodynamics of complex media,thermal system,control theory,signal and image processing models.Therefore,the fractional differential equations are widely concerned and studied by many scholars.This paper is devoted to discussing a class of nonlinear Caputo-type fractional differential equations with two-point type boundary value conditions.The conclusions of existence and uniqueness for the boundary value problems of fractional differential equations are achieved by the relevant fixed point theorems.Additionally,the Hyers-Ulam stability of the fractional differential equations with the boundary value conditions y（0）+y（1）=yo is studied as well.The Hyers-Ulam stability of differential equations with initial value conditions is easy to prove.However,the situation is quite different when the restriction conditions are changed from the initial value problems to the boundary value problems.Due to the limitation of boundary value conditions,the research of the Hyers-Ulam stability becomes more complicated.Generally,we make use of the Laplace transform and the classical Gronwall inequality to verify the Hyers-Ulam stability of the equation system.In this article,it is difficult to give a concise proof of the Hyers-Ulam stability at the point t=1.Therefore,the existing Gronwall inequality is not sufficient to solve this kind of boundary value problem,which means that we need to find a suitable method to solve this type of question.In order to solve the Hyers-Ulam stability,we construct an novel integral-type Gronwall inequality,which can be considered as a generalization of the classical Gronwall inequality.Additionally,a class of nonlinear fractional differential equations with infinite delay is dicussed by us.Our approach is largely based on the alternative of Leray-Schauder and Banach fixed point theorem.Due to the characteristic of delay equations,we need to give the proper form of the solutions when discussing the existence and uniqueness,which is one of the key and difficult points to solve the problem.Generally,delay differential equations can be transformed into integral equations.Under the definition of phase space,the solutions of the integral equations can be appropriately extended,and the constructed equations are still continuous at the point t=1.What’s more,we further study the Hyers-Ulam stability of fractional differential equations with infinite delay.In this paper,we verify the Hyers-Ulam stability of delay differential equation by using the related properties of phase space,and obtain the stability conclusion by means of a class of Gronwall inequalities and a comparative property of fractional integrals.