非线性二阶差分方程三点边值问题的研究

为了拓展非线性离散边值问题的基本理论,研究了一类非线性二阶差分方程三点边值问题正解存在性的充分条件。首先,给出了相应的二阶差分方程三点边值问题解的表达式并证明其性质;其次,在Banach空间中构造合适的锥和积分算子,运用锥上的Krasnoselskii’s不动点定理,在非线性项允许变号的条件下,获得非线性二阶差分方程三点边值问题正解存在性的充分条件;最后,通过2个例子证明主要定理和结果的有效性。结果表明,定理条件得证且离散边值问题满足正解的存在性。所研究的方法在二阶离散边值问题理论证明方面效果良好,对探究非线性高阶多点离散边值问题具有一定的借鉴意义。

Research of the three-point boundary value problems for nonlinear second-order difference equation

In order to extend the basic theory of nonlinear discrete boundary value problems,this paper studied the sufficient conditions for the existence of positive solutions for a class of nonlinear second-order difference equations with three-point boundary value problems.Firstly,the expressions of the solutions for the corresponding three-point boundary value problems for second-order difference equations were given and their properties were proved; Secondly,by constructing suitable cone and integral operator in Banach space and utilizing Krasnoselskii''s fixed point theorem in cones,the sufficient conditions for the existence of positive solutions of three-point boundary value problems for nonlinear second-order difference equations were obtained under the condition that the nonlinear term was allowed to change sign.Finally,two examples were given to illustrate the validity of the main theorems and results.The results show that the conditions of the theorem are proved and the discrete boundary value problems satisfies the existence condition of positive solutions.The method is effective in the theoretical proof of the second-order discrete boundary value problem,and has reference for the study of the nonlinear high-order multi-point discrete boundary value problems.