带有完全非线性项的四阶边值问题的多正解性

Existence and multiplicity of positive solutions for fourth-order boundary value problems with a fully nonlinear term

本文讨论四阶两点边值问题{u(4)(t)=f (t, u(t), u′(t), u″(t), u′′′(t)), t∈(0, 1),u(0)=u′(0)=u″(1)=u′′′(1)=0.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LeggettWilliams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.

In this paper, we discuss the fourth-order two-point boundary value problem { u(4)(t) = f(t, u(t), u′(t), u′′(t), u′′′(t)), t ∈(0, 1),u(0) = u′(0) = u′′(1) = u′′′(1)= 0.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.