Existence And Stability Of Solutions Of Stieltjes Differential Equations

Stieltjes differential equation is suitable for describing the discontinuous differential equation at the right end and impulsive differential equation.It is widely used in biology and physics.In this paper,we mainly study the existence,continuous dependence and stability of solutions of Stieltjes differential equations.Firstly,the result of uniform convergence of g-function and continuity of limit func-tion,the sufficient conditions of commutative limit and integral of monotone function,and the properties of g-exponential function corresponding to linear Stieltjes differential equation are given.Secondly,we study the nonlinear Cauchy problem.By choosing workspace BC.q([O,H],R)and using the thought of the existence and uniqueness theo-rem of classical ordinary differential equation for reference,we construct the sequence of Picard successive approximation function,and obtain the existence and uniqueness of the solution under Lipschitz condition.Furthermore,under the condition of growth,using Schauder fixed point theorem,we obtain the existence result of Caratheodory solution.At the same time,we prove the continuous dependence of solution on the initial value.Thirdly,the nonlinear nonlocal problems are studied,and the existence and uniqueness of Caratheodory solutions are obtained by using the principle of com-pressed image in the sense of different norms.Fourth,select workspace ACg([O,H],R)and SCg([O,H],R),the reasonable definitions of Ulam-Hyers(UH)stability and gener-alized Ulam-Hyers-Rassias(UHR)stability for linear and nonlinear Stieltjes differential equations are given,and the UH stability and generalized UHR stability of linear and nonlinear Stieltjes differential equations are obtained by using techniques such as of di-rect estimation and Lebesgue-Stieltjes integral inequality.Finally,a numerical example is given to verify the validity of the above theoretical results.