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Optimal existence theory for single and multiple positive periodic solutions to functional difference equations. (English) Zbl 1068.39009
For any $$(a,b)\in \mathbb{Z}^2$$, $$a< b$$, denote $\begin{gathered} Z[a, b]= \{a,a+1,\dots, b\},\quad Z(a, b)= \{a-1,\dots, b-1\},\\ Z[a, b)= \{a,a+1,\dots, b-1\},\quad Z(a, b]= \{a-1,\dots, b\},\\ Z[a,+\infty)= \{a,a+1,\dots\},\quad Z(-\infty, a]= \{\dots,a-1,a\},\\ Z(a,+\infty)= \{a+1,\dots\},\quad Z(-\infty, a)= \{\dots, a-1\},\end{gathered}$ therefore $$Z[1,+\infty)= N$$ and $$Z(-\infty,+\infty)= Z$$, let $$T\in N$$.
The authors consider the general periodic logistic difference equation $\Delta y(i)= y(i)(a(i)- f(i,y(i- \tau_1(i)),\dots, y(i-\tau_n(i)))),\quad i\in \mathbb{Z},\tag{1}$ where $$y: Z\to [0,+\infty)$$, $$\Delta y(i)= y(i+ 1)- y(i)$$, $$i\in \mathbb{Z}$$, the function $$f: \mathbb{Z}\times [0,+\infty)^n\to [0,+\infty)$$ is continuous, the function $$a: \mathbb{Z}\to (0,+\infty)$$ is continuous, $$a(i)= a(i+ T)$$, $$i\in \mathbb{Z}$$, the functions $$\tau_m: \mathbb{Z}\to \mathbb{Z}$$, $$1\leq m\leq n$$, satisfying $$\tau_m(i)= \tau_m(i+ t)$$, $$i\in \mathbb{Z}$$, $$1\leq m\leq n$$, and $f(i,u_1,\dots, u_n)= f(i+ T,u_1,\dots, u_n),\quad i\in \mathbb{Z},\quad (u_1,\dots, u_n)\in [0,+\infty)^n.$ They also consider the general periodic functional difference equation $\Delta y(i)= -a(i)y(i)+ g(i,y(i-\tau(i))),\quad i\in \mathbb{Z},\tag{2}$ where $$a\in C(\mathbb{Z},(0,1))$$, $$g\in C(\mathbb{Z}\times [0,+\infty),[0,+\infty))$$, $$\tau\in C(\mathbb{Z},\mathbb{Z})$$, $$a(i)= a(i+ T)$$, $$i\in\mathbb{Z}$$, $$\tau(i)= \tau(i+ T)$$, $$i\in\mathbb{Z}$$ and $$g(i,u)= g(i+ T,u)$$, $$i\in\mathbb{Z}$$, $$u\in [0,+\infty)$$.
Let $X= \{y;\,y\in C(\mathbb{Z},\mathbb{R}),\;y(i)= y(i+ T),\;i\in\mathbb{Z}\}$ be the Banach space endowed with the norm $\| y\|= \sup_{i\in\mathbb{Z}[0,T-1]}\,|y(i)|,\quad y\in X$ and the cone in $$X$$ defined by $K= \{y\in X;\;(\forall i\in \mathbb{Z})(y(i)\geq 0\wedge y(i)\geq \sigma\| y\|\},$ where $\sigma= \Biggl(\prod^{T-1}_{i=0} (1+ a(i))\Biggr)^{-1}.$ Using a fixed point theorem in cones the authors establish the existence of positive $$T$$-periodic solutions $$y$$ to equation (1) and the existence for single and multiple positive $$T$$-periodic solutions $$y$$ to equation (2).
Reviewer: D. M. Bors (Iaşi)

##### MSC:
 39A11 Stability of difference equations (MSC2000)
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##### References:
 [1] Kelley, W.G.; Peterson, A.C., Difference equations: an introduction with applications, (1991), Academic Press New York · Zbl 0733.39001 [2] Li, Y.K., Existence and global attractivity of positive periodic solution for a class of delay differential equations, Sci. China (series A), 28, 108-118, (1998), (in Chinese) [3] Zhang, R.Y.; Wang, Z.C.; Cheng, Y.; Wu, J., Periodic solutions of a single species discrete population model with periodic harvest/stock, Comput. math. appl, 39, 77-90, (2000) · Zbl 0970.92019 [4] Jiang, D.Q.; Agarwal, R.P., Existence of positive solutions for a class of difference equations with several deviating arguments, Advance in difference equation IV of the J. comput. math. appl, 45, 1303-1309, (2003) · Zbl 1052.39008 [5] Jiang, D.Q.; Wei, J.J., Existence of positive periodic solution for nonautonomous delay differential equations, Chin. ann. math. (series A), 20A, 716-720, (1999), (in Chinese) [6] Jiang, D.Q.; Wei, J.; Zhang, B., Positive periodic solutions of functional differential equations and population models, Electron. J. differ. equat, 2002, 71, 1-13, (2002) [7] Kuang, Y., Global stability for a class of nonlinear nonautonomous delay logistic equations, Nonlinear anal, 17, 627-634, (1991) · Zbl 0766.34053 [8] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Publishers Boston · Zbl 0752.34039 [9] Gopalsamy, K.; Lalli, B.S., Oscillatory and asymptotic behavior of a multiplicative delay logistic equation, Dynam. stability syst, 7, 35-42, (1992) · Zbl 0764.34049 [10] Pielou, E.C., Mathematics ecology, (1977), Wiley-Interscience New York · Zbl 0259.92001 [11] Lenhart, S.; Travis, C., Global stability of a biological model with time delay, Proc. amer. math. soc, 96, 75-78, (1986) · Zbl 0602.34044 [12] Mallet-Paret, J.; Nussbaum, R., Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation, Ann. math. pure. appl, 145, 33-128, (1986) · Zbl 0617.34071 [13] Luo, J.; Yu, J., Global asymptotic stability of nonautonomous mathematical ecological equations with distributed deviating arguments, Acta math. sin, 41, 1273-1282, (1998), (in Chinese) · Zbl 1027.34088 [14] Weng, P.; Liang, M., The existence and behavior of periodic solution of hematopoiesis model, Math. appl, 8, 4, 434-439, (1995) · Zbl 0949.34517 [15] Weng, P., Existence and global attractivity of periodic solution of integrodifferential equation in population dynamics, Acta appl. math, 12, 4, 427-434, (1996) · Zbl 0886.45005 [16] Gurney, W.S.C.; Blythe, S.P.; Nisbet, R.M., Nicholson’s blowfies revisited, Nature, 287, 17-20, (1980) [17] Gopalsamy, K.; Weng, P., Global attractivity and level crossing in model of hoematcpoiesis, Bull. inst. math., acad. sin, 22, 4, 341-360, (1994) · Zbl 0829.34067 [18] Joseph, W.; So, H.; Yu, J., Global attractivity and uniformly persistence in Nicholson’s blowfies, Differ. equat. dynam. syst, 2, 1, 11-18, (1994) · Zbl 0869.34056 [19] Mackey, M.C.; Galass, Oscillations and chaos in phycological control systems, Sciences, 197, 2, 287-289, (1987) [20] Jiang, D.Q.; Wei, J.J., Existence of positive periodic solutions for Volterra integro-differential equations, Acta math. sci, 21B, 4, 553-560, (2002) · Zbl 1035.45003 [21] Wan, A.Y.; Jiang, D.Q., Existence of positive periodic solutions for functional differential equations, Kyushu J. math, 56, 1, 193-202, (2002) · Zbl 1012.34068 [22] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag New York · Zbl 0559.47040
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