一类非线性四阶常微分方程边值问题解的存在唯一性

Existence and uniqueness of solutions for a class of boundary value problems of nonlinear fourth-order ordinary differential equations

研究了非线性四阶常微分方程边值问题{u^(4)(t)=f(t,u(t),u′(t),u″(t)),a.e.t∈(0,1),u(0)=u″(0)=u(1)=u″(1)=0,{其中非线性项f:[0,1]×R 3→R为Carathéodory函数。运用Leray-Schauder原理,在f满足适当的至多线性增长性条件时,获得了该问题解的存在性。进一步,在f满足Lipschitz条件时,得到了该问题解的存在唯一性。

This article considers the existence and uniqueness of solutions of boundary value problems of nonlinear fourth-order ordi-nary differential equations {u^(4)(t)=f(t,u(t),u′(t),u″(t)),a.e.t∈(0,1),u(0)=u″(0)=u(1)=u″(1)=0,{where nonlinearity f:[0,1]×R 3→R is a Carathéodory function.The existence of solutions is obtained when f satisfies the condition of proper utmost linear growth by using the Leray-Schauder principle.Furthermore,the uniqueness of solutions is proved when f sat-isfies the Lipschitz condition.