The Existence Of Positive Solutions For The Difference Equations Boundary Value Problems With Sum Forms Boundary Conditions

Difference equations are mathematical models which have been used to describe thechanging process on the discrete time in the actual world. It is applied to many fields. Forexample, it can be applied to Economical and financial insurance, the number ofpopulation structure analysis, the prevention and control of disease and pests, the study ofgenetic development, and so on. As long as it is about the variable regulation andproperties, we can use the difference equation model to perform and analysis it. That is tosay, although the mathematical model is continuous in the actual world, it always need toswitch the continuous variable to be discrete variable under some certain conditions, andthen the continuous model is transformed into discrete model. At last, all of it can comedown to study the problem which is about the solutions to discrete model of differenceequations. As is known to all, difference equations are the discrimination of differentialequations. We may be able to work out an accurate solution of a differential equation, butif it is transformed into a difference equation, we can work out an approximate solution.So the study of qualitative theory of discrete system attracts many scholars in recent years.And the research of difference equation boundary value problem becomes moremeaningful and valuable.This article mainly talks about the existence of three positive solutions for m-pointboundary value problem with P-Laplacian and the existence of symmetric positivesolutions for a boundary value problem with summation boundary conditions. The paper isdivided into three chapters. In the first chapter, we introduce the scientific significance andapplication prospect of difference equations and boundary value problems. At the sametime we recommend the development history of the differential equation boundary valueproblem, and the research status of difference equation boundary value problem at homeand abroad in recent years. In the second chapter, using the Avery-Peterson fixed pointtheorem and giving some sufficient conditions, we get that the m-point discrete boundaryvalue problem which with P-Laplacian has at least three positive solutions. In the thirdchapter, we apply the H lder inequality and the fixed point theorem to establish theexistence of symmetric positive solutions for the second order difference equationboundary value problem. In the fourth chapter, making use of expansion and compressionfixed point theorem in a cone, H lder inequality and the properties of Green’s function,we prove conditions under which there exist positive solutions for the fourth order difference equation boundary value problem.