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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  1. Adivar M., Islam M.N., Raffoul Y.N., Separate contraction and existence of periodic solution in totally nonlinear delay differential equations, Hacet. J. Math. Stat. 41 (2012) no. 1, 1–13. MR2976906
  2. Ardjouni A., Djoudi A., 10.4067/S0716-09172013000400006, Proyecciones 32 (2013), no. 4, 377–391. Zbl1293.34085MR3145042DOI10.4067/S0716-09172013000400006
  3. Ardjouni A., Djoudi A., Existence of periodic solutions in totally nonlinear neutral dynamic equations with variable delay on a time scale, Mathematics in engineering, science and aerospace MESA, Vol. 4, No. 3, pp. 305–318, 2013. CSP - Cambridge, UK; I&S - Florida, USA, 2013. Zbl1297.34080
  4. Ardjouni A., Djoudi A., Existence of positive periodic solutions for two kinds of nonlinear neutral differential equations with variable delay, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 20 (2013), 357–366. Zbl1268.34127MR3098458
  5. Ardjouni A., Djoudi A., Existence of positive periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale, Malaya J. Matematik 1 (2013), no. 2, 60–67. 
  6. Ardjouni A., Djoudi A., Existence and positivity of solutions for a totally nonlinear neutral periodic differential equation, Miskolc Math. Notes 14 (2013), no. 3, 757–768. Zbl1299.34230MR3153963
  7. Ardjouni A., Djoudi A., 10.1515/anona-2012-0024, Adv. Nonlinear Anal. 2 (2013), no. 2, 151–161, DOI 10.1515/anona-2012-0024. Zbl1278.34077MR3055532DOI10.1515/anona-2012-0024
  8. Ardjouni A., Djoudi A., Existence of periodic solutions for a second order nonlinear neutral differential equation with functional delay, Electronic J. Qual. Theory Differ. Equ. 2012, no. 31, 1–9. MR2904111
  9. Ardjouni A., Djoudi A., Existence of periodic solutions for totally nonlinear neutral differential equations with variable delay, Sarajevo J. Math. 8 (2012), no. 1, 107–117. Zbl1260.34134MR2977530
  10. Ardjouni A., Djoudi A., Existence of positive periodic solutions for a nonlinear neutral differential equation with variable delay, Appl. Math. E-Notes 12 (2012), 94–101. Zbl1254.34098MR2988223
  11. Ardjouni A., Djoudi A., Periodic solution in totally nonlinear dynamic equations with functional delay on a time scale, Rend. Semin. Mat. Univ. Politec. Torino 68 (2010), no. 4, 349–359. MR2815207
  12. Becker L.C., Burton T.A., Stability, fixed points and inverse of delays, Proc. Roy. Soc. Edinburgh Set. A 136 (2006), 245–275. MR2218152
  13. Burton T.A., Liapunov functionals, fixed points and stability by Krasnoselskii's theorem, Nonlinear Stud. 9 (2002), no. 2, 181–190. Zbl1084.47522MR1898587
  14. Burton T.A., Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006. Zbl1160.34001MR2281958
  15. Burton T.A., 10.1016/S0893-9659(97)00138-9, Appl. Math. Lett. 11 (1998), 85–88. Zbl1127.47318MR1490385DOI10.1016/S0893-9659(97)00138-9
  16. Burton T.A., Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, NY, 1985. Zbl1209.34001MR0837654
  17. Chen F., 10.1016/j.amc.2004.03.009, Appl. Math. Comput. 162 (2005), no. 3, 1279–1302. Zbl1125.93031MR2113969DOI10.1016/j.amc.2004.03.009
  18. Deham H., Djoudi A., Periodic solutions for nonlinear differential equation with functional delay, Georgian Math. J. 15 (2008), no. 4, 635–642. Zbl1171.47061MR2494962
  19. Deham H., Djoudi A., Existence of periodic solutions for neutral nonlinear differential equations with variable delay, Electron. J. Differential Equations 2010, no. 127, 1–8. Zbl1203.34110MR2685037
  20. Hale J., 10.1007/978-1-4612-9892-2_3, second edition, Applied Mathematical Sciences, 3, Springer, New York-Heidelberg, 1977. Zbl1092.34500MR0508721DOI10.1007/978-1-4612-9892-2_3
  21. Hale J.K., Verduyn Lunel S.M., 10.1007/978-1-4612-4342-7_3, Applied Mathematical Sciences, 99, Springer, New York, 1993. Zbl0787.34002MR1243878DOI10.1007/978-1-4612-4342-7_3
  22. Fan M., Wang K., Wong P.J.Y., Agarwal R.P., 10.1007/s10114-003-0311-1, Acta Math. Sin. (Engl. Ser.) 19 (2003), no. 4, 801–822. Zbl1047.34080MR2023372DOI10.1007/s10114-003-0311-1
  23. Smart D.R., Fixed Point Theorems, Cambridge Tracts in Mathematics, 66, Cambridge University Press, London-New York, 1974. Zbl0427.47036MR0467717
  24. Wang Y., Lian H., Ge W., 10.1016/j.aml.2006.02.028, Appl. Math. Lett. 20 (2007), 110–115. MR2273618DOI10.1016/j.aml.2006.02.028
  25. Yankson E., 10.5817/AM2012-4-261, Arch. Math. (Brno) 48 (2012), no. 4, 261–270. Zbl1274.34230MR3007609DOI10.5817/AM2012-4-261

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