二阶微分方程三点边值问题定号解的存在性

doi:10.6040/j.issn.1671-9352.0.2023.038

摘要/Abstract

摘要： 研究二阶微分方程三点边值问题{u″+a(t)f(u)=0,t∈［0,1］,u(0)=0, u(1)=u(ε)的定号解的存在性,其中ε∈(0,1), a∈C(［0,1］,(0,∞)), f∈C(R,R)且当s≠0时,sf(s)>0,λ₁为线性特征问题u″+λa(t)u=0, u(0)=0, u(1)=u(ε), t∈［0,1］的主特征值。当(λ₁)/(f_∞)<1<(λ₁)/(f₀)或(λ₁)/(f₀)<1<(λ₁)/(f_∞)时,问题至少存在一个正解u(t)和一个负解v(t)。主要结果的证明基于分歧理论。

关键词: 二阶微分方程, 三点边值问题, 格林函数, 分歧理论, 定号解

Abstract: In this paper, we study the existence of one-signed solutions for three-point boundary value problems of nonlinear second-order differential equations{u″+a(t)f(u)=0, t∈［0,1］,u(0)=0, u(1)=u(ε)where ε∈(0,1), a∈C(［0,1］,(0,∞)), f∈C(R,R)with sf(s)>0 for s≠0, λ1 is the principal eigenvalue of the linear eigenvalue problem: u″+λa(t)u=0, u(0)=0, u(1)=u(ε), t∈［0,1］. Assume that either(λ1)/(f_∞)<1<(λ1)/(f0) or (λ1)/(f0)<1<(λ1)/(f_∞), the problem has at least one positive solution u(t) and one negative solution v(t). The proof of main results is based on bifurcation techniques.

Key words: second-order differential equation, three-point boundary value problem, Greens function, bifurcation technique, one-signed solution

中图分类号:

O175.8

刘慧娟Symbol`@@. 二阶微分方程三点边值问题定号解的存在性[J]. 《山东大学学报(理学版)》, 2024, 59(12): 79-86.

LIU Huijuan. Existence of one-signed solutions for three-point boundary value problems of second-order differential equations[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2024, 59(12): 79-86.

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