Multi-point boundary value problems of second-order differential equations. I
The authors study boundary value problem of second-order nonlinear differential equations with nonhomogeneous multipoint boundary conditions of the form \[ x''+f(t,x,x')=0,\quad t\in (0,1),\tag{\(*\)} \] \[ u'(0)-\sum_{i=1}^m a_iu'(t_i)=\lambda_1,\qquad u(1)-\sum_{i=1}^m b_iu(t_i)=\lambda_2. \tag{\(**\)} \] Results concerning the existence of solutions of \((*)\) and \((**)\) are established using the method of lower and upper solutions. Explicit ranges of values of \(\lambda_1\) and \(\lambda_2\) are determined for the existence of no solution, one solution and a positive solution. The plane of \(\lambda_1\) and \(\lambda_2\) is divided by a continuous curve \(\Gamma\) into two disjoint connected regions \(\Lambda_E\) and \(\Lambda_N\) such that the problem \((*)\) and \((**)\) has a solution for each \((\lambda_1,\lambda_2)\in \Lambda_E\) and has no solution for each \((\lambda_1,\lambda_2)\in \Lambda_N\). Examples are given to generate motivation.
scientific article; zbMATH DE number 2105455