Existence of periodic solutions for first-order totally nonlinear neutral differential equations with variable delay

Abdelouaheb Ardjouni; Ahcène Djoudi

Commentationes Mathematicae Universitatis Carolinae (2014)

  • Volume: 55, Issue: 2, page 215-225
  • ISSN: 0010-2628

Abstract

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We use a modification of Krasnoselskii’s fixed point theorem due to Burton (see [Liapunov functionals, fixed points and stability by Krasnoselskii’s theorem, Nonlinear Stud. 9 (2002), 181–190], Theorem 3) to show that the totally nonlinear neutral differential equation with variable delay x ' ( t ) = - a ( t ) h ( x ( t ) ) + c ( t ) x ' ( t - g ( t ) ) Q ' ( x ( t - g ( t ) ) ) + G ( t , x ( t ) , x ( t - g ( t ) ) ) , has a periodic solution. We invert this equation to construct a fixed point mapping expressed as a sum of two mappings such that one is compact and the other is a large contraction. We show that the mapping fits very nicely for applying the modification of Krasnoselskii’s theorem so that periodic solutions exist.

How to cite

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Ardjouni, Abdelouaheb, and Djoudi, Ahcène. "Existence of periodic solutions for first-order totally nonlinear neutral differential equations with variable delay." Commentationes Mathematicae Universitatis Carolinae 55.2 (2014): 215-225. <http://eudml.org/doc/261852>.

@article{Ardjouni2014,
abstract = {We use a modification of Krasnoselskii’s fixed point theorem due to Burton (see [Liapunov functionals, fixed points and stability by Krasnoselskii’s theorem, Nonlinear Stud. 9 (2002), 181–190], Theorem 3) to show that the totally nonlinear neutral differential equation with variable delay \begin\{equation*\} x^\{\prime \}(t) = -a(t)h (x(t)) + c(t)x^\{\prime \}(t-g(t))Q^\{\prime \} (x(t-g(t))) + G (t,x(t),x(t-g(t))), \end\{equation*\} has a periodic solution. We invert this equation to construct a fixed point mapping expressed as a sum of two mappings such that one is compact and the other is a large contraction. We show that the mapping fits very nicely for applying the modification of Krasnoselskii’s theorem so that periodic solutions exist.},
author = {Ardjouni, Abdelouaheb, Djoudi, Ahcène},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {periodic solution; nonlinear neutral differential equation; large contraction; integral equation; nonlinear neutral differential equation; existence of periodic solutions; Burton-Krasnoselskii fixed point theorem},
language = {eng},
number = {2},
pages = {215-225},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Existence of periodic solutions for first-order totally nonlinear neutral differential equations with variable delay},
url = {http://eudml.org/doc/261852},
volume = {55},
year = {2014},
}

TY - JOUR
AU - Ardjouni, Abdelouaheb
AU - Djoudi, Ahcène
TI - Existence of periodic solutions for first-order totally nonlinear neutral differential equations with variable delay
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2014
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 55
IS - 2
SP - 215
EP - 225
AB - We use a modification of Krasnoselskii’s fixed point theorem due to Burton (see [Liapunov functionals, fixed points and stability by Krasnoselskii’s theorem, Nonlinear Stud. 9 (2002), 181–190], Theorem 3) to show that the totally nonlinear neutral differential equation with variable delay \begin{equation*} x^{\prime }(t) = -a(t)h (x(t)) + c(t)x^{\prime }(t-g(t))Q^{\prime } (x(t-g(t))) + G (t,x(t),x(t-g(t))), \end{equation*} has a periodic solution. We invert this equation to construct a fixed point mapping expressed as a sum of two mappings such that one is compact and the other is a large contraction. We show that the mapping fits very nicely for applying the modification of Krasnoselskii’s theorem so that periodic solutions exist.
LA - eng
KW - periodic solution; nonlinear neutral differential equation; large contraction; integral equation; nonlinear neutral differential equation; existence of periodic solutions; Burton-Krasnoselskii fixed point theorem
UR - http://eudml.org/doc/261852
ER -

References

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