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Journal of Applied Mathematics and Computing
Published in: Journal of Applied Mathematics and Computing 1-2/2013

01-07-2013 | Applied mathematics

Almost automorphy profile of solutions for difference equations of Volterra type

Authors: Ravi P. Agarwal, Claudio Cuevas, Filipe Dantas

Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2013

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Abstract

This work deals with the almost automorphic profile of solutions of the nonlinear Volterra difference equation \(u(n+1) = \lambda\sum_{j=-\infty}^{n}a(n-j)u(j) + f(n,u(n))\), n∈ℤ, for λ in a distinguished subset of the complex plane, where a(n) is a complex summable sequence and the perturbation f is a non-Lipschitz nonlinearity. Many illustrating remarks and examples are considered.
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Footnotes
1
This seems reasonable and necessary since the uniform continuity condition is the main condition needed for the composition theorems of almost periodic functions, asymptotically almost periodic functions and pseudo-almost periodic functions (see [2, 32]).
 
2
The convex hull of a set K is the set of all convex combinations of point in K: co(K):={θ 1 x 1+⋯+θ k x k :x i ∈K, θ i ≥0, i=1,…,k; θ 1+⋯+θ k =1}. As the name suggests, the convex hull co(K) is always convex. It is the smallest convex set that contain K.
 
