Existence of Periodic Solutions for Nonlinear Neutral Dynamic Equations with Functional Delay on a Time Scale

Ardjouni, Abdelouaheb, and Djoudi, Ahcène. "Existence of Periodic Solutions for Nonlinear Neutral Dynamic Equations with Functional Delay on a Time Scale." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 52.1 (2013): 5-19. <http://eudml.org/doc/260734>.

@article{Ardjouni2013,
abstract = {Let $\mathbb \{T\}$ be a periodic time scale. The purpose of this paper is to use a modification of Krasnoselskii’s fixed point theorem due to Burton to prove the existence of periodic solutions on time scale of the nonlinear dynamic equation with variable delay $x^\{\triangle \}\left( t\right) =-a\left( t\right) h\left( x^\{\sigma \}\left( t\right) \right) +c(t)x^\{\widetilde\{\triangle \}\}\left( t-r\left( t\right) \right) +G\left( t,x\left( t\right) ,x\left( t-r\left( t\right) \right) \right)$, $t\in \mathbb \{T\}$, where $f^\{\triangle \}$ is the $\triangle $-derivative on $\mathbb \{T\}$ and $f^\{\widetilde\{\triangle \}\}$ is the $\triangle $-derivative on $(id-r)(\mathbb \{T\})$. We invert the given equation to obtain an equivalent integral equation from which we define a fixed point mapping written as a sum of a large contraction and a compact map. We show that such maps fit very nicely into the framework of Krasnoselskii–Burton’s fixed point theorem so that the existence of periodic solutions is concluded. The results obtained here extend the work of Yankson [Yankson, E.: Existence of periodic solutions for totally nonlinear neutral differential equations with functional delay Opuscula Mathematica 32, 3 (2012), 617–627.].},
author = {Ardjouni, Abdelouaheb, Djoudi, Ahcène},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {fixed point; large contraction; periodic solutions; time scales; nonlinear neutral dynamic equations; fixed point; large contraction; periodic solutions; time scales; nonlinear neutral dynamic equations},
language = {eng},
number = {1},
pages = {5-19},
publisher = {Palacký University Olomouc},
title = {Existence of Periodic Solutions for Nonlinear Neutral Dynamic Equations with Functional Delay on a Time Scale},
url = {http://eudml.org/doc/260734},
volume = {52},
year = {2013},
}

TY - JOUR
AU - Ardjouni, Abdelouaheb
AU - Djoudi, Ahcène
TI - Existence of Periodic Solutions for Nonlinear Neutral Dynamic Equations with Functional Delay on a Time Scale
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2013
PB - Palacký University Olomouc
VL - 52
IS - 1
SP - 5
EP - 19
AB - Let $\mathbb {T}$ be a periodic time scale. The purpose of this paper is to use a modification of Krasnoselskii’s fixed point theorem due to Burton to prove the existence of periodic solutions on time scale of the nonlinear dynamic equation with variable delay $x^{\triangle }\left( t\right) =-a\left( t\right) h\left( x^{\sigma }\left( t\right) \right) +c(t)x^{\widetilde{\triangle }}\left( t-r\left( t\right) \right) +G\left( t,x\left( t\right) ,x\left( t-r\left( t\right) \right) \right)$, $t\in \mathbb {T}$, where $f^{\triangle }$ is the $\triangle $-derivative on $\mathbb {T}$ and $f^{\widetilde{\triangle }}$ is the $\triangle $-derivative on $(id-r)(\mathbb {T})$. We invert the given equation to obtain an equivalent integral equation from which we define a fixed point mapping written as a sum of a large contraction and a compact map. We show that such maps fit very nicely into the framework of Krasnoselskii–Burton’s fixed point theorem so that the existence of periodic solutions is concluded. The results obtained here extend the work of Yankson [Yankson, E.: Existence of periodic solutions for totally nonlinear neutral differential equations with functional delay Opuscula Mathematica 32, 3 (2012), 617–627.].
LA - eng
KW - fixed point; large contraction; periodic solutions; time scales; nonlinear neutral dynamic equations; fixed point; large contraction; periodic solutions; time scales; nonlinear neutral dynamic equations
UR - http://eudml.org/doc/260734
ER -