Second-order boundary value problems with nonhomogeneous boundary conditions. I. (English) Zbl 1060.34005

Summary: We study the nonlinear boundary value problem with nonhomogeneous multipoint boundary condition \[ \begin{gathered} u''+ f(t,u,u')= 0,\quad t\in (0,1),\\ u(0)- \sum^m_{i=1} a_i u(t_i)= \lambda_1,\quad u(1)- \sum^m_{i=1} b_i u(t_i)= \lambda_2.\end{gathered} \] Sufficient conditions are found for the existence of solutions of the problem based on the existence of lower and upper solutions with certain relation. Using this existence result, under some assumptions, we obtain explicit ranges of the values of \(\lambda_1\) and \(\lambda_2\) with which the problem has a solution, has a positive solution, and has no solution, respectively. Furthermore, we prove that the whole plane for \(\lambda_1\) and \(\lambda_2\) can be divided into two disjoint connected regions \(\Lambda^E\) and \(\Lambda^N\) such that the problem has a solution for \((\lambda_1,\lambda_2)\in \Lambda^E\) and has no solution for \((\lambda_1,\lambda_2)\in \Lambda^N\). We also show that under different assumptions, the problem has a solution for all \((\lambda_1,\lambda_2)\in \mathbb{R}^2\).


34B08 Parameter dependent boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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