International Journal of Difference Equations ISSN 0973-6069, Volume 9, Number 2, pp. 233 242 (204) http://campus.mst.edu/ijde Global Attractivity in a Nonlinear Difference Equation and Applications to a Biological Model Chuanxi Qian Mississippi State University Department of Mathematics and Statistics Mississippi State, MS 39762, USA qian@math.msstate.edu Abstract Consider the following difference equation of order k + x n+ = f(x n ) + g(x n k ), n = 0,,..., where f, g : [0, ) [0, ) are continuous functions, and k is a nonnegative integer. We establish a sufficient condition for the global attractivity of positive solutions of this equation. Our result can be applied to the following biological model e p qxn x n+ = ax2 n + c, n = 0,,..., p qxn b + x n + e where 0 < a <, b, c, p and q are positive constants, whose dynamics of solutions has been studied in [8] recently. A new global stability result for the positive solutions of this model is obtained, which is a significant improvement of the corresponding result obtained in [8]. AMS Subject Classifications: 39A0, 92D25. Keywords: Higher order difference equation, biological model, global attractivity, positive equilibrium. Introduction The dynamics of positive solutions of the difference equation x n+ = e p qxn ax2 n + c, n = 0,,..., (.) p qxn b + x n + e Received March 3, 204; Accepted June, 204 Communicated by Gerasimos Ladas
234 Chuanxi Qian where 0 < a <, b, c, p and q are positive constants, is studied in [8] recently. Eq. (.) is a biological model derived from the evaluation of a perennial grass [3]. The boundedness and the persistence of positive solutions, the existence, the attractivity and the global asymptotic stability of the unique positive equilibrium and the existence of periodic solutions have been discussed in [8]. Motivated by the work in [8], we study here the asymptotic behavior of positive solutions of the following general higher order difference equation x n+ = f(x n ) + g(x n k ), n = 0,,..., (.2) where f, g : [0, ) [0, ) are continuous functions with f nondecreasing and k is a nonnegative integer. Clearly, Eq. (.) is a special case of Eq. (.2) with f(x) = ax2 b + x, g(x) = c e p qx + e p qx and k = 0. In this paper, we will establish a sufficient condition for the global attractivity of positive solutions of Eq. (.2). In particular, by applying our result to Eq. (.), we will obtain a new result on the globally asymptotic stability of positive solutions of Eq. (.), which is a significant improvement of the corresponding result obtained in [8]. Because of its theoretical interest and many applications, the study of difference equations has become an active area of research, see, for example, [ 3] and references cited therein. Recently, the asymptotic behavior of positive solutions of the higher order difference equation x n+ x n = p n [f(x n k ) g(x n+ )], n = 0,,..., (.3) where k {0,,...}, f, g [[0, ), [0, )] with g nondecreasing, and {p n } is a nonnegative sequence, is studied and some global attractivity results are obtained in [7]. These results may be applied to several difference equations derived from mathematical biology. However, Eq. (.) can not be written in the form (.3) and so the results obtained in [7] cannot be applied to this case. 2 Main Results In this section, we establish a sufficient condition for the global attractivity of positive solutions of Eq. (.2). In the following discussion, we assume that Eq. (.2) has a unique positive equilibrium x, that is, x is the only positive solution of the equation n f(x) + g(x) = x. In addition, we adopt the notation a n = 0 whenever m > n. Theorem 2.. Assume that f(x) is nondecreasing and there is a positive number a < such that f(x) ax and f(x) ax is nonincreasing for x 0. Suppose also that (x x)(f(x) + g(x) x) < 0 for x > 0 and x x, (2.) i=m
Global Attractivity in a Nonlinear Difference Equation 235 and that g is L-Lipschitz with [ ] L <. Then every positive solution {x n } of Eq. (.2) converges to x as n. Proof. First, we show that {x n } is bounded. Otherwise, there is a subsequence {x ni } of {x n } such that x ni > max{x n : k n < n i }, i =, 2,..., and lim i x ni =. From Eq. (.2) we see that and so it follows that g(x ni k) + f(x ni ) x ni = f(x ni ) f(x ni ) 0 g(x ni k) x ni f(x ni ). (2.2) Since f(x) ax is nondecreasing, x f(x) is increasing. Then (2.2) yields Hence, g(x ni k) > x ni k f(x ni k ). f(x ni k) + g(x ni k ) x ni k > 0 which, in view of (2.), implies that x ni k < x. Then it follows that {g(x ni k )} is bounded which clearly contradicts (2.2) since x ni f(x ni ) x ni ax ni as i. Hence, {x n } must be bounded. Now, we are ready to show that x n x as n. First, we assume that {x n } is a nonoscillatory (about x) solution. Suppose that x n x is eventually nonnegative. The proof for the case that x n x is eventually nonpositive is similar and will be omitted. Let lim sup n x n = A. Then x A <. Clearly, it suffices to show that A = x. First, we assume that {x n } is nonincreasing eventually. Then lim x n = A. If A > x, n then by noting (2.), it follows from Eq. (.2) that x n+ x 0 = n (f(x i ) + g(x i k ) x i ) as n, i=0 which is a contradiction. Hence, A = x.
