Existence of Periodic Solutions for Totally Nonlinear Neutral Differential Equations With Variable Delay
DOI:
https://doi.org/10.5644/SJM.08.1.08Keywords:
Periodic solution, nonlinear neutral differential equation, large contraction, integral equationAbstract
We use a modification of Krasnoselskii's fixed point theorem introduced by T. A. Burton (see $\left[ 1\right] $ Theorem $3$) to show that the totally nonlinear neutral differential equation with variable delay
\begin{multline*}
x^{\prime }(t)=-a(t)x^{3}(t)+c(t)x^{\prime }(t-g(t))Q^{\prime
}\left( x\left( t-g(t)\right) \right)\\ +G\left(
t,x^{3}(t),x^{3}(t-g(t))\right) ,
\end{multline*}
has a periodic solution. We invert this equation to construct a sum of a compact map and a large contraction which is suitable for applying the modification of Krasnoselskii's theorem. The results of $\left[ 5\right]$ are generalized. Finally, an example is given to illustrate our result.
2000 Mathematics Subject Classification. 34K20, 45J05, 45D05
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References
T. A. Burton, Liapunov functionals, fixed points and stability by Krasnoseskii’s theorem, Nonlinear Stud., 9 (2002), 181–190.
T. A. Burton, A fixed point theorem of Krasnoselskii, Appl. Math. Lett., 11 (1998), 85–88.
T. A. Burton, Integral equations, implicit relations and fixed points, Proc. Amer. Math. Soc., 124 (1996), 2383–2390.
T. A. Burton, Stability and Periodic Solutions of Ordinary Functional Differential Equations, Academic Press. NY, 1985.
H. Deham, A. Djoudi, Existence of periodic solutions for neutral nonlinear differential equations with variable delay, Electron. J. Differ. Equ., 127 (2010), 1–8.
Y. M. Dib, M.R. Maroun, Y.N. Raffoul, Periodicity and stability in neutral nonlinear differential equations with functional delay, Electron. J. Differ. Equ., 142 (2005), 1–11.
L. Y. Kun, Periodic solution of a periodic neutral delay equation, J. Math. Anal. Appl., 214 (1997), 11–21.
Y. N. Raffoul, Periodic solutions for neutral nonlinear differential equations with functional delays, Electron. J. Differ. Equ., 102 (2003), 1–7.
D. R. Smart, Fixed Points Theorems, Cambridge University Press, Cambridge. 1980.
