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

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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.

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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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Correspondence to Manuel Zamora.

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Godoy, J., Zamora, M. A General Result to the Existence of a Periodic Solution to an Indefinite Equation with a Weak Singularity. J Dyn Diff Equat 31, 451–468 (2019). https://doi.org/10.1007/s10884-018-9704-9

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  • DOI: https://doi.org/10.1007/s10884-018-9704-9

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