The Upper And Lower Solutions Method For Discrete Boundary Value Problem

In recent years, the difference equation with boundary value problems has attracted many researchers due to its application to many fields, such as biological, physical and mechanical. Especially, with the rapid development of computer technology, the fitting of discrete data promotes the theoretical analysis and application process of discrete equation.In this paper, the existence of solutions to second order discrete equation with Sturm-Liouville bvp has been investigated on the basic of the upper and lower solutions method and fixed point theorem, critical point theory etc. It has four parts.In section 1, a brief introduction is presented for the history of the upper and lower solutions method, as well as the present work on difference equation using such meth-ods. We then propose the problem discussed in this thesis and illustrate the theoretical significance of our work.In section 2, we investigate a bvp of second order difference equation where â–³xk=xk+1-xk is the forward difference operator. N={1,2, …,∞}, and f:N × R2â†' R is continuous, a> 0, B,C ∈R, â–³x∞= limkâ†'∞ â–³xk. Firstly, we introduce a new Banach space. Then the discrete Areza-Ascoli lemma is established. Lastly, we will show that in the presence of a pair of upper and lower solutions, the boundary value problem has at least one solution by applying the Schauder fixed point theorem. The solution is admitted to be unbounded, which is an interesting point.In section 3, we consider a Sturm-Liouville boundary value problem of second order difference equation where [1, T]={1,2.....T}, α,γ> 0, β,δ≥0,f:[0, T+1] × Râ†' R is continuous. We construct the variational framework and correspond the solvability of the problem to the existence of critical points of the functional. The existence of four solutions are obtained by applying lower and upper solutions and multi-critical point theorems.The conclusions of the work done here and expectations of next study are provided in the last.