Periodicity of solutions for a system of nonlinear integro-differential equations

Mouataz Billah Mesmouli; Abdelouaheb Ardjouni; Ahcene Djoudi

doi:10.5644/SJM.11.1.04

Authors

Mouataz Billah Mesmouli Applied Mathematics Laboratory, Faculty of Sciences, Department of Mathematics, University Annaba, Annaba, Algeria
Abdelouaheb Ardjouni Applied Mathematics Laboratory, Faculty of Sciences, Department of Mathematics, University Annaba, Annaba, Algeria
Ahcene Djoudi Applied Mathematics Laboratory, Faculty of Sciences, Department of Mathematics, University Annaba, Annaba, Algeria

DOI:

https://doi.org/10.5644/SJM.11.1.04

Keywords:

Krasnoselskii's theorem, contraction, neutral differential equation, integral equation, periodic solution, fundamental matrix solution, Floquet theory

Abstract

In this paper, we study the existence of periodic solutions of the nonlinear system of integro-differential equations
\begin{equation*}
\frac{d}{dt}x\left( t\right) =-\int_{t-\tau \left( t\right)
}^{t}A(t,s)x(s)ds+Q\left( t,x(t),x(t-\tau \left( t\right) )\right) ,\ t\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
.
\end{equation*}
In the process we use the fundamental matrix solution to convert the given integro-differential equation into an equivalent integral equation. Then by using Krasnoselskii's fixed point theorem we obtain the existence of periodic solution and by contraction mapping principle we obtain the uniqueness. An application in two dimension is given.

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References

[1] T. A. Burton, A fixed point theorem of Krasnoselskii, Appl. Math. Lett., 11 (1) (1998), 85--88.

[2] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, New York, 2006.

[3] T. R. Ding, R. Iannacci and F. Zanolin, On periodic solutions of sublinear Duffing equations, J. Math. Anal. Appl., 158 (1991) 316--332.

[4] J. Hale, Ordinary Differential Equations, Robert E. Krieger Publishing Company, New York, 1980.

[5] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Verlag, New York,1993.

[6] K. Gopalsamy, X. He, and L. Wen, On a periodic neutral logistic equation, Glasgow Math., J. 33 (1991), 281--286.

[7] K. Gopalsamy and B. G. Zhang, On a neutral delay-logistic equation, Dynam. Stability Systems, 2 (1988), 183--195.

[8] I. Gyori and F. Hartung, Preservation of stability in a linear neutral differential equation under delay perturbations, Dyn. Syst. Appl., 10 (2001), 225--242.

[9] I. Gyori and G. Ladas, Positive solutions of integro-differential equations with unbounded delay, J. Integral Equations Appl., 4 (1992), 377--390.

[10] M. N. Islam and Y. N. Raffoul, Periodic solutions of neutral nonlinear system of differential equations with functional delay, J. Math. Anal. Appl., 331 (2007) 1175--1186.

[11] C. H. Jin and J. W. Luo, Stability of an integro-differential equation}, Comput. Math. Appl., 57 (2009) 1080--1088.

[12] C. H. Jin and J. W. Luo, Stability in functional differential equations established using fixed point theory, Nonlinear Anal., 68 (2008) 3307--3315.

[13] C. H. Jin and J. W. Luo, Fixed points and stability in neutral differential equations with variable delays, Proc. Amer. Math. Soc., 136 (3) (2008), 909--918.

[14] L. Y. Kun, Periodic solutions of a periodic neutral delay equation, J. Math. Anal. Appl., 214 (1997), 11--21.

[15] V. Lakshmikantham and S. G. Deo, Methods of Variation of Parameters for Dynamical Systems, Gordon and Breach Science Publishers, Australia, 1998.

[16] M. Maroun and Y. N. Raffoul, Periodic solutions in nonlinear neutral difference equations with functional delay, J. Korean Math. Soc., 42 (2005), 255--268.

[17] D. R. Smart, Fixed Point Theorems, Cambridge University Press, 1980. 2.4

[18] R. Weikard, Floquet theory for linear differential equations with meromorphic solutions, Electron. J. Qual. Theory Differ. Equ., 8 (2000), 1--6.

[19] E. Yankson, Periodicity in a system of differential equations with finite delay, Casp. J. Math. Sci., In press.