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Existence of periodic solutions of a scalar functional differential equation via a fixed point theorem. (English) Zbl 1145.34041
Consider the periodic scalar functional differential equation $$\dot{y}(t)=-a(t)y(t)+f(t,y(t-\tau_1(t)),\dots, y(t-\tau_n(t))).$$ Using a fixed point theorem, the authors establishes a variety of sufficient conditions on the existence of single and multiple periodic solutions. These results improve and generalized some existing ones. Moreover, analogous results can be obtained similarly for the following periodic scalar functional differential equation, $$\dot{y}(t)=a(t)y(t)-f(t,y(t-\tau_1(t)),\dots, y(t-\tau_n(t))).$$

MSC:
 34K13 Periodic solutions of functional differential equations 47N20 Applications of operator theory to differential and integral equations
Full Text:
References:
 [1] Cheng, S. S.; Zhang, G.: Existence of positive periodic solutions for non-autonomous functional differential equations. Electron. J. Differential equations, No. 59, 1-8 (2001) · Zbl 1003.34059 [2] Chow, S. N.: Existence of periodic solutions of autonomous functional differential equations. J. differential equations, No. 15, 350-378 (1974) · Zbl 0295.34055 [3] Fan, M.; Wang, K.: Optimal harvesting policy for single population with periodic coefficients. Math. biosci. 152, No. 2, 165-177 (1998) · Zbl 0940.92030 [4] Freedman, H. I.; Wu, J. H.: Periodic solutions of single-species models with periodic delay. SIAM J. Math. anal., No. 23, 689-701 (1992) · Zbl 0764.92016 [5] Gopalsamy, K.; Trofimchuk, S. I.: Almost periodic solutions of lasota--wazewska type delay differential equations. J. math. Anal. appl., No. 237, 106-127 (1999) · Zbl 0936.34058 [6] Gurney, W. S.; Blythe, S. P.; Nisbet, R. M.: Nicholson’s blowflies revisited. Nature, No. 287, 17-21 (1980) [7] Jiang, D. Q.; Wei, J. J.: Existence of positive periodic solutions for nonautonomous delay differential equations. Chinese ann. Math. 20A, No. 6, 715-720 (1999) · Zbl 0948.34046 [8] Jiang, D. Q.; Wei, J. J.: Existence of positive periodic solutions for Volterra intergo-differential equations. Acta math. Sinica 21B, No. 4, 553-560 (2001) · Zbl 1035.45003 [9] Krasnoselskii, M. A.: Positive solution of operator equations. (1964) [10] Li, Y. K.: Existence and global attractivity of positive periodic solutions for a class of delay differential equations. Sci. China 28A, No. 2, 108-118 (1998) [11] Li, J. W.; Cheng, S. S.: Global attractivity in a survival model of wazewska and lasota. Quart. appl. Math. 60, No. 3, 477-483 (2002) · Zbl 1022.92008 [12] Nicholson, A. J.: The self-adjustment of populations to change. Cold spring harbor symp. Quant. biol., No. 22, 153-173 (1957) [13] Nicholson, A. J.: The balance of animal population. J. animal ecol., No. 2, 132-178 (1993) [14] Richards, F. J.: A flexible growth function for empirical use. J. exp. Botany 10, No. 29, 290 (1959) · Zbl 0095.26602 [15] Deimling, K.: Nonlinear functional analysis. (1985) · Zbl 0559.47040 [16] Lan, K.; Jeffry, K.; Webb, J. R. L.: Positive solutions of semilinear differential equations with singularities. J. differential equations, No. 148, 407-421 (1998) · Zbl 0909.34013 [17] Joseph, W.; So, H.; Yu, J. S.: Global attractivity and uniformly persistence in Nicholson blowfies. Differential equations dynam. Systems, No. 2, 11-18 (1994) [18] Gopalsamy, K.; Weng, P.: Global attractivity and level crossing in model of haematopoiesis. Bull. inst. Math. acad. Sinica, No. 22, 341-360 (1994) · Zbl 0829.34067 [19] Hale, J. K.: Theory of functional differential equations. (1977) · Zbl 0352.34001 [20] Weng, P. X.; Liang, M. L.: The existence and behavior of periodic solution of a hematopoiesis model. Math. appl., No. 8, 434-439 (1995) · Zbl 0949.34517 [21] Wan, A.; Jiang, D.; Xu, X.: A new existence theory for positive periodic solutions to functional differential equations. Comput. math. Appl., No. 47, 1257-1262 (2004) · Zbl 1073.34082 [22] Kuang, Y.: Delay differential equations with applications in population dynamics. (1993) · Zbl 0777.34002 [23] Ruan, S.: Delay differential equations in single species dynamics. Delay differential equations and applications, 433-515 (2006) · Zbl 1130.34059