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Nonresonance and existence for singular boundary-value problems. (English) Zbl 0827.34010
The author studies the problem of the existence of solutions of the differential equation \((1/p(t))\) \((p(t) y')'+ cy= q(t) f(t, y, py')\) for different types of singular boundary value problems (Sturm-Liouville, Neumann, periodic). Let \(\lambda_i\) be the eigenvalues of the associated linear second-order self-adjoint differential equation. The author addresses the interesting case when \(\lambda_k< c< \lambda_{k+ 1}\). The two main differences with respect to the known literature are that \(p\) is not assumed to be equal to 1 and that the nonlinear term depends also on the first derivative. The results obtained not only generalize known results to this new setting but also improve them. The existence is obtained in the framework of the nonlinear alternative of Leray-Schauder and by the properties of the spectrum of compact self-adjoint operator.
Reviewer: P.Zezza (Firenze)

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
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