A necessary and sufficient condition of positive solutions for nonlinear singular differential systems with four-point boundary conditions

作者：Zhang, Xinguang^*; Liu, Lishan

来源：Applied Mathematics and Computation, 2010, 215(10): 3501-3508.

DOI：10.1016/j.amc.2009.10.044

• Necessary and sufficient condition
• Positive solutions
• Differential systems
• Schauder fixed point theorem

In this paper, we concert with the existence of positive solution for the following nonlinear singular differential system with four-point boundary conditions
{-x '' = f(t, y), -y '' = g(t, x),
ax(0) - beta x'(0) = delta x(1) + gamma x'(1) = 0, y(0) = ay(xi(1)), y(1) = by(xi(2)),
where 0 < xi(1) < xi(2) < 1, alpha, beta, delta, gamma, a, b are nonnegative constants such that rho = beta gamma + alpha gamma + alpha delta > 0. By structuring upper and lower solution and using Schauder fixed point theorem, a necessary and sufficient condition for the existence of positive solutions is established. An example is worked out to illustrate our main result.

• 出版日期2010-1-15
• 单位曲阜师范大学; 烟台大学