Existence and multiplicity of positive solutions for fourth-order boundary value problems with a fully nonlinear term
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摘要:
本文讨论四阶两点边值问题 $\left\{ {\begin{array}{*{20}{l}} {{u^{(4)}}(t) = f(t,u(t),u'(t),u''(t),u'''(t)), t \in (0,1), }\\ {u(0) = u'(0) = u''(1) = u'''(1){\rm{ = 0}}. } \end{array}} \right.$ 这里非线性项 $f$ 中含有项$u'$ ,$u''$ 和$u'''$ , 因而该问题为带有完全非线性项的四阶边值问题. 运用Leggett-Williams型的两个不动点定理, 在$f$ 满足一定条件的情况下, 获得了该问题至少存在两个或者三个正解的结果. 最后举例验证了所获定理的有效性.-
关键词:
- 完全非线性项 /
- 多正解 /
- Leggett-Williams不动点定理
Abstract:In this paper, we discuss the fourth-order two-point boundary value problem $\left\{ {\begin{array}{*{20}{l}} {{u^{(4)}}(t) = f(t,u(t),u'(t),u''(t),u'''(t)),t \in (0,1), }\\ {u(0) = u'(0) = u''(1) = u'''(1){\rm{ = 0}}. } \end{array}} \right.$ Here, the nonlinear term $f$ contains$u'$ ,$u''$ and$u'''$ ; therefore, the problem is a fourth-order boundary value problem with a fully nonlinear term. By using the two fixed point theorems of Leggett-Williams type, the existence of at least two or at least three positive solutions are obtained under the term$f$ that satisfies certain conditions. Finally, two examples are given to verify the theorems. -
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