Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations

Jun Zhou; Jun Shen

doi:10.3934/dcdsb.2021198

Article Contents

2022, Volume 27, Issue 7: 3605-3624. Doi: 10.3934/dcdsb.2021198

This issue Previous Article Practical partial stability of time-varying systems Next Article Positive solutions of a diffusive two competitive species model with saturation

Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations

Jun Zhou^1, and
Jun Shen^2, ,

1.
College of Mathematical Sciences, Sichuan Normal University, Chengdu, Sichuan 610066, China
2.
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

^* Corresponding author: Jun Shen, junshen85@163.com
^* Corresponding author: Jun Shen, junshen85@163.com

Received: April 2021

Revised: June 2021

Early access: August 2021

Published: July 2022

This work was supported by NSFC #11701400, #11831012, #12090013 and #12071317, and Sichuan Science and Technology Program #2020YJ0328.

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Abstract

In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for the general iterative functional differential equation $ \dot{x}(t) = f(t,x(t),x^{[2]}(t),...,x^{[n]}(t)). $ As $ n = 2 $, this equation can be regarded as a mixed-type functional differential equation with state-dependence $ \dot{x}(t) = f(t,x(t),x(T(t,x(t)))) $ of a special form but, being a nonlinear operator, $ n $-th order iteration makes more difficulties in estimation than usual state-dependence. Then we apply our results to the existence, uniqueness, boundedness, asymptotics and continuous dependence of solutions for the mixed-type functional differential equation. Finally, we present two concrete examples to show the boundedness and asymptotics of solutions to these two types of equations respectively.

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Mathematics Subject Classification: Primary: 34K25, 45M20; Secondary: 39B62, 26D15.

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