Nonlinear second-order periodic boundary value problems. (English) Zbl 0678.34022

Nieto, Juan J.

J. Math. Anal. Appl. 130, No. 1, 22-29 (1988).

For the nonlinear boundary value problem (BVP) with periodic boundary conditions \(u''+f(t,u)=0\), \(u(0)=u(2\pi)\), \(u'(0)=u'(2\pi)\), the existence of lower and upper solutions \(\alpha\) and \(\beta\) with \(\alpha\leq \beta\) on [0,2\(\pi\) ] guarantees the existence of a solution u(t) of the BVP with \(\alpha\leq u\leq \beta\). In this paper, the case of reversed lower and upper solutions \(\alpha\) and \(\beta\) with \(\alpha\geq \beta\) is considered. Examples are given to show that the existence of reversed upper and lower solutions will not imply the existence of a solution of the BVP in between, and indeed, the BVP may not have a solution. Conditions are given which imply the existence of solutions of the BVP when reversed lower and upper solutions exist.

Reviewer: L.Grimm

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
34A30	Linear ordinary differential equations and systems

Keywords:

lower solutions; periodic boundary conditions; Examples; upper solutions

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