1008-1542 2021 42 4 360 368 10.7535/hbkd.2021yx04006 article 非線性二階差分方程三點邊值問題的研究 Research of the three-point boundary value problems for nonlinear second-order difference equation 為了拓展非線性離散邊值問題的基本理論，研究了一類非線性二階差分方程三點邊值問題正解存在性的充分條件。首先，給出了相應的二階差分方程三點邊值問題解的表達式并證明其性質；其次，在Banach空間中構造合適的錐和積分算子，運用錐上的Krasnoselskii’s不動點定理，在非線性項允許變號的條件下，獲得非線性二階差分方程三點邊值問題正解存在性的充分條件；最后，通過2個例子證明主要定理和結果的有效性。結果表明，定理條件得證且離散邊值問題滿足正解的存在性。所研究的方法在二階離散邊值問題理論證明方面效果良好，對探究非線性高階多點離散邊值問題具有一定的借鑒意義。 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. 差分方程；離散邊值問題；不動點定理；錐；正解；存在性 difference equation; discrete boundary value problem; fixed point theorem; cone; positive solution; existence 魏文英,紀玉德,郭彥平 WEI Wenying, JI Yude, GUO Yanping hbkjdx/article/abstract/b202104006