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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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