# 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

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topArdjouni, 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 -

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