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Bounded solutions of a class of difference equations in Banach spaces producing controlled chaos. (English) Zbl 1142.39013
The author studies the equivalence between convergent and slowly varying sequences defined in a complex Banach space $$X$$. The main result of the paper says that given $$\alpha_j$$, $$j=1,\dots,k$$, real numbers such that $$\sum_{j=1}^k\alpha_j=1$$, and $$P_k(z)=z^k-\alpha_1z^{k-1}-\cdots-\alpha_k$$. If $$(x_n)_{n\in \mathbb{N}} \subset X$$ is such that $$\lim_{n\to\infty}\| x_{n+k}-\sum_{j=1}^k\alpha_jx_{n+k-j}\| =0$$, then the boundedness of the sequence $$(x_n)_{n\in\mathbb{N}}$$ implies $$\lim_{n\to\infty}\| x_{n+1}-x_n\| =0$$ if and only if all the zeros of the polynomial $$P_k(z)$$ belong to the set $${\mathbb C}\backslash\{z: | z| =1,z\not=1\}$$.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A10 Additive difference equations 40A05 Convergence and divergence of series and sequences
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##### References:
 [1] Beverton, R.J.; Holt, S.J., On the dynamics of exploited fish populations, Fish invest London, 19, 1-53, (1957) [2] Berenhaut, K.S.; Bandyopadhyay, D., Monotone convex sequences and Cholesky decomposition of symmetric Toeplitz matrices, Linear algebra appl, 403, 75-85, (2005) · Zbl 1076.15014 [3] Berg, L., Inclusion theorems for non-linear difference equations with applications, J differ equations appl, 10, 4, 399-408, (2004) · Zbl 1056.39003 [4] Berg, L., Corrections to “inclusion theorems for non-linear difference equations with applications,” from [3], J differ equations appl, 11, 2, 181-182, (2005) · Zbl 1080.39002 [5] Berg, L.; Wolfersdorf, L.v., On a class of generalized autoconvolution equations of the third kind, Z anal anwend, 24, 2, 217-250, (2005) · Zbl 1104.45001 [6] Bibby, J., Axiomatisations of the average and a further generalisation of monotonic sequences, Glasgow math J, 15, 63-65, (1974) · Zbl 0291.40003 [7] Borwein, D., Convergence criteria for bounded sequences, Proc Edinburgh math soc, 18, 2, 99-103, (1972) · Zbl 0247.40001 [8] Copson, E.T., On a generalisation of monotonic sequences, Proc Edinburgh math soc, 17, 2, 159-164, (1970) · Zbl 0223.40001 [9] Cushing, J.M.; Kuang, Y., Global stability in a nonlinear difference delay equation model of flour beetle population growth, J differ equations appl, 2, 31-37, (1996) · Zbl 0862.39005 [10] El-Owaidy, H.M.; Ahmed, A.M.; Mousa, M.S., On asymptotic behaviour of the difference equation $$x_{n + 1} = \alpha + \frac{x_{n - 1}^p}{x_n^p}$$, J appl math comput, 12, 1-2, 31-37, (2003) · Zbl 1052.39005 [11] Fan, Y.W.; Berenhaut, K.S., A bound for linear recurrence relations with unbounded order, Comput math appl, 50, 3-4, 509-518, (2005) · Zbl 1085.11006 [12] Fisher, M.E.; Goh, B.S., Stability results for delayed-recruitment models in population dynamics, J math biol, 19, 147-156, (1984) · Zbl 0533.92017 [13] Ishiyama, K.; Saiki, Y., Unstable periodic orbits and chaotic economic growth, Chaos, solitons & fractals, 26, 33-42, (2005) · Zbl 1073.65146 [14] Karakostas, G.L., Convergence of a difference equation via the full limiting sequences method, Differ equ dyn syst, 1, 4, 289-294, (1993) · Zbl 0868.39002 [15] Karakostas, G.L., Asymptotic 2-periodic difference equations with diagonally self-invertible responces, J differ equations appl, 6, 329-335, (2000) · Zbl 0963.39020 [16] Mitrinović, D.S.; Adamović, D.D., Sequencies and series, (1990), Naučna knjiga Beograd [17] Pielou, E.C., An introduction to mathematical ecology, (1969), Wiley-Interscience · Zbl 0259.92001 [18] Pielou, E.C., Population and community ecology, (1974), Gordon and Breach · Zbl 0349.92024 [19] Páles, Z., Hyers-Ulam stability of the Cauchy functional equation on square-symmetric grupoids, Publ math debrecen, 58, 4, 651-666, (2001) · Zbl 0980.39022 [20] Polya, G.; Szego, G., Aufgaben und lehrsatze aus der analysis, (1925), Verlag von Julius Springer Berlin · JFM 51.0173.01 [21] Popa, D., Hyers-Ulam stability of the linear recurrence with constant coefficients, Adv difference equat, 2005, 2, 101-107, (2005) · Zbl 1095.39024 [22] Russell, D.C., On bounded sequences satisfying a linear inequality, Proc Edinburgh math soc, 19, 11-16, (1973) · Zbl 0287.40002 [23] Szydowski, M., Time to-built in dynamics of economic models I: kalecki’s model, Chaos, solitons & fractals, 14, 697-703, (2002) [24] Szydowski, M., Time to-built in dynamics of economic models II: models of economic growth, Chaos, solitons & fractals, 18, 355-364, (2003) [25] Stević, S., Behaviour of the positive solutions of the generalized beddington – holt equation, Panamer math J, 10, 4, 77-85, (2000) · Zbl 1039.39005 [26] Stević, S., A note on bounded sequences satisfying linear inequality, Indian J math, 43, 2, 223-230, (2001) · Zbl 1035.40002 [27] Stević, S., A generalization of the copson’s theorem concerning sequences which satisfy a linear inequality, Indian J math, 43, 3, 277-282, (2001) · Zbl 1034.40002 [28] Stević, S., A global convergence result, Indian J math, 44, 3, 361-368, (2002) · Zbl 1034.39002 [29] Stević, S., Asymptotic behaviour of a sequence defined by iteration with applications, Colloq math, 93, 2, 267-276, (2002) · Zbl 1029.39006 [30] Stević, S., On the recursive sequence xn+1=xn−1/g(xn), Taiwanese J math, 6, 3, 405-414, (2002) [31] Stević, S., A note on periodic character of a difference equation, J differ equations appl, 10, 10, 929-932, (2004) · Zbl 1057.39005 [32] Stević, S., On the recursive sequence $$x_{n + 1} = \alpha + \frac{x_{n - 1}^p}{x_n^p}$$, J appl math comput, 18, 1-2, 229-234, (2005) · Zbl 1078.39013 [33] Stević, S., On positive solutions of a (k+1)-th order difference equation, Appl math lett, 19, 5, 427-431, (2006) · Zbl 1095.39010
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