二阶非线性常微分方程组边值问题的正解

Positive Solutions of Boundary Value Problems for Systems of Nonlinear Second Order Differential Equations

利用锥上的不动点指数理论,研究了二阶非线性常微分方程组边值问题:.在较为广泛的条件下,证明了边值问题正解的存在性和多解的存在性,改进和推广了文献[4]中的主要结果.主要创新之处是:非线性项既可以是超线性的和次线性的,也可以是混合非线性的(即在f和g中,一个是超线性的,另一个是次线性的).主要思路运用凹函数的有关性质和Jensen不等式对正解做先验估计.

In this paper, by using fixed point theory in a cone, we are concerned with the existence and multiplicity of positive solutions of boundary value problem for systems of nonlinear second order ordinary differential equations： {-u″=f（x,u,v）, -v″=g（x,u,v）, u（0）=u（1）=0, v（0）=v（1）=0. The existence and multiplicity of positive solutions of the above problem are proved under fairly general conditions. Therefore, the results in [4] are considerably improved and generalized.