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A generalized multi-point boundary value problem for second order ordinary differential equations. (English) Zbl 0910.34032
The author deals with the existence of a solution to generalized multipoint boundary value problems \[ x''(t)=f \bigl(t,x(t), x'(t)\bigr) +e(t),\;0<t<1, \] \[ x(0)=\sum^{m-2}_{i=1} a_i x(\xi_i), \quad x(1)= \sum^{n-2}_{j=1} b_jx (\tau_j) \] and \[ x''(t)= f\bigl(t,x(t), x'(t)\bigr) +e(t), \quad 0<t <1, \] \[ x(0)= \sum^{m-2}_{i=1} a_ix(\xi_i), \quad x'(1)= \sum^{n-2}_{j=1} b_jx'(\tau_j), \] where \(f:[0,1] \times \mathbb{R}^2\to \mathbb{R}\) satisfies Carathéodory conditions, \(e\in L^1[0,1]\), \(a_i,b_j\in\mathbb{R}\), \(1\leq i\leq m-2\), \(1\leq j\leq n-2\), \(0<\xi_1<\xi_2< \cdots <\xi_{m-2}<1\), \(0<\tau_1 <\tau_2< \cdots <\tau_{n -2} <1\). The present work improves an earlier one due to the author et al. when all \(a_i\)’s have the same sign and all \(b_j\)’s have the same sign. In this case all \(a_i\)’s and \(b_j\)’s do not necessarily have the same sign (nonresonance case).

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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