带非线性边界条件的二阶奇异微分系统正解的存在性

研究了带非线性边界条件的二阶奇异微分系统边值问题-u″=ΛG(t)F(u),0<t<1,u(0)=0,u'(1)+C(u(1))u(1)=0正解的存在性,其中u=(u1,u2,?,un)T,G(t)=diag[g1(t),g2(t),?,gn(t)],且gi(t)(i=1,2,?,n)在t=0处允许有奇性F(u)=(f1(u),f2(u),?,fn(u))T,C=diag(c1,c2,?,cn),Λ=diag(λ1,λ2,?,λn),λi(i=1,2,?,n)为正参数。在非线性项F分别满足超线性、次线性和渐近线性的增长条件下,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结论。

Existence of positive solutions for boundary value problems of second-order systems with nonlinear boundary conditions

The existence of positive solutions for boundary value problems of second-order singular differentialsystem swith nonlinear boundary conditions -u″=ΛG(t)F(u),0<t<1,u(0)=0,u'(1)+C(u(1))u(1)=0 is studied,where u=(u1,u2,?,un)T,G(t)=diag[g1(t),g2(t),?,gn(t)],gi(t)(i=1,2,?,n), allows singularity at t=0,F(u)=(f1(u),f2(u),?,fn(u))T,C=diag(c1,c2,?,cn),Λ=diag(λ1,λ2,?,λn),λi(i=1,2,?,n) is a positive parameter. Under the condition that the nonlinearity term F satisfies superlinear, sublinear and asymptotically linear growth respectively, the existence of positive solutions of the problems is obtained by using the fixed-point theorem of cone expansion-compression.