236 Chuanxi Qian Next, assume that {x n } is not eventually nonincreasing. Then, there is a subsequence {x nm } of {x n } such that Hence, it follows from Eq. (.2) that and so lim x n m = A and x nm > x m nm, m = 0,,.... g(x nm k) + f(x nm ) x nm = f(x nm ) f(x nm ) 0, m = 0,,..., Since x nm k x, and it follows that g(x nm k) x nm f(x nm ), m = 0,,.... (2.3) g(x nm k) + f(x nm k) x nm k 0 Hence, in view of (2.3) and (2.4), we see that g(x nm k) x nm k f(x nm k). (2.4) x nm k f(x nm k) x nm f(x nm ). By noting that x f(x) is increasing, we see that x nm k x nm which yields lim m x n m k = A. Then by taking limit on both sides of (2.3), we find that g(a) + f(a) A 0 which implies that A x. Hence, A = x. Finally, assume that {x n } is a solution of Eq. (.2) and oscillates about x. Let y n = x n x. Then {y n } satisfies y n+ = f(y n + x) f( x) + g(y n k + x) g( x), n = 0,,..., (2.5) and {y n } oscillates about zero. Since {x n } is bounded, there is a positive constant M such that y n = x n x M, n = 0,,.... Then by noting the Lipschitz property of g, we see that g(y n k + x) g( x) L y n k LM, n k. Let y l and y s be two consecutive members of the solution {y n } with N 0 < l < s such that y l 0, y s+ 0 and y n > 0 for l + n s. (2.6) Let y r = max{y l+, y l+2,..., y s } (2.7)
Global Attractivity in a Nonlinear Difference Equation 237 where y r is chosen as the first one to reach the maximum among y l+, y l+2... y s. We claim that r (l + ) k. (2.8) Suppose, for the sake of contradiction, that r (l + ) > k. Then, y r > y r k > 0. By noting y r k + x > x and (2.), we see that Since f(x) x is decreasing, g(y r k + x) + f(y r k + x) (y r k + x) < 0. (2.9) f(y r k + x) (y r k + x) > f(y r + x) (y r + x) and then it follows from (2.9) that However, on the other hand, (2.5) yields g(y r k + x) + f(y r + x) (y r + x) < 0. g(y r k + x) + f(y r + x) (y r + x) = f(y r + x) f(y r + x) 0 which contradicts (2.9). Hence, (2.8) holds. Now, observe that y n+y n = f(y n + x) f( x) ay n + g(y n k + x) g( x) and so it follows that y n+ a y n n+ a = n a [f(y n+ n + x) f( x) ay n ] + a [g(y n+ n k + x) g( x)]. Summing up from l to r, we see that y r a r y l a l = r and so ( y r = a r y l a + r l a [f(y r j+ j + x) f( x) ay j ] + a [f(y r j+ j + x) f( x) ay j ] + a j+ [g(y j k + x) g( x)] ) a [g(y j+ j k + x) g( x)] ( = a r y l a + l a [f(y r 2 l+ l + x) f( x) ay l ] + a [f(y j+ j + x) f( x) ay j ] ) + r a j+ [g(y j k + x) g( x)] ( = a r a [f(y l+ l + x) f( x)] + + r r 2 a [f(y j+ j + x) f( x) ay j ] ) a j+ [g(y j k + x) g( x)]. (2.0)
