带变号格林函数的四阶三点边值问题的多个正解的存在性

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国家自然科学基金项目

详细信息

摘要: 应用~Leggett-Williams~不动点定理研究了四阶三点边值问题 $u^{(4)}(t)=f(t,u(t))\quad ~(t\in [0,1]),$ $u'(0)=u''(\eta)=u'''(0)=u(1)=0$ 多个正解的存在性.~其中~ $f:[0,1]\times[0,+\infty )\rightarrow[0,+\infty)$ ~连续,~ $\eta\in[\frac{\sqrt{3}}{3},1]$ ~为常数.~尽管~Green~函数是变号的,~对任意的正整数~ $m,$ ~该问题~仍有正解且至少有~2 $m$ -1~个正解.
- 多个正解
Abstract: By applying Leggett-Williams fixed point theorem,the fourth-order three-point boundary value problem is studied: $u^{(4)}(t)=f(t,u(t))\quad ~(t\in [0,1]),$ $u'(0)=u''(\eta)=u'''(0)=u(1)=0,$ where~ $f:[0,1]\times[0,+\infty )\rightarrow[0,+\infty)$ ~is continuous,~ $\eta\in[\frac{\sqrt{3}}{3},1].$ ~The existence of at least~2 $m$ -1~positive solutions for arbitrary positive integer $m$ is obtained while the problem has the sign-changing Green's function.