一类非线性二阶系统周期边值问题正解的存在性

Existence of positive solutions for a class of periodic boundary value problems of nonlinear second-order systems

考察了非线性二阶系统周期边值问题u″+ A( t) u =ΛG( t) F( u),0 < t < 1,{u( 0)= u( 1),u'( 0)= u'( 1)正解的存在性,其中 u =( u1,…,un ) T,A( t)= diag[a1 ( t),…,an ( t)],ai ( t)可变号( i = 1,…,n),G( t)= diag[g1 ( t),…, gn ( t)],F( u)=( f1 ( u),…,fn ( u)) T,Λ= diag(λ1,…,λn ),λi 为正参数( i = 1,…,n)。在非线性项 F 满足超线性,次线性和渐近线性的条件下,本文运用锥拉伸与压缩不动点定理获得了该问题正解的存在性,所得结论推广和改进了已有的相关结果。

We consider the existence of positive solutions for the periodic boundary value problems of nonlinear second-order systems u″+ A( t) u =ΛG( t) F( u),0 < t < 1,{u( 0)= u( 1),u'( 0)= u'( 1) where u =( u1,…,un ) T,A( t)= diag[a1 ( t),…,an ( t)],ai ( t) can change the sign in[0,1]( i = 1,…,n),G( t)= diag[g1 ( t),…,gn ( t)],F( u)=( f1 ( u),…,fn ( u)) T,Λ= diag(λ1,…,λn ),λi is a positive parameter ( i = 1,…,n). Under the assumption that the nonlinear term F satisfies superlinear,sublinear and asymptotic growth condition,the existence of positive solutions of the problem are obtained by using the fixed-point theorem of cone expansion-compression. The conclusions in this paper generalize and improve the related results.