Literature
1.
Agarwal, R.P.: Difference Equations and Inequalities. Monographs and Textbooks in Pure and Applied Mathematics, vol. 155. Dekker, New York (1992) MATH
2.
Agarwal, R.P., de Andrade, B., Cuevas, C.: Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Nonlinear Anal., Real World Appl. 11(5), 3532–3554 (2010) MathSciNetCrossRefMATH
3.
Agarwal, R.P., Cuevas, C., Frasson, M.: Semilinear functional difference equations with infinite delay. Math. Comput. Model. 55(3–4), 1083–1105 (2012) MathSciNetCrossRefMATH
4.
Allouche, J.-P., Bacher, R.: Toeplitz sequences, paperfolding, towers of Hanoi and progression-free sequences of integers. Enseign. Math. (2) 38(3–4), 315–327 (1992) MathSciNetMATH
5.
Araya, D., Castro, R., Lizama, C.: Almost automorphic solutions of difference equations. Adv. Differ. Equ. 2009, 591380 (2009) MathSciNetCrossRef
6.
Arrieta, J.M., Carvalho, A.N.: Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations. Trans. Am. Math. Soc. 352(1), 285–310 (1999) MathSciNetCrossRef
7.
8.
Bochner, S., Von Neumann, J.: On compact solutions of operational-differential equations. I. Ann. Math. (2) 36(1), 255–291 (1935) CrossRef
9.
Cardoso, F., Cuevas, C.: Exponential dichotomy and boundedness for retarded functional difference equations. J. Differ. Equ. Appl. 15(3), 261–290 (2009) MathSciNetCrossRefMATH
10.
Castro, A., Cuevas, C.: Perturbation theory, stability, boundedness and asymptotic behavior for second order evolution in discrete time. J. Differ. Equ. Appl. 17(3), 327–358 (2011) MathSciNetCrossRefMATH
11.
Corduneanu, C.: Almost periodicity in functional equations, differential equations, chaos and variational problems. Prog. Nonlinear Differ. Equ. Appl. 75, 157–163 (2008) MathSciNetCrossRef
12.
Cuevas, C., de Souza, J.C.: A perturbation theory for the discrete harmonic oscillator equation. J. Differ. Equ. Appl. 16(12), 1413–1428 (2010) CrossRefMATH
13.
Cuevas, C., Lizama, C.: Semilinear evolution equations of second order via maximal regularity. Adv. Differ. Equ. 2008, 316207 (2008) MathSciNetCrossRef
14.
15.
Cuevas, C., Lizama, C.: Semilinear evolution equations on discrete time and maximal regularity. J. Math. Anal. Appl. 361(1), 234–245 (2010) MathSciNetCrossRefMATH
16.
Cuevas, C., Pinto, M.: Convergent solutions of linear functional difference equations in phase space. J. Math. Anal. Appl. 277(1), 324–341 (2003) MathSciNetCrossRefMATH
17.
18.
De Andrade, B., Cuevas, C., Henríquez, E.: Asymptotic periodicity and almost automorphy for a class of Volterra integro-differential equations. Math. Methods Appl. Sci. (2012). doi:10.​1002/​mma.​1607
19.
Diagana, T., N’Guérékata, G., Minh, N.V.: Almost automorphic solutions of evolution equations. Proc. Am. Math. Soc. 132(11), 3289–3298 (2004) CrossRefMATH
20.
Elaydi, S.: An Introduction to Difference Equations, 3rd edn. Undergraduate Texts in Mathematics. Springer, New York (2005) MATH
21.
Fink, A.M.: Almost Differential Equations. Lecture Notes in Mathematics, vol. 377. Springer, Berlin (1974) MATH
22.
Gjerde, R., Johansen, O.: Bratteli-Vershik models for Cantor minimal systems: applications to Toeplitz flows. Ergod. Theory Dyn. Syst. 20(6), 1687–1710 (2000) MathSciNetCrossRefMATH
23.
Goldstein, J.A., N’Guérékata, G.M.: Almost automorphic solutions of semilinear evolution equations. Proc. Am. Math. Soc. 133(8), 2401–2408 (2005) CrossRefMATH
24.
Granas, A., Dugundji, J.: Fixed Point Theory. Springer Monographs in Mathematics. Springer, New York (2003) CrossRefMATH
25.
Halanay, A.: Solutions périodiques et presque-périodiques des systèmes d’équations aux différences finies. Arch. Ration. Mech. Anal. 12, 134–149 (1963) MathSciNetCrossRefMATH
26.
27.
Istrăţescu, V.I.: Fixed Point Theory, An Introduction. Mathematics and Its Applications, vol. 7. Reidel, Dordrecht (1981) CrossRefMATH
28.
Kolmanovskii, V., Shaikhet, L.: Some conditions for boundedness of solutions of difference Volterra equations. Appl. Math. Lett. 16(6), 857–862 (2003) MathSciNetCrossRefMATH
29.
Kolmanovskii, V.B., Myshkis, A.D.: Stability in the first approximation of some Volterra difference equations. J. Differ. Equ. Appl. 3(5–6), 563–569 (1998) MathSciNetMATH
30.
Kolmanovskii, V.B., Myshkis, A.D., Richard, J.-P.: Estimate of solutions for some Volterra difference equations. Nonlinear Anal. 40(1–8), 345–363 (2000) MathSciNetCrossRefMATH
31.
Kolmanovskii, V.B., Castellanos-Velasco, E., Torres-Muñoz, J.A.: A survey: stability and boundedness of Volterra difference equations. Nonlinear Anal. 53(7–8), 861–928 (2003) MathSciNetCrossRefMATH
32.
Li, H.-X., Huang, F.-L., Li, J.-Y.: Composition of pseudo almost-periodic functions and semilinear differential equations. J. Math. Anal. Appl. 255(2), 436–446 (2001) MathSciNetCrossRefMATH
33.
Liu, J.-h., Song, X.-q.: Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations. J. Funct. Anal. 258(1), 196–207 (2010) MathSciNetCrossRefMATH
34.
Minh, N.V., Naito, T., N’Guérékata, G.: A spectral countability condition for almost automorphy of solutions of differential equations. Proc. Am. Math. Soc. 134(11), 3257–3266 (2006) CrossRefMATH
35.
N’Guérékata, G.M.: Almost Automorphic and Almost Periodic Functions in Abstract Spaces. Kluwer Academic/Plenum, New York (2001) CrossRefMATH
36.
N’Guérékata, G.M.: Topics in Almost Automorphy. Springer, New York (2005) MATH
37.
38.
Song, Y.: Periodic and almost periodic solutions of functional difference equations with finite delay. Adv. Differ. Equ. 2007, 68023 (2007)
39.
Song, Y.: Asymptotically almost periodic solutions of nonlinear Volterra difference equations with unbounded delay. J. Differ. Equ. Appl. 14(9), 971–986 (2008) CrossRefMATH
40.
Song, Y.: Positive almost periodic solutions of nonlinear discrete systems with finite delay. Comput. Math. Appl. 58(1), 128–134 (2009) MathSciNetCrossRefMATH
41.
Song, Y., Tian, H.: Periodic and almost periodic solutions of nonlinear discrete Volterra equations with unbounded delay. J. Comput. Appl. Math. 205(2), 859–870 (2007) MathSciNetCrossRefMATH
42.
43.
Walther, A.: Fastperiodische Folgen und ihre Fouriersche Analysis. Alti Congresso Inst. Mat. Bologna 2, 289–298 (1928)
44.
Walther, A.: Fastperiodische Folgen und Potenzreihen mit fastperiodischen Koeffizienten. Abh. Math. Sem. Hamburg Univ. 6, 217–234 (1928) CrossRefMATH
45.
Yi, Y.: On almost automorphic oscillations. In: Differences and Differential Equations. Fields Institute Communications, vol. 42, pp. 75–99. Am. Math. Soc., Providence (2004)
46.
Zaidman, S.: Almost automorphic solutions of some abstract evolution equations. Rend. - Ist. Lomb., Accad. Sci. Lett. A 110(2), 578–588 (1976) MathSciNet
47.
Zaidman, S.: Behavior of trajectories of C 0-semigroups. Rend. - Ist. Lomb., Accad. Sci. Lett. A 114, 205–208 (1980–1982). MathSciNet
48.
49.
Zaidman, S.: Topics in Abstract Differential Equations. II. Pitman Research Notes in Mathematics, vol. 321. Longman Scientific & Technical, Harlow (1995) MATH
Metadata
Title
Almost automorphy profile of solutions for difference equations of Volterra type
Authors
Ravi P. Agarwal
Claudio Cuevas
Filipe Dantas
Publication date
01-07-2013
Publisher
Springer-Verlag
Published in
Journal of Applied Mathematics and Computing / Issue 1-2/2013
Print ISSN: 1598-5865
Electronic ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-012-0615-3

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