Existence of asymptotically periodic solutions of partial functional differential equations with state-dependent delay

ABSTRACT

In this paper, we are concerned with systems determined by partial differential equations with infinite and state-dependent delay. To study the system, we model the equation as an abstract retarded functional differential equations with infinite and state-dependent delay. We establish the existence of mild solutions, and the existence of S-asymptotically periodic solutions and asymptotically periodic solutions using local Lipschitz conditions on the involved functions. We apply our results to study the existence of asymptotically periodic solutions of a partial differential equation with infinite and state-dependent delay.

KEYWORDS:

• Retarded functional differential equations
• abstract functional differential equations
• equations with state-dependent delay
• mild solutions
• asymptotically periodic solutions

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary: 34K30
• 34K13
• Secondary: 34G20

Acknowledgments

The authors are grateful to the referees for the useful suggestions and also for bringing to our attention the papers [5–8].

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 We note that how $y\in B_R\left(0\right)$ then $\parallel y-0\parallel <R$. Taking any r>R we has $0\in B_r\left(y\right)$, then $B_r\left(y\right)\cap SA{P}_{\omega }\left(X\right)$ is non-empty, without r be much large than R.

Additional information

Funding

Claudio Cuevas is partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnologico CNPq-Brazil [grant number 407952/2016-0]. Hernán R. Henríquez is partially supported by CONICYT [grant number FONDECYT 1130144] and DICYT-USACH.