# 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

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topSoulahia, 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

top- 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
- Ardjouni A., Djoudi A., 10.4067/S0716-09172013000400006, Proyecciones 32 (2013), no. 4, 377–391. Zbl1293.34085MR3145042DOI10.4067/S0716-09172013000400006
- 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
- 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
- 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.
- 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
- 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
- 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
- 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
- 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
- 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
- Becker L.C., Burton T.A., Stability, fixed points and inverse of delays, Proc. Roy. Soc. Edinburgh Set. A 136 (2006), 245–275. MR2218152
- Burton T.A., Liapunov functionals, fixed points and stability by Krasnoselskii's theorem, Nonlinear Stud. 9 (2002), no. 2, 181–190. Zbl1084.47522MR1898587
- Burton T.A., Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006. Zbl1160.34001MR2281958
- 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
- Burton T.A., Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, NY, 1985. Zbl1209.34001MR0837654
- 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
- Deham H., Djoudi A., Periodic solutions for nonlinear differential equation with functional delay, Georgian Math. J. 15 (2008), no. 4, 635–642. Zbl1171.47061MR2494962
- 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
- 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
- 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
- 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
- Smart D.R., Fixed Point Theorems, Cambridge Tracts in Mathematics, 66, Cambridge University Press, London-New York, 1974. Zbl0427.47036MR0467717
- 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
- 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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