## 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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### References:

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