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A new existence theory for positive periodic solutions to functional differential equations with impulse effects. (English) Zbl 1156.34053
The authors study the existence of positive periodic solutions for the nonautonomous functional differential equation $$\dot{y}(t)=-a(t) y(t)+g(t,y(t-\tau(t))), t \neq t_j, j \in \mathbb{Z}, y(t_j^+)=y(t_j^-)+I_j(y(t_j)), \tag1$$ where $a \in C(\mathbb{R},(0,\infty))$, $\tau \in C(\mathbb{R},\mathbb{R})$, $g \in C(\mathbb{R}\times [0,\infty),[0,\infty))$, $I_j \in C([0,\infty),[0,\infty))$, $a(t)$, $\tau(t)$, $g(t,y)$ are $w$-periodic functions, and $w>0$ is a constant. Here, $y(t_j^+)$, $y(t_j^-)=y(t_j)$ denote, respectively, the right and the left-hand side limits of $y$ at $t_j$, and the following hypotheses are assumed $t_{j+p}=t_j+w$, $I_{j+p}=I_j$, $j \in \mathbb{Z}$, for a certain positive integer $p$ in such a way that $[0,w)\cap \{t_j : j \in \mathbb{Z}\}=\{t_1,t_2,\dots,t_p\}$. The results included in this paper are connected with those given in the papers of {\it A. Wan, D. Jiang} and {\it X. Xu} [Comput. Math. Appl. 47, 1257--1262 (2004; Zbl 1073.34082)], and {\it A. Wan} and {\it D. Jiang} [Kyushu J. Math. 56, 193--202 (2002; Zbl 1012.34068)] for the nonimpulsive case. The new results extend those in [Comput. Math. Appl. 47, 1257--1262 (2004; Zbl 1073.34082)], and their proof is based on the application of fixed-point theorems in cones [see {\it K. Deimling}, Nonlinear Functional Analysis. Springer-Verlag, New York (1985; Zbl 0559.47040), and {\it K. Lan} and {\it J. R. L. Webb}, J. Differ. Equations 148, 407--421 (1998; Zbl 0909.34013)]. The calculus of the Green’s function associated with problem (1) is also essential in their procedure. To this purpose, the authors consider an impulsive `linear’ problem related to (1) which is written on its equivalent integral form. This expression is useful to write problem (1) as an integral equation. As a particular case of the main result, it is analyzed the case of sublinear and superlinear behavior. Finally, the authors include some examples which illustrate the applicability of the new results to some problems with biological meaning.

MSC:
34K13Periodic solutions of functional differential equations
34K45Functional-differential equations with impulses
47N20Applications of operator theory to differential and integral equations
WorldCat.org
Full Text: DOI
References:
[1] Bainov, D. D.; Simeonov, P. S.: Impulsive differential equations: periodic solutions and applications. (1993) · Zbl 0815.34001
[2] Bainov, D. D.; Simeonov, P. S.: Systems with impulse effect. (1989) · Zbl 0671.34052
[3] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989) · Zbl 0719.34002
[4] Liz, E.: Boundary value problems for new types of differential equations (in Spanish). Ph.d. thesis, univ. Vigo-Spain (1994)
[5] Samoilenko, A. M.; Perestyuk, N. A.: Differential equations with impulse effect. (1987)
[6] Anokhin, A. V.; Berezansky, L.; Braverman, E.: Exponential stability of linear delay impulsive differential equations. J. math. Anal. appl. 193, 923-941 (1995) · Zbl 0837.34076
[7] Bainov, D. D.; Covachev, V.; Stamova, I.: Stability under persistent disturbances of impulsive differential-difference equations of neutral type. J. math. Anal. appl. 187, 790-808 (1994) · Zbl 0811.34057
[8] Shen, J. H.: On some asymptotic stability results of impulsive integro-differential equations. Chinese math. Ann. 17A, 759-765 (1996) · Zbl 0877.34051
[9] Shen, J. H.: The existence of non-oscillatory solutions of delay differential equations with impulses. Appl. math. Comput. 77, 156-165 (1996) · Zbl 0861.34044
[10] Shen, J. H.; Yan, J. R.: Razumikhin type stability theorems for impulsive functional differential equations. Nonlinear anal. TMA 33, 519-531 (1998) · Zbl 0933.34083
[11] Yu, J. S.; Zhang, B. G.: Stability theorems for delay differential equations with impulses. J. math. Anal. appl. 199, 162-175 (1996) · Zbl 0853.34068
[12] Nieto, J. J.: Basic theory for nonresonance impulsive periodic problems of first order. J. math. Anal. appl. 205, 423-433 (1997) · Zbl 0870.34009
[13] Zhang, W.; Fan, M.: Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays. Mathl. comput. Modelling. 39, No. 4/5, 479-493 (2004) · Zbl 1065.92066
[14] Wan, A.; Jiang, D.; Xu, X.: A new existence theory for positive periodic solutions to functional differential equations. Computers math. Applic. 47, No. 8/9, 1257-1262 (2004) · Zbl 1073.34082
[15] Wan, A.; Jiang, D.: Existence of positive periodic solutions for functional differential equations. Kyushu journal of mathematics 56, No. 1, 193-202 (2002) · Zbl 1012.34068
[16] Luo, J.; Yu, J.: Global asymptoticstability of nonautonomous mathematical ecological equations with distributed deviating arguments (in chinese). Acta Mathematica sinica 41, 1273-1282 (1998) · Zbl 1027.34088
[17] Weng, P.; Liang, M.: The existence and behavior of periodic solution of hematcpoiesis model. Mathematica applicate 8, No. 4, 434-439 (1995) · Zbl 0949.34517
[18] Gopalsamy, K.; Weng, P.: Global attractivity and level crossing in model of hoematcpoiesis. Bulletin of the institute of mathematics, academia sinica 22, No. 4, 341-360 (1994) · Zbl 0829.34067
[19] Mackey, M. C.; Glass, L.: Oscillations and chaos in phycological control systems. Sciences 197, No. 2, 287-289 (1987)
[20] Weng, P.: Existence and global attractivity of periodic solution of interodifferential equation in population dynamics. Acta. appl. Math. 12, No. 4, 427-434 (1996) · Zbl 0886.45005
[21] Gurney, W. S. C.; Blythe, S. P.; Nisbet, R. M.: Nicholson’s blowfies revisited. Nature 287, 17-20 (1980)
[22] Joseph, W.; So, H.; Yu, J.: Global attractivity and uniformly persistence in Nicholson’s blowfies. Differential equation and dynamics systems 2, No. 1, 11-18 (1994) · Zbl 0869.34056
[23] Deimling, K.: Nonlinear functional analysis. (1985) · Zbl 0559.47040
[24] Lan, K.; Webb, J. R. L.: Positive solutions of semilinear differential equations with singularities. J. differential equations 148, 407-421 (1998) · Zbl 0909.34013