zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Global stability of a difference equation with maximum. (English) Zbl 1167.39007

The main results of the paper are the following two theorems.

Theorem 1. Every positive solution to the difference equation

x n =max1 x n-1 α 1 ,1 x n-2 α 2 ,...,1 x n-k α k ,n 0 ,

where k, α i >0, i=1, ..., k and α 1 α k (0,1), converges to one.

Theorem 2. Assume that k, α i (0,1), i=1, ..., k and A i >0, i=1, ..., k. Then every positive solution to

x n =maxA 1 x n-1 α 1 ,A 2 x n-2 α 2 ,...,A k x n-k α k ,n 0

converges to max 1ik A i 1 α i +1 .

Moreover, Theorem 2 solves the conjecture proposed by the author of this paper [Appl. Math. Comput. 210, No. 2, 525–529 (2009; Zbl 1167.39007)] and by F. Sun [Discrete Dyn. Nat. Soc. 2008, Article ID 243291, 6 p. (2008; Zbl 1155.39008)].

39A11Stability of difference equations (MSC2000)
39A20Generalized difference equations
[1]Berenhaut, K.; Foley, J.; Stević, S.: Boundedness character of positive solutions of a MAX difference equation, J. differ. Equations appl. 12, No. 12, 1193-1199 (2006) · Zbl 1116.39001 · doi:10.1080/10236190600949766
[2]&ccedil, C.; Inar; Stević, S.; Yalçinkaya, I.: On positive solutions of a reciprocal difference equation with minimum, J. appl. Math. comput. 17, No. 1 – 2, 307-314 (2005)
[3]El-Morshedy, H. A.: New explicit global asymptotic stability criteria for higher order difference equations, J. math. Anal. appl. 336, No. 1, 262-276 (2007) · Zbl 1186.39022 · doi:10.1016/j.jmaa.2006.12.049
[4]E.M. Elsayed, S. Stević, On the max-type equation xn+1=maxAxn,xn-2, Nonlinear Anal. TMA, 2008, in press, doi:10.1016/j.na.2008.11.016.
[5]Feuer, J.: On the eventual periodicity of xn+1=max1xn,Anxn-1 with a period-four parameter, J. difference equ. Appl. 12, No. 5, 467-486 (2006) · Zbl 1095.39016 · doi:10.1080/10236190600574002
[6]Grove, E. A.; Ladas, G.: Periodicities in nonlinear difference equations, (2005)
[7]Kent, C. M.; Radin, M. A.: On the boundedness nature of positive solutions of the difference equation xn+1=maxAnxn,Bnxn-1, with periodic parameters, Dyn. contin. Discrete impuls. Syst. ser. B appl. Algorithms, No. suppl., 11-15 (2003)
[8]Mishev, D. P.; Patula, W. T.; Voulov, H. D.: A reciprocal difference equation with maximum, Comput. math. Appl. 43, No. 8 – 9, 1021-1026 (2002) · Zbl 1050.39015 · doi:10.1016/S0898-1221(02)80010-4
[9]Mishkis, A. D.: On some problems of the theory of differential equations with deviating argument, Uspekhi mat. Nauk 32:2, No. 194, 173-202 (1977)
[10]Patula, W. T.; Voulov, H. D.: On a MAX type recurrence relation with periodic coefficients, J. difference equ. Appl. 10, No. 3, 329-338 (2004) · Zbl 1050.39017 · doi:10.1080/10236190310001659741
[11]Popov, E. P.: Automatic regulation and control, (1966)
[12]Szalkai, I.: On the periodicity of the sequence xn+1=maxA0xn,A1xn-1,...,Akxn-k, J. differ. Equations appl. 5, 25-29 (1999)
[13]Stević, S.: Behavior of the positive solutions of the generalized beddington – Holt equation, Panamer. math. J. 10, No. 4, 77-85 (2000) · Zbl 1039.39005
[14]S. Stević, Some open problems and conjectures on difference equations, http://www.mi.sanu.ac.yu/colloquiums/mathcollprograms/mathcoll.apr2004.htm.
[15]S. Stević, Boundedness character of a max-type difference equation, in: Conference in Honour of Allan Peterson, Book of Abstracts, Novacella, Italy, July 26 – August 02, 2007, p. 28.
[16]S. Stević, On the recursive sequence xn+1=A+xnpxn-1r, Discrete Dyn. Nat. Soc., vol. 2007, Article ID 40963, 2007, 9 pages. · Zbl 1151.39011 · doi:10.1155/2007/40963
[17]S. Stević, On behavior of a class of difference equations with maximum, Mathematical Models in Engineering, Biology and Medicine, in: Conference on Boundary Value Problems, Book of abstracts, Santiago de Compostela, Spain, September 16 – 19, 2008, p. 35.
[18]Stević, S.: On the recursive sequence xn+1=maxc,xnpxn-1p, Appl. math. Lett. 21, No. 8, 791-796 (2008)
[19]Stević, S.: Boundedness character of a class of difference equations, Nonlinear anal. TMA 70, 839-848 (2009) · Zbl 1162.39011 · doi:10.1016/j.na.2008.01.014
[20]F. Sun, On the asymptotic behavior of a difference equation with maximum, Discrete Dyn. Nat. Soc., vol. 2008, Article ID 243291, 2008, 6 pages. · Zbl 1155.39008 · doi:10.1155/2008/243291
[21]Voulov, H. D.: Periodic solutions to a difference equation with maximum, Proc. amer. Math. soc. 131, No. 7, 2155-2160 (2003) · Zbl 1019.39005 · doi:10.1090/S0002-9939-02-06890-9
[22]Voulov, H. D.: On the periodic nature of the solutions of the reciprocal difference equation with maximum, J. math. Anal. appl. 296, No. 1, 32-43 (2004) · Zbl 1053.39023 · doi:10.1016/j.jmaa.2004.02.054
[23]Voulov, H. D.: On a difference equation with periodic coefficients, J. difference equ. Appl. 13, No. 5, 443-452 (2007) · Zbl 1121.39011 · doi:10.1080/10236190701264651
[24]I. Yalçinkaya, B.D. Iričanin, C. Ccedil;inar, On a max-type difference equation, Discrete Dyn. Nat. Soc., vol. 2007, Article ID 47264, 2007, 11 pages.
[25]Yang, Y.; Yang, X.: On the difference equation xn+1=pxn-s+xn-tqxn-s+xn-t, Appl. math. Comput. 203, No. 2, 903-907 (2008) · Zbl 1162.39015 · doi:10.1016/j.amc.2008.03.023
[26]Yang, X.; Liao, X.: On a difference equation with maximum, Appl. math. Comput. 181, 1-5 (2006) · Zbl 1148.39303 · doi:10.1016/j.amc.2006.01.005