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Positive solutions and nonlinear multipoint conjugate eigenvalue problems. (English) Zbl 0888.34013

Summary: Values of \(\lambda\) are determined for which there exist solutions in a cone of the \(n\)-th order nonlinear differential equation, \(u^{(n)} = \lambda a(t) f(u)\), \(0 < t < 1\), satisfying the multipoint boundary conditions, \(u^{(j)}(a_i) = 0\), \(0 \leq j \leq n_i -1\), \(1 \leq i \leq k\), where \(0 = a_1 < a_2 < \cdots < a_k = 1\), and \(\sum _{i=1}^k n_i = n\), where \(a\) and \(f\) are nonnegative valued, and where both \(\lim_{|x|\to 0^+} f(x)/|x|\) and \(\lim_{|x|\to\infty} f(x)/|x|\) exist.

MSC:

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