Ulam Stability Of N-th Order Delay Nonlinear Differential Equation

Differential equation plays an important role in the development of engineering and physics.As an important problem in the study of differential equations,the stability theory of differential equation is a hot issue.Different from the traditional and classical Lyapunov stability,the problem of Ulam stability is more simple.It mainly considers the degree of approximation between the exact solution of the given equation and the solution of the approximate function equation under the given conditions.In this paper,we mainly study the Ulam stability of two kinds of delay nonlinear equations,including the Ulam stability of n-th order delay nonlinear integro-differential equation and the Ulam stability of n-th order delay nonlinear differential equation.There are four chapters in this paper.The structure is as follows:In Chapter 1,firstly,the research significance of Ulam stability of the following n-th order delay integro-differential equation is given,(?) where u(j)=u(j)(s),u(j)(?)=u(j)(?(s)),j=0,…,n-1.Then,the research significance of using disconjugate differential operators to study Ulam stability of the following n-th order delay differential equation is also discussed,(?) where (?),n?2 is disconjuage differential operator;Lju=Lju(s),Lju(?)=Lju(?(s)),j=0,…,n-1.In Chapter 2,we introduce preliminary knowledge,in the basic definition part,the solutions of the above two kinds of differential equations are given in detail,in the basic lemma part,we give some inequality lemmas which are needed in the main theorem proving part of Chapter 3 and Chapter 4.In Chapter 3,we introduce the basic definition and main theorems of Ulam stability for n-th order delay nonlinear integro-differential equations.In the basic definition part,the definitions of Hyers-Ulam stability and Hyers-Ulam-Rassias stability of n-th order de-lay nonlinear integro-differential equations are given in detail.Then,in the main theorem part,we give the conditions,when the n-th order delay nonlinear integro-differential equa-tion satisfies these conditions,it has Hyers-Ulam stability and Hyers-Ulam-Rassias stability.Theorem proving mainly uses the inequality result of Fanwei Meng in[32],it simplifies the research process.In Chapter 4,we introduce the basic definition and main theorems of Ulam stability for n-th order delay nonlinear differential equations.In the basic definition part,the definitions of Hyers-Ulam stability and Hyers-Ulam-Rassias stability of n-th order delay nonlinear dif-ferential equations are given.Then,in the main theorem part,we introduce the conditions,when the n-th order delay nonlinear differential equation satisfies these conditions,it has Hyers-Ulam stability and Hyers-Ulam-Rassias stability.In theorem proving part,by using the expression of the solution of the disconjugate differential equation given by Fanwei Meng in[4],it simplifies the research process.