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

  • José Godoy
  • Manuel ZamoraEmail author


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.


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

Mathematics Subject Classification

34C25 34B16 47H11 



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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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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