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A General Result to the Existence of a Periodic Solution to an Indefinite Equation with a Weak Singularity

  • José Godoy
  • Manuel ZamoraEmail author
Article
  • 84 Downloads

Abstract

Efficient conditions guaranteeing the existence of a T-periodic solution to the second order differential equation
$$\begin{aligned} u''=\frac{h(t)}{u^{\lambda }},\quad \lambda \in (0,1), \end{aligned}$$
are established. Here, \(h\in L(\mathbb {R}/T\mathbb {Z})\) is a rather general sign-changing function with \(\overline{h}<0\). In contrast with the results in Godoy and Zamora (Proc R Soc Edinb Sect A Math) and Hakl and Zamora (J Differ Equ 263:451–469, 2017), the key ingredient to solve the aforementioned problem seems to be connected more with the oscillation and the symmetry aspects of the weight function h than with the multiplicity of its zeroes. Roughly speaking, the solvability for the above-mentioned problem can be guaranteed when \(H_+\approx H_-\) and \(H_+\) is large enough.

Keywords

Singular differential equations Weak-indefinite singularity Periodic solutions Degree theory Leray–Schauder continuation theorem 

Mathematics Subject Classification

34C25 34B16 47H11 

Notes

Acknowledgements

M. Zamora gratefully acknowledge support from FONDECYT, Project No. 11140203. J. Godoy was supported by a CONICYT fellowship (Chile) in the Program Doctorado en Matemática Aplicada, Universidad Del Bío-Bío No. 21161131.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Grupo de Investigación en Sistemas Dinámicos y Aplicaciones (GISDA), Departamento de MatemáticaUniversidad del Bío-BíoConcepciónChile
  2. 2.Departamento de Matemáticas, Grupo de Investigación en Sistemas Dinámicos y Aplicaciones (GISDA)Universidad de OviedoOviedoSpain

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