Exsitence Of Positive Solutions For Boundary Value Problem Of Some Nonlinear Second-order Difference Equations

The existence theory of difference equation is one of important branch of difference equations. In the field of modern applied mathematics, it has made considerable headway in recent years, because all the structures of its emergence have deep physical background and realistic mathematical models. Many scholars take on the research of this field, they have achieved many good results. With the increasing development of science and technology, there are many problems relating to difference equation derived from lots of real applications and practice, In very resent years, great changes of this field have taken place. Especially, the second order nonlinear difference equation has been paid more attentions and investigated in various classes by using different methods(see [l]-[32]).The present paper employs fixed point theorem, monotone of functions to investigate some class of existence theoxy of difference equations, the results of which generalized and improved some known existence theory.The thesis is divided into five sections according to contents.In Chapter 1, Preface, we introduce the main contents of this paper.In Chapter 2, we mainly study the existence of positive solutions of the second-order nonlinear difference equation,satisfy boundary problem we mainly employed Krasnoselskii's fixed points theorem in a cone to gain one positive solutions of above difference equation.In Chapter 3, we mainly study the existence of positive solutions of the second-order nonlinear difference equation,satisfy boundary problemwhereφ_p(s) is a p-Laplacian operator,φ_p(s) andφ_q(s) are defined as above, and f:R⁺→R⁺ is continuous, a(t) is apositive valued funtion defined on [1:T+1].First, we mainly introduce the notation p(δ, d)(?)P(δ, d),(?) and a fixed points theorem.Second, we employ fixed points theorem to get the main resultIn Chapter 4, we study the following equationsatisfy boundary problemwhereWe give the following assumptions(H₁) f :R→[0, +∞) is a continuous function. (H₂) {a(k)} is a sequence of positive numbers.(H₃)φ_p(s) andφ_q(s) are defined as above, N is a positive integer,l∈{0,1,... N+1} is a constant.(H₄) B₀:R→R is continuous and satisfies that there areβ≥α≥0 suchthatαx≤B₀(x)≤β(x) for x∈R⁺.First, we mainly introduce some definitions and a fixed point theorem.Second, we employ the fixed points theorem to get the main result.In Chapter 5, we mainly study the following difference equationwhere...