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On a class of third-order nonlinear difference equations. (English) Zbl 1178.39011
The topic of the paper is the third order rational difference (or iteration) equation \[ x_{n+1} = A + \frac{x_n^p}{x_{n-1}^q x_{n-2}^r}, \quad n \in \mathbb{N}_0. \] The authors present five theorems containing some conditions on the positive parameters \(p,q,r\) and \(A\) that assure either unboundedness of at least one solution or boundedness of all solutions.

MSC:
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
39A22 Growth, boundedness, comparison of solutions to difference equations
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[1] Berenhaut, K.; Foley, J.; Stević, S., The global attractivity of the rational difference equation \(y_n = 1 +(y_{n - k} / y_{n - m})\), Proc. am. math. soc., 135, 1133-1140, (2007) · Zbl 1109.39004
[2] Berenhaut, K.; Foley, J.; Stević, S., The global attractivity of the rational difference equation \(y_n = A +(y_{n - k} / y_{n - m})^p\), Proc. am. math. soc., 136, 103-110, (2008) · Zbl 1134.39002
[3] Berenhaut, K.S.; Stević, S., A note on positive nonoscillatory solutions of the difference equation \(x_{n + 1} = \alpha +(x_{n - k}^p / x_n^p)\), J. differ. eqs. appl., 12, 5, 495-499, (2006) · Zbl 1095.39004
[4] Berenhaut, K.; Stević, S., The behaviour of the positive solutions of the difference equation \(x_n = A +(x_{n - 2} / x_{n - 1})^p\), J. differ. eqs. appl., 12, 9, 909-918, (2006) · Zbl 1111.39003
[5] Berenhaut, K.; Stević, S., The difference equation \(x_{n + 1} = \alpha + \left(x_{n - k} / \sum_{i = 0}^{k - 1} c_i x_{n - i}\right)\) has solutions converging to zero, J. math. anal. appl., 326, 1466-1471, (2007) · Zbl 1113.39003
[6] Berezansky, L.; Braverman, E., On impulsive beverton – holt difference equations and their applications, J. differ. eqs. appl., 10, 9, 851-868, (2004) · Zbl 1068.39005
[7] Berezansky, L.; Braverman, E., Sufficient conditions for the global stability of nonautonomous higher order difference equations, J. differ. eqs. appl., 11, 9, 785-798, (2005) · Zbl 1078.39005
[8] Berg, L., On the asymptotics of nonlinear difference equations, Z. anal. anwendungen, 21, 4, 1061-1074, (2002) · Zbl 1030.39006
[9] Berg, L., Inclusion theorems for non-linear difference equations with applications, J. differ. eqs. appl., 10, 4, 399-408, (2004) · Zbl 1056.39003
[10] Berg, L., Oscillating solutions of rational difference equations, Rostock. math. kolloq., 58, 31-35, (2004) · Zbl 1096.39002
[11] Berg, L., Nonlinear difference equations with periodic solutions, Rostock. math. kolloq., 61, 12-19, (2005)
[12] Berg, L., On the asymptotics of difference equation \(x_{n - 3} = x_n(1 + x_{n - 1} x_{n - 2})\), J. differ. eqs. appl., 14, 1, 105-108, (2008) · Zbl 1138.39003
[13] Berg, L.; Stević, S., Periodicity of some classes of holomorphic difference equations, J. differ. eqs. appl., 12, 8, 827-835, (2006) · Zbl 1103.39004
[14] Berg, L.; Stević, S., Linear difference equations mod 2 with applications to nonlinear difference equations, J. differ. eqs. appl., 14, 7, 693-704, (2008) · Zbl 1156.39003
[15] De la Sen, M., About the properties of a modified generalized beverton – holt equation in ecology models, Discrete dyn. nat. soc., 2008, 23, (2008), Article ID 592950 · Zbl 1148.92031
[16] De la Sen, M.; Alonso-Quesada, S., A control theory point of view on beverton – holt equation in population dynamics and some of its generalizations, Appl. math. comput., 199, 2, 464-481, (2008) · Zbl 1137.92034
[17] DeVault, R.; Kent, C.; Kosmala, W., On the recursive sequence \(x_{n + 1} = p +(x_{n - k} / x_n)\), J. differ. eqs. appl., 9, 8, 721-730, (2003) · Zbl 1049.39026
[18] Feuer, J., On the behaviour of solutions of \(x_{n + 1} = p +(x_{n - 1} / x_n)\), Appl. anal., 83, 6, 599-606, (2004) · Zbl 1053.39009
[19] Gutnik, L.; Stević, S., On the behaviour of the solutions of a second-order difference equation, 2007, 14, (2007), Article ID 27562 · Zbl 1180.39002
[20] Hu, L.H.; Li, W.T.; Stević, S., Global asymptotic stability of a second order rational difference equation, J. differ. eqs. appl., 14, 8, 779-797, (2008) · Zbl 1153.39015
[21] Iričanin, B., A global convergence result for a higher-order difference equation, Discrete dyn. nat. soc., 2007, 7, (2007), Article ID 91292 · Zbl 1180.39003
[22] Karakostas, G.L., Convergence of a difference equation via the full limiting sequences method, Differ. eqs. dyn. syst., 1, 4, 289-294, (1993) · Zbl 0868.39002
[23] Karakostas, G.L., Asymptotic 2-periodic difference equations with diagonally self-invertible responses, J. differ. eqs. appl., 6, 329-335, (2000) · Zbl 0963.39020
[24] Karakostas, G.L., Asymptotic behavior of the solutions of the difference equation \(x_{n + 1} = x_n^2 f(x_{n - 1})\), J. differ. eqs. appl., 9, 6, 599-602, (2003) · Zbl 1045.39006
[25] Mishkis, A.D., On some problems of the theory of differential equations with deviating argument, Uspekhi mat. nauk, 32:2, 194, 173-202, (1977)
[26] Ozban, A.Y., On the system of rational difference equations \(x_n = a / y_{n - 3}, y_n = \mathit{by}_{n - 3} / x_{n - q} y_{n - q}\), Appl. math. comput., 188, 1, 833-837, (2007) · Zbl 1123.39006
[27] Ozen, S.; Ozturk, I.; Bozkurt, F., On the recursive sequence \(y_{n + 1} = (\alpha + y_{n - 1}) /(\beta + y_n) + y_{n - 1} / y_n\), Appl. math. comput., 188, 1, 180-188, (2007) · Zbl 1123.39007
[28] Pielou, E.C., An introduction to mathematical ecology, (1969), Wiley Interscience · Zbl 0259.92001
[29] Pielou, E.C., Population and community ecology, (1974), Gordon and Breach · Zbl 0349.92024
[30] E.P. Popov, Automatic Regulation and Control, Nauka, Moscow, Russia, 1966, Russian.
