## Existence of solutions of nonlinear $$m$$-point boundary-value problems.(English)Zbl 0988.34009

Let $$\xi_i \in (0,1)$$, and let $$a_i, b_i, i=1,\dots ,m-2$$, be positive numbers satisfying the conditions $$\sum_{i=1}^{m-2} a_i < 1$$, $$\sum_{i=1}^{m-2} b_i < 1$$. Here, the existence of a positive solution to the multipoint boundary value problem $u''(t) + a(t) f(u) = 0, \quad t\in (0,1), \qquad u'(0) = \sum_{i=1}^{m-2}b_i u'(\xi_i), \quad u(1)=\sum_{i=1}^{m-2} a_i u(\xi_i),$ is studied. By using a fixed-point theorem, the existence of a solution is proved if $$f(u)$$ is either superlinear or sublinear.

### 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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