中山大学学报自然科学版 ›› 2017, Vol. 56 ›› Issue (1): 66-73.

• 研究论文 • 上一篇    下一篇

离散Φ-Laplace问题的正解

李延明1,白定勇2   

  1. 1. 兰州石化职业技术学院信息处理与控制工程系, 甘肃 兰州 730060;
    2. 广州大学数学与信息科学学院, 广东 广州 510006
  • 收稿日期:2016-05-18 出版日期:2017-01-25 发布日期:2017-01-25
  • 通讯作者: 白定勇(1972年生),男;研究方向:泛函微分方程;Email:baidy@gzhu.edu.cn

Positive solutions of discrete Φ-Laplacian problems

LI Yanming1, BAI Dingyong2   

  1. 1. Department of Information Processing and Control Engineering,Lanzhou Petrochemical College of Vocational Technology, Lanzhou 730060, China;
    2. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
  • Received:2016-05-18 Online:2017-01-25 Published:2017-01-25

摘要:

考虑如下离散的Φ-Laplace边值问题〖JB({〗Δ((Δu(k-1)))+λp(k)f(u(k))=0,k∈{1,2,…,T},u(0)=u(T+1)=0〖JB)〗 其中, T>5是给定的正整数, λ是正参数, 是从〖WTHZ〗R〖WTBZ〗到〖WTHZ〗R〖WTBZ〗上的单调递增的奇同胚映射。 在较弱的条件liminf〖DD(X〗u→∞〖DD)〗〖SX(〗f(u)〖〗(u)〖SX)〗∈(0,∞]下,利用锥上的不动点定理,建立了问题至少存在一个正解的结果, 并给出了参数λ的显式开区间。

关键词: 离散Φ-Laplace问题, 正解, 半正问题

Abstract:

The discrete Φ-Laplacian boundary problem 〖JB({〗〖HL(1〗Δ((Δu(k-1)))+λp(k)f(u(k))=0,k∈{1,2,…,T},u(0)=u(T+1)=0〓〓〓〓〓〓〓〓〖HL)〗〖JB)〗  is studied, where T>5 is a given positive integer, λ is a nonnegative parameter and Φ is an odd and increasing homeomorphism from 〖WTHZ〗R〖WTBZ〗 onto 〖WTHZ〗R〖WTBZ〗. Applying the fixed point theorem in cones, it is proved under the weakened condition liminf〖DD(X〗u→∞〖DD)〗〖SX(〗f(u)〖〗(u)〖SX)〗∈(0,∞] that the problem has at least one positive solution for λ belonging to an explicit open interval.

Key words: discrete Φ-Laplacian problem, positive solution, semipositone problem

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