Positive periodic solution to an indefinite singular equation. (English) Zbl 1476.34101
The paper is concerned with the singular equation \[ x'' + q(t)x = \frac{b(t)}{x^\rho} + e(t), \] where \(\rho > 0\) and \(q,b,e\) are \(\omega\)-periodic continuous functions, with \(q(t), e(t) > 0\) for all \(t\).
Under suitable assumptions on the Green’s function of the associated linear equation and on the nonlinear term, the existence of a positive \(\omega\)-periodic solution is proved by the use of a fixed point theorem in cones.
In principle, the result is applicable both in the case of weak singularities (that is, \(0 < \rho < 1\)) and strong singularities (\(\rho > 1\)). An example of application is provided.
Under suitable assumptions on the Green’s function of the associated linear equation and on the nonlinear term, the existence of a positive \(\omega\)-periodic solution is proved by the use of a fixed point theorem in cones.
In principle, the result is applicable both in the case of weak singularities (that is, \(0 < \rho < 1\)) and strong singularities (\(\rho > 1\)). An example of application is provided.
Reviewer: Alberto Boscaggin (Collegno)
MSC:
| 34C25 | Periodic solutions to ordinary differential equations |
| 34B27 | Green’s functions for ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
Keywords:
fixed point theorem in cones; indefinite singularity; Green’s function; positive periodic solutionReferences:
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