238 Chuanxi Qian Then by noting (2.6), f(x) is nondecreasing and f(x) ax is nonincreasing, we see that f(y l + x) f( x) 0 and f(y j + x) f( x) ay j 0, j = l +,..., r. Hence, it follow from (2.0) that and so r y r a r a [g(y j+ j k + x)) g( x))] [ ] y n LM, l n s. [ ] LM Since y l and y s are two arbitrary members of the solution with property (2.6), we see that there is a positive integer N N 0 such that y n [ ]ML, n N. Then, by a similar argument, it can be shown that there is a positive integer N N 0 such that y n [ ]ML, n N. Hence, there is a positive integer N N 0 such that [ ] y n LM, n N. (2.) Now, by noting the Lipschitz property of f(x) and (2.), we see that [ ] g(y n k + x) g( x) L y n k L 2 M, n N + k. Let y l and y s be two consecutive members of the solution {y n } with t 0 l < s such that (2.6) holds. Let y r be defined by (2.7). By a similar argument, we may show that (2.8) holds and ([ ] 2 y r L) M. Then it follows that y n ( [ ]L)2 M, l n s and so again by noting y l and y s are two arbitrary members of the solution with property (2.6), there is a positive integer N 2 N + k such tat y n ([ ]L)2 M, n N 2. Similarly, it can be shown that there is a positive integer N 2 N + K such that y n ([ ]L)2 M, n N 2. Hence, there is a positive integer N 2 N + k such that y n ([ ]L)2 M, n N 2. Finally, by induction, we find that for any positive integer m, there is a positive integer N m with N m as m such that ([ ] m y n L) M, n N m.
Global Attractivity in a Nonlinear Difference Equation 239 [ ] Then, by noting the hypotheses L <, we see that y n 0 as n, and so it follows that x n x as n. The proof is complete. 3 Applications Consider the difference equation x n+ = e p qx n k ax2 n + c, n = 0,,..., (3.) b + x n + e p qx n k where 0 < a <, b, c, p and q are positive constants, and k is a nonnegative integer. Eq. (3.) is in the form of (.2) with f(x) = ax2 b + x Clearly, f is increasing and f(x) ax. By noting we see that f(x) ax is decreasing. Let p qx e and g(x) = c + e. p qx (f(x) ax) = ab2 (b + x) < 0 2 e p qx F (x) = ax2 b + x + c x. + ep qx It is easy to check that F (0) > 0, lim x F (x) = and F (x) < 0. Hence, there is a unique positive number x such that F ( x) = 0, that is, Eq. (3.) has a unique positive equilibrium x, and Now, observe that (x x)(f(x) + g(x) x) < 0, x > 0 and x x. e p qx g (x) = cq ( + e p qx ) and 2 g (x) = cq 2 ep qx ( e p qx ). ( + e p qx ) 3 By noting that x = p q is the only point such that g (x) = 0, we see that g (x) takes maximum when x = p q and g ( p q ) = cq 4. Hence, g is L-Lipschitz with L = cq 4. Then by Theorem 2., we have that following conclusion immediately. Theorem 3.. Assume that [ ] cq 4 <. (3.2) Then every positive solution {x n } of Eq. (3.) tends to its positive equilibrium x as n.
