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Keywords:
periodic solution; nonlinear neutral differential equation; large contraction; integral equation
periodic solution; nonlinear neutral differential equation; large contraction; integral equation
Summary:
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'(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))), \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.
References:
[1] Adivar M., Islam M.N., Raffoul Y.N.: Separate contraction and existence of periodic solutions in totally nonlinear delay differential equations. Hacet. J. Math. Stat. 41 (2012), no. 1, 1–13. MR 2976906 | Zbl 1260.34132
[2] Ardjouni A., Djoudi A.: Existence of periodic solutions for totally nonlinear neutral differential equations with variable delay. Sarajevo J. Math. 8 (20) (2012), 107–117. MR 2977530 | Zbl 1260.34134
[3] Ardjouni A., Djoudi A.: Existence of periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale. Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 3061–3069. DOI 10.1016/j.cnsns.2011.11.026 | MR 2880475 | Zbl 1254.34128
[4] Ardjouni A., Djoudi A.: Periodic solutions in totally nonlinear difference equations with functional delay. Stud. Univ. Babeş-Bolyai Math. 56 (2011), no. 3, 7–17. MR 2869710 | Zbl 1274.39029
[5] Ardjouni A., Djoudi A.: Periodic solutions for a second-order nonlinear neutral differential equation with variable delay. Electron. J. Differential Equations 2011, no. 128, 1–7. MR 2853014 | Zbl 1278.34077
[6] Ardjouni A., Djoudi A.: Periodic solutions in totally nonlinear dynamic equations with functional delay on a time scale. Rend. Semin. Mat. Univ. Politec. Torino 68 (2010), no. 4, 349–359. MR 2815207 | Zbl 1226.34062
[7] Burton T.A.: Liapunov functionals, fixed points and stability by Krasnoselskii's theorem. Nonlinear Stud. 9 (2002), 181–190. MR 1898587 | Zbl 1084.47522
[8] Burton T.A.: A fixed point theorem of Krasnoselskii. Appl. Math. Lett. 11 (1998), 85–88. DOI 10.1016/S0893-9659(97)00138-9 | MR 1490385 | Zbl 1127.47318
[9] Burton T.A.: Integral equations, implicit relations and fixed points. Proc. Amer. Math. Soc. 124 (1996), 2383–2390. DOI 10.1090/S0002-9939-96-03533-2 | MR 1346965
[10] Burton T.A.: Stability and Periodic Solutions of Ordinary Functional Differential Equations. Academic Press, Orlando, FL, 1985. MR 0837654
[11] Derrardjia I., Ardjouni A., Djoudi A.: Stability by Krasnoselskii's theorem in totally nonlinear neutral differential equations. Opuscula Math. 33 (2013), no. 2, 255–272. DOI 10.7494/OpMath.2013.33.2.255 | MR 3023531
[12] 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. MR 2685037 | Zbl 1203.34110
[13] Dib Y.M., Maroun M.R., Raffoul Y.N.: Periodicity and stability in neutral nonlinear differential equations with functional delay. Electron. J. Differential Equations 2005, no. 142, 1–11. MR 2181286 | Zbl 1097.34049
[14] Kang S., Zhang G.: Existence of nontrivial periodic solutions for first order functional differential equations. Appl. Math. Lett. 18 (2005), 101–107. DOI 10.1016/j.aml.2004.07.018 | MR 2121560 | Zbl 1075.34064
[15] Kun L.Y.: Periodic solution of a periodic neutral delay equation. J. Math. Anal. Appl. 214 (1997), 11–21. DOI 10.1006/jmaa.1997.5576 | MR 1645495 | Zbl 0894.34075
[16] Raffoul Y.N.: Periodic solutions for neutral nonlinear differential equations with functional delays. Electron. J. Differential Equations 2003, no. 102, 1–7. MR 2011575
[17] Smart D.R.: Fixed Points Theorems. Cambridge University Press, Cambridge, 1980.
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