Positive periodic solutions for neutral functional diﬀerential systems.
ResumenWe study the existence of positive periodic solutions of a system of neutral diﬀerential equations. In the process we construct two map- pings in which one is a contraction and the other compact. A Kras- noselskii’s ﬁxed point theorem is then used in the analysis.
T. A. Burton, Stability by Fixed Point Theory for functional Diﬀerential Equations, Dover, New York, (2006).
E. Beretta, F. Solimano, Y. Takeuchi, A mathematical model for drug administration by using the phagocytosis of red blood cells, J. Math Biol. 10 Nov; 35 (1), pp. 1-19, (1996).
Y. Chen, New results on positive periodic solutions of a periodic integro-diﬀerential competition system, Appl. Math. Comput., 153 (2), pp. 557-565, (2004).
F. D. Chen, Positive periodic solutions of neutral Lotka-Volterra system with feedback control, Appl. Math. Comput., 162 (3), pp. 1279- 1302, (2005).
F. D. Chen, Periodicity in a nonlinear predator-prey system with state dependent delays, Acta Math. Appl. Sinica English Series, 21 (1) (2005), pp. 49-60, (2005).
F. D. Chen, S. J. Lin, Periodicity in a logistic type system with several delays, Comput. Math. Appl., 48 (1-), pp. 35-44, (2004).
F. D. Chen, F. X. Lin, X. X. Chen, Suﬃcient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control, Appl. Math. Comput., 158 (1), pp. 45-68, (2004).
M. Fan, K. Wang, Global periodic solutions of a generalized n-species Gilpin-Ayala competition model, Comput. Math. Appl., 40, pp. 1141- 1151, (2000).
M. Fan, P. J. Y. Wong, Periodicity and stability in a periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments, Acta Math. Sinica,, 19 (4), pp. 801-822, (2003).
M. E. Gilpin, F. J. Ayala, Global Models of Growth and Competition, Proc. Natl. Acad. Sci., USA 70, pp. 3590-3593, (1973).
A. Datta and J. Henderson, Diﬀerences and smoothness of solutions for functional diﬀerence equations, Proceedings Diﬀerence Equations 1, pp. 133-142, (1995).
J. Henderson and A. Peterson, Properties of delay variation in solutions of delay diﬀerence equations, Journal of Diﬀerential Equations 1, pp. 29-38, (1995).
D. Jiang, J. wei, B. Zhang, Positive periodic solutions of functional diﬀerential equations and population models, Electron. J. Diﬀ.Eqns., Vol. No. 71, pp. 1-13, (2002).
M. A. Krasnosel’skii, Positive solutions of operator Equations Noordhoﬀ, Groningen, (1964).
L. Y. Kun, Periodic solution of a periodic neutral delay equation, J. Math. Anal. Appl.,, 319, pp. 315-325, (2006).
Y. Li, Y. Kuang, Periodic solutions of periodic delay Lotka-Volterra equations and systems, J. Math. Anal. Appl., 255, pp. 260-280, (2001).
M. Maroun and Y. Raﬀoul, Periodic solutions in nonlinear neutral diﬀerence equations with functional delay, Journal of Korean Mathematical Society 42, pp. 255-268, (2005).
Y. Raﬀoul, Periodic solutions for scaler and vector nonlinear diﬀerence equations, Pan-American Journal of Mathematics 9, pp. 97-111, (1999).
Y. N. Raﬀoul, Periodic solutions for neutral nonlinear diﬀerential equations with delay, Electron. J. Diﬀ. Eqns., Vol. No. 102, pp. 1-7, (2003).
Y. Raﬀoul, Positive periodic solutions in neutral nonlinear diﬀerential equations, Electronic Journal of Qualitative Theory of Diﬀerential Equations 16, pp. 1-10, (2007).
Y. Raﬀoul, Existence of positive periodic solutions in neutral nonlinear equations with functional delay, Rocky Mount. Journal of Mathematics 42(6), pp. 1983-1993, (2012).
N. Zhang, B. Dai, Y. Chen, Positive periodic solutions of nonautonomous functional diﬀerential systems, J. Math. Anal., 333, pp. 667- 678, (2007).