240 Chuanxi Qian We claim that (3.2) is also a sufficient condition for the positive equilibrium x to be globally asymptotically stable when k = 0 or k =. In fact, when k = 0, (3.2) reduces to cq < 4 and the linearized equation of (3.) about the equilibrium x is ( ) a x 2 + 2ab x x n+ = cq x ( x + b) 2 ( + ) 2 n. (3.3) It is obvious that since By noting cq < 4, we see that a x 2 + 2ab x ( x + b) 2 a x 2 + 2ab x ( x + b) 2 cq ( + ) < 2 < x2 + 2b x ( x + b) 2 <. cq ( + ) < 4 2 ( + ) < + a x2 + 2ab x 2 ( x + b) 2 and so Hence, it follows that < a x2 + 2ab x ( x + b) 2 a x 2 + 2ab x ( x + b) 2 cq ( + ). 2 cq ( + ) 2 < and so the zero solution of Eq. (3.3) is asymptotically stable. By the linearization stability theory, the equilibrium x of Eq. (3.) is locally asymptoticly stable. Then by combining this fact and Theorem 3., we see that the equilibrium x of Eq. (3.) is globally asymptoticly stable. When k =, (3.2) reduces to cq < 4 and the linearized equation of (3.) about + a the equilibrium x is x n+ = a x2 + 2ab x ( x + b) 2 It is well known (see, for example [6]) for the linear equation x n cq ( + ) x n. (3.4) 2 y n+ + αy n + βy n = 0, where α and β are constants, a necessary and sufficient condition for the asymptotic stability is α < + β < 2.
Global Attractivity in a Nonlinear Difference Equation 24 Hence, by noting that under the condition cq < 4 + a, a x 2 + 2ab x ( x + b) 2 < + cq ( + ) < + 4 2 ( + ) 2, 2 we see that the zero solution of Eq. (3.4) is asymptoticly stable. Then it follows that the equilibrium x of Eq. (3.) is locally asymptotically stable, which together with Theorem 3. implies that x is globally asymptotically stable. Remark 3.2. In particular, when k = 0, Eq. (3.) reduces to the first order difference equation x n+ = ax2 n e p qxn + c, n = 0,,... (3.5) p qxn b + x n + e which is a biological model mentioned in Section. It has been shown in [8] that if cq < 2(), (3.6) then the positive equilibrium x of Eq. (3.5) is globally asymptotically stable. Clearly, our condition cq < 4 is a significant improvement of (3.6). References [] H. A. El-Morshedy and E. Liz, Convergence to equilibria in discrete population models, J. Diff. Equ. Appl. 2(2005), 7 3. [2] H. A. El-Morshedy and E. Liz, Globally attracting fixed points in higher order discrete population models, J. Math. Biol. 53(2006), 365 384. [3] J. R. Graef and C. Qian, Global stability in a nonlinear difference equation, J. Diff. Equ. Appl. 5(999), 25 270. [4] A. F. Ivanov, On global stability in a nonlinear discrete model, Nonlinear Anal. 23(994), 383 389. [5] G. Karakostas, Ch. G. Philos and Y. G. Sficas, The dynamics of some discrete population models, Nonlinear Anal. 7(99), 069 084. [6] V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 993. [7] S. Padhi and C. Qian, Global attractivity in a higher order nonlinear difference equation, Dyn. Contin. Discrete Impuls. Syst. 9(202), 95 06. [8] G. Papaschinopoulos, C. J. Schinas and G. Ellina, On the dynamics of the solutions of a biological model, J. Diff. Equ. Appl., to appear.
242 Chuanxi Qian [9] C. Qian, Global attractivity in a higher order difference equation with applications, Int. J. Qual. Theory Diff. Equ. Appl. (2007), 23 222. [0] C. Qian, Global attractivity in a higher order difference equation with variable coefficients, J. Diff. Equ. Appl. 8(202), 2 32. [] S. Stević, On positive solutions of a (k + ) th order difference equation, Appl. Math. Lett 9(2006), 427 43. [2] S. Stević, On a class of higher-order difference equations, Chaos Solutions Fractals 42(2009), 38 45. [3] D. Tilman and D. Wedin, Oscillations and chaos in the dynamics of a perennial grass, Nature 353(99), 653 655.