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Periodic solutions for a second-order differential equation with indefinite weak singularity

Part of: Boundary value problems Qualitative theory

Published online by Cambridge University Press:  15 January 2019

José Godoy  and
Manuel Zamora
José Godoy
Affiliation:
Departamento de Matemática, Grupo de investigación en Sistemas Dinámicos y Aplicaciones (GISDA), Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile (jgodoy@ubiobio.cl; mzamora@ubiobio.cl)
Manuel Zamora
Affiliation:
Departamento de Matemática, Grupo de investigación en Sistemas Dinámicos y Aplicaciones (GISDA), Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile (jgodoy@ubiobio.cl; mzamora@ubiobio.cl)
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Abstract

As a consequence of the main result of this paper efficient conditions guaranteeing the existence of a T −periodic solution to the second-order differential equation

$${u}^{\prime \prime} = \displaystyle{{h(t)} \over {u^\lambda }}$$
are established. Here, h ∈ L(ℝ/Tℤ) is a piecewise-constant sign-changing function and the non-linear term presents a weak singularity at 0 (i.e. λ ∈ (0, 1)).

Keywords

Singular differential equationsweak-indefinite singularityperiodic solutionsdegree theoryLeray-Schauder continuation theorem

MSC classification

Primary: 34C25: Periodic solutions
Secondary: 34B18: Positive solutions of nonlinear boundary value problems 34B30: Special equations (Mathieu, Hill, Bessel, etc.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 5 , October 2019 , pp. 1135 - 1152
Copyright
Copyright © Royal Society of Edinburgh 2019 

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