Existence Of Solutions For Two Class Of Second-order Difference Equation Boundary Value Problems

Difference equation is applied to many fields.For example,it can be applied to modern medical, biology mathematics, ecology, physics, chemistry,and so on. So the study of them attracts many scholars. A variety of methods and techniques research on the problems of the differential equation boundary value on infinite interval attracts many scholars in recent year. but the problems of difference equation boundary value on infinite interval relative lacks enough theories and methods.Recently, the differential equation boundary value on infinite interval problems have been gotten lots of profound conclusions. Overall, but the difference equation boundary value on infinite interval problems have not been gotten findings,the theory is far from perfect, and in dire need of further research and discussion.The q-difference equations initiated in the beginning of the 20th century, is a very interesting field in difference equations.it has invoved into a multidisciplinary subject and plays an important role in several fields in physics,such as cosmic strings and black holes, conformal quantum mechanics, nuclear and high energy physics. However, the theory of boundary value problems (BVPs) for nonlinear q-difference equations is still in the initial stages and many aspects of this theory need to be explored. Henceforth, impulsive difference equations have gained considerable importance due to their application in various sciences, such as control theory, population dynamics and medicine and so on. To the best of our knowledge, the study of BVPs for nonlinear impulsive qk -difference equation with Sturm-Liouville type is yet to be initiated, and in dire need of further research and discussion.This article mainly talks about the existence of solutions for second order difference equation boundary value problems and Solvability for a Class impulsive difference equation.The paper is divided into four chapters, Main contents are as follow:the first chapter, we introduce the history background and development of boundary value problems for difference equations and impulsive q-difference equations boundary value problems. and set forth the purpose and meaning of this paper and the current situations of domestic and foreign investigations.the second chapter, we obtain the property by constructing its Green function which we obtain the existence of solutions for second-order three point difference equation boundary value problems on infinite intervals through the Leray-Schauder fixed point theorem.three chapter, we obtain the property by constructing its Green function which we obtain the existence of solutions for second-order three point difference equation boundary value problems on infinite intervals through the Leray-Schauder fixed point theorem.then give an example to illustrate our results. the forth chapter, we first recall some basic concepts of qk -derivative and qk -integral, then we obtaion the solvability for nonlinear second-order impulsive qk -difference equations through the Banach's and Schaefer'sfixed theorem, finally,we give an example to illustrate our results.