## 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

### Keywords:

multipoint; nonlinear eigenvalue problem; cone
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