Existence of nonnegative periodic solutions in neutral integro-differential equations with functional delay

Imene Soulahia; Abdelouaheb Ardjouni; Ahcene Djoudi

Commentationes Mathematicae Universitatis Carolinae (2015)

  • Volume: 56, Issue: 1, page 23-44
  • ISSN: 0010-2628

Abstract

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The fixed point theorem of Krasnoselskii and the concept of large contractions are employed to show the existence of a periodic solution of a nonlinear integro-differential equation with variable delay x ' ( t ) = - t - τ ( t ) t a ( t , s ) g ( x ( s ) ) d s + d d t Q ( t , x ( t - τ ( t ) ) ) + G ( t , x ( t ) , x ( t - τ ( t ) ) ) . We transform this equation and then invert it to obtain a sum of two mappings one of which is completely continuous and the other is a large contraction. We choose suitable conditions for τ , g , a , Q and G to show that this sum of mappings fits into the framework of a modification of Krasnoselskii’s theorem so that existence of nonnegative T-periodic solutions is concluded.

How to cite

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Soulahia, Imene, Ardjouni, Abdelouaheb, and Djoudi, Ahcene. "Existence of nonnegative periodic solutions in neutral integro-differential equations with functional delay." Commentationes Mathematicae Universitatis Carolinae 56.1 (2015): 23-44. <http://eudml.org/doc/269879>.

@article{Soulahia2015,
abstract = {The fixed point theorem of Krasnoselskii and the concept of large contractions are employed to show the existence of a periodic solution of a nonlinear integro-differential equation with variable delay \begin\{equation*\} x^\{\prime \}(t)= -\int \_\{t-\tau (t)\}^\{t\}a(t,s) g(x(s))\,ds + \frac\{d\}\{dt\}Q (t,x(t-\tau (t)))+ G(t,x(t),x(t-\tau (t))). \end\{equation*\} We transform this equation and then invert it to obtain a sum of two mappings one of which is completely continuous and the other is a large contraction. We choose suitable conditions for $\tau $, $g$, $a$, $Q$ and $G$ to show that this sum of mappings fits into the framework of a modification of Krasnoselskii’s theorem so that existence of nonnegative T-periodic solutions is concluded.},
author = {Soulahia, Imene, Ardjouni, Abdelouaheb, Djoudi, Ahcene},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Krasnoselskii's fixed points; periodic solution; large contraction; Krasnoselskii's fixed points; periodic solution; large contraction},
language = {eng},
number = {1},
pages = {23-44},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Existence of nonnegative periodic solutions in neutral integro-differential equations with functional delay},
url = {http://eudml.org/doc/269879},
volume = {56},
year = {2015},
}

TY - JOUR
AU - Soulahia, Imene
AU - Ardjouni, Abdelouaheb
AU - Djoudi, Ahcene
TI - Existence of nonnegative periodic solutions in neutral integro-differential equations with functional delay
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2015
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 56
IS - 1
SP - 23
EP - 44
AB - The fixed point theorem of Krasnoselskii and the concept of large contractions are employed to show the existence of a periodic solution of a nonlinear integro-differential equation with variable delay \begin{equation*} x^{\prime }(t)= -\int _{t-\tau (t)}^{t}a(t,s) g(x(s))\,ds + \frac{d}{dt}Q (t,x(t-\tau (t)))+ G(t,x(t),x(t-\tau (t))). \end{equation*} We transform this equation and then invert it to obtain a sum of two mappings one of which is completely continuous and the other is a large contraction. We choose suitable conditions for $\tau $, $g$, $a$, $Q$ and $G$ to show that this sum of mappings fits into the framework of a modification of Krasnoselskii’s theorem so that existence of nonnegative T-periodic solutions is concluded.
LA - eng
KW - Krasnoselskii's fixed points; periodic solution; large contraction; Krasnoselskii's fixed points; periodic solution; large contraction
UR - http://eudml.org/doc/269879
ER -

References

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