[31] Stević, S., Behaviour of the positive solutions of the generalized beddington – holt equation, Panamer. math. J., 10, 4, 77-85, (2000) · Zbl 1039.39005
[32] Stević, S., Asymptotic behaviour of a sequence defined by iteration with applications, Colloq. math., 93, 2, 267-276, (2002) · Zbl 1029.39006
[33] Stević, S., A global convergence results with applications to periodic solutions, Indian J. pure appl. math., 33, 1, 45-53, (2002) · Zbl 1002.39004
[34] Stević, S., Asymptotic behaviour of a nonlinear difference equation, Indian J. pure appl. math., 34, 12, 1681-1687, (2003) · Zbl 1049.39012
[35] Stević, S., On the recursive sequence \(x_{n + 1} = \left(A / \prod_{i = 0}^k x_{n - i}\right) + \left(1 / \prod_{j = k + 2}^{2(k + 1)} x_{n - j}\right)\), Taiwanese J. math., 7, 2, 249-259, (2003)
[36] Stević, S., A note on periodic character of a difference equation, J. differ. eqs. appl., 10, 10, 929-932, (2004) · Zbl 1057.39005
[37] Stević, S., On the recursive sequence \(x_{n + 1} = \alpha +(x_{n - 1}^p / x_n^p)\), J. appl. math. comput., 18, 1-2, 229-234, (2005) · Zbl 1078.39013
[38] Stević, S., On the recursive sequence \(x_{n + 1} = (\alpha + \beta x_{n - k}) / f(x_n, \ldots, x_{n - k + 1})\), Taiwanese J. math., 9, 4, 583-593, (2005) · Zbl 1100.39014
[39] Stević, S., A short proof of the cushing – henson conjecture, Discrete dyn. nat. soc., 2006, 5, (2006), Article ID 37264 · Zbl 1149.39300
[40] Stević, S., Asymptotic behaviour of a class of nonlinear difference equations, Discrete dyn. nat. soc., 2006, 10, (2006), Article ID 47156
[41] Stević, S., Global stability and asymptotics of some classes of rational difference equations, J. math. anal. appl., 316, 60-68, (2006) · Zbl 1090.39009
[42] Stević, S., On positive solutions of a \((k + 1)\)th order difference equation, Appl. math. lett., 19, 5, 427-431, (2006) · Zbl 1095.39010
[43] Stević, S., Asymptotics of some classes of higher order difference equations, Discrete dyn. nat. soc., 2007, 20, (2007), Article ID 56813 · Zbl 1180.39009
[44] Stević, S., Existence of nontrivial solutions of a rational difference equation, Appl. math. lett., 20, 28-31, (2007) · Zbl 1131.39009
[45] Stević, S., On the recursive sequence \(x_{n + 1} = A +(x_n^p / x_{n - 1}^r)\), Discrete dyn. nat. soc., 2007, 9, (2007), Article ID 40963 · Zbl 1151.39011
[46] Stević, S., On the recursive sequence \(x_n = 1 + \left(\sum_{i = 1}^k \alpha_i x_{n - p_i} / \sum_{j = 1}^m \beta_j x_{n - q_j}\right)\), Discrete dyn. nat. soc., 2007, 7, (2007), Article ID 39404
[47] Stević, S., Boundedness and global stability of a higher-order difference equation, J. differ. eqs. appl., 14, 10-11, 1035-1044, (2008) · Zbl 1161.39011
[48] Stević, S., On the difference equation \(x_{n + 1} = \alpha +(x_{n - 1} / x_n)\), Comput. math. appl., 56, 5, 1159-1171, (2008) · Zbl 1155.39305
[49] Stević, S., Boundedness character of a class of difference equations, Nonlin. anal. TMA, 70, 839-848, (2009) · Zbl 1162.39011
[50] Yang, Y.; Yang, X., On the difference equation \(x_{n + 1} = (\mathit{px}_{n - s} + x_{n - t}) /(\mathit{qx}_{n - s} + x_{n - t})\), Appl. math. comput., 203, 2, 903-907, (2008) · Zbl 1162.39015
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