一类二阶非线性微分系统正解的存在性及多解性  

Existence and multiplicity of positive solutions for a class of second-order nonlinear differential systems

作　　者：程虎文 CHENG Huwen(College of Mathematics and Statistics,Northwest Normal University,Lanzhou 730070,China)

机构地区：[1]西北师范大学数学与统计学院,甘肃兰州730070

出　　处：《厦门大学学报(自然科学版)》2025年第2期361-366,共6页Journal of Xiamen University:Natural Science

基　　金：国家自然科学基金(12061064)。

摘　　要：[目的]考察二阶非线性微分系统u″+θh 1(t)f(t,u,v,u′,v′)=0,t∈(0,1),v″+μh 2(t)ɡ(t,u,v,u′,v′)=0,t∈(0,1),u(0)=u(1)=v(0)=v(1)=0正解的存在性和多解性,其中θ,μ为正参数,f,ɡ:[0,1]×[0,∞)^(2)×R^(2)→[0,∞)是连续函数.[方法]主要结果的证明基于锥上的不动点定理.[结果]当f,ɡ满足适当条件时,存在正数λ_(1),λ_(2),若θ,μ≥λ_(1)或θ,μ≤λ_(2),则该问题有两个解.[结论]当微分系统中的非线性项包含增长不受限制的一阶导数项时,可以通过构造特殊的锥从而获得正解的存在性和多解性.[Objective]In this study,we consider the existence and the multiplicity of positive solutions for second-order nonlinear systems given as u″+θh 1(t)f(t,u,v,u′,v′)=0,t∈(0,1),v″+μh 2(t)ɡ(t,u,v,u′,v′)=0,t∈(0,1),u(0)=u(1)=v(0)=v(1)=0 whereθandμdenote positive parameters;f,ɡ:[0,1]×[0,∞)^(2)×R^(2)→[0,∞)denote continuous functions.[Methods]The proof is based on the fixed point theorem on cones.[Results]When f andɡsatisfy suitable conditions,positive numbersλ_(1) andλ_(2) exist.Ifθ,μ≥λ_(1) orθ,μ≤λ_(2),then the problem secures two solutions.[Conclusion]When the nonlinear term in the differential system contains a first-order derivative term with unrestricted growth,the existence and the multiplicity of positive solutions can be obtained by constructing a special cone.

关 键 词：正解 多解 不动点定理 DIRICHLET边界条件 

分 类 号：O175.8[理学—数学]