# zbMATH — the first resource for mathematics

##### Examples
 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.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 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)
On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence. (English) Zbl 1250.39002

The authors study asymptotic properties of solutions of a linear Volterra difference equation

${x}_{n+1}=a\left(n\right)+b\left(n\right)x\left(n\right)+\sum _{i=0}^{n}K\left(n,i\right)x\left(i\right),$

where $n\in {ℕ}_{0}=\left\{0,1,2,\cdots \right\}$, $x:{ℕ}_{0}\to ℝ$, $a:{ℕ}_{0}\to ℝ$, $K:{ℕ}_{0}×{ℕ}_{0}\to ℝ$, $b:{ℕ}_{0}\to ℝ\setminus \left\{0\right\}$ is $\omega$-periodic, $\omega \in ℕ=\left\{1,2,\cdots \right\}$. Schauder’s fixed point technique is applied to obtain sufficient conditions for the validity of a property of solutions that, for every admissible constant $c\in ℝ$, there exists a solution $x=x\left(n\right)$ such that

$x\left(n\right)\sim \left(c+\sum _{i=0}^{n-1}\frac{a\left(i\right)}{\beta \left(i+1\right)}\right)\beta \left(n\right),$

where $\beta \left(n\right)={\prod }_{j=0}^{n-1}b\left(j\right)$, for $n\to \infty$, and inequalities for solutions are derived. Relevant comparisons and illustrative examples are given as well.

##### MSC:
 39A10 Additive difference equations 39A06 Linear equations (difference equations) 39A22 Growth, boundedness, comparison of solutions (difference equations) 39A23 Periodic solutions (difference equations)
##### References:
 [1] R.P. Agarwal, Difference equations and inequalities, Theory, methods, and applications, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 228. Marcel Dekker, Inc., New York, 2000. · Zbl 0952.39001 [2] Appleby, J.; Györi, I.; Reynolds, D.: On exact convergence rates for solutions of linear systems of Volterra difference equations, J. diff. Equat. appl. 12, 1257-1275 (2006) · Zbl 1119.39003 · doi:10.1080/10236190600986594 [3] Diblík, J.; Růžičková, M.; Schmeidel, E.: Asymptotically periodic solutions of Volterra difference equations, Tatra mt. Math. publ. 43, 43-61 (2009) · Zbl 1212.39020 [4] Diblík, J.; Schmeidel, E.; Růžičková, M.: Existence of asymptotically periodic solutions of system of Volterra difference equations, J. diff. Equat. appl. 15, 1165-1177 (2009) · Zbl 1180.39022 · doi:10.1080/10236190802653653 [5] Diblík, J.; Schmeidel, E.; Růžičková, M.: Asymptotically periodic solutions of Volterra system of difference equations, Comput. math. Appl. 59, 2854-2867 (2010) · Zbl 1202.39013 · doi:10.1016/j.camwa.2010.01.055 [6] J. Diblík, M. Růžičková, E. Schmeidel, M. Zbacedil;szyniak, Weighted asymptotically periodic solutions of linear Volterra difference equations, Abstract and Applied Analysis, Article ID 370982, 2011, 14 pages. [7] S.N. Elaydi, An introduction to difference equations, 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, 2005. [8] Elaydi, S. N.; Murakami, S.: Uniform asymptotic stability in linear Volterra difference equations, J. differ. Equat. appl. 3, 203-218 (1998) · Zbl 0891.39013 · doi:10.1080/10236199808808097 [9] Gil, M.; Medina, R.: Nonlinear Volterra difference equations in space lp, Discrete dyn. Nat. soc. 2, 301-306 (2004) · Zbl 1071.39006 · doi:10.1155/S1026022604312021 [10] I. Györi, L. Horváth, Asymptotic representation of the solutions of linear Volterra difference equations, Adv. Difference Equ., Art. ID 932831, (2008), 22 pages. · Zbl 1146.39011 · doi:10.1155/2008/932831 [11] Györi, I.; Reynolds, D.: On asymptotically periodic solutions of linear discrete Volterra equations, Fasc. math. 44, 53-67 (2010) · Zbl 1210.39009 [12] Kocić, V. L.; Ladas, G.: Global behavior of nonlinear difference equations of higher order with applications, mathematics and its applications, (1993) [13] Medina, R.: Asymptotic behavior of solutions of Volterra difference equations with finite linear part, Nonlinear stud. 8, 87-95 (2001) · Zbl 0994.39006 [14] Medina, R.: Asymptotic behavior of Volterra difference equations, Comput. math. Appl. 41, 679-687 (2001) · Zbl 0985.39013 · doi:10.1016/S0898-1221(00)00312-6 [15] Medina, R.: Asymptotic equivalence of Volterra difference systems, Int. J. Differ. equat. Appl. 1, 53-64 (2000) · Zbl 0953.39003 [16] Medina, R.: The asymptotic behavior of the solutions of a Volterra difference equation, Comput. math. Appl. 34, 19-26 (1997) · Zbl 0880.39011 · doi:10.1016/S0898-1221(97)00095-3 [17] Messina, E.; Muroya, Y.; Russo, E.; Vecchio, A.: Convergence of solutions for two delays Volterra integral equations in the critical case, Appl. math. Lett. 23, No. 10, 1162-1165 (2010) · Zbl 1197.45006 · doi:10.1016/j.aml.2010.05.002 [18] Morchało, J.: Volterra summation equations and second order difference equations, Math. bohem. 135, 41-56 (2010) · Zbl 1224.39004 · doi:http://www.emis.de/journals/MB/135.1/index.html [19] Morchało, J.; Szmańda, A.: Asymptotic properties of solutions of some Volterra difference equations and second-order difference equations, nonlinear anal., theory methods appl., ser. A, Theory methods 63, e801-e811 (2005) · Zbl 1224.39022 · doi:10.1016/j.na.2005.02.007 [20] J. Musielak, Wstep do analizy funkcjonalnej, PWN, Warszawa, 1976 (in Polish). [21] Z. Šmarda, Singular Cauchy initial value problem for certain classes of integro-differential equations, Adv. Diff. Equat., 2010 (2010), Article ID 810453, 13 p. · Zbl 1192.45002 · doi:10.1155/2010/810453 [22] Song, Y.; Baker, C. T. H.: Admissibility for discrete Volterra equations, J. diff. Equat. appl. 12, 433-457 (2006) · Zbl 1107.39012 · doi:10.1080/10236190600563260 [23] S. Stević, On global periodicity of a class of difference equations, Discrete Dyn. Nat. Soc., 2007 (2007), Article ID 23503, (2007), 10 pages. · Zbl 1180.39005 · doi:10.1155/2007/23503 [24] Werbowski, J.: An estimation of the solution of the system of nonlinear equations of Volterra type with delay, Fasc. math. 9, 67-75 (1975) · Zbl 0362.45002 [25] Zeidler, E.: Nonlinear functional analysis and its application I, fixed-point theorems, (1986) · Zbl 0583.47050