A multiplicity result for a periodic boundary value problem. (English) Zbl 0859.34016

Rudolf, Boris

Nonlinear Anal., Theory Methods Appl. 28, No. 1, 137-144 (1997).

The author considers the differential equation \[ x''+f(x) \text{ sgn}(x')|x'|^m+ g(t,x)=s(t) \] with periodic boundary conditions on \(I=[0,2\pi]\), where \(f\), \(g\), \(s\) are continuous functions. He proves a multiplicity result of Abrosetti-Prodi type and presents the conditions under which the above problem has at least three solutions. The method is based on the existence of lower and upper solutions and the topological degree.
In the case of \(g\) unbounded, decreasing in \(x\) at infinity the results are obtained under the assumptions \(0<m\leq 2\) and \(g(t,x_1)< g(t,x_2)\) for some \(x_1<x_2\). In the case of \(g\) unbounded, increasing in \(x\) at infinity the author assumes \(m\leq 1\) and \(g(t,x_1)> g(t,x_2)\) for some \(x_1<x_2\). Some examples are given.

Reviewer: I.Rachůnková (Olomouc)

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
34C25	Periodic solutions to ordinary differential equations

Keywords:

multiplicity result; least three solutions; lower and upper solutions; topological degree

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