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A functional-analytical approach to the asymptotics of recursions

Author: Christian Pötzsche
Journal: Proc. Amer. Math. Soc. 137 (2009), 3297-3307
MSC (2000): Primary 39A11; Secondary 46T20, 47H09, 47H10, 47J05, 65Q05
Published electronically: May 13, 2009
MathSciNet review: 2515399
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Abstract: We propose a functional-analytical method to investigate the long-term behavior of recursions (difference equations). It is based on a formulation of given (implicit) recursions as abstract operator equations in sequence spaces. Solving them using appropriate tools from nonlinear analysis yields quantitative convergence results and equips us with a method to verify summable or subexponential decay.

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  • [ADL97] J. M. Ayerbe Toledano, T. Domínguez Benavides, and G. López Acedo, Measures of noncompactness in metric fixed point theory, Operator Theory: Advances and Applications 99, Birkhäuser Verlag, Basel, 1997. MR 1483889 (99e:47070)
  • [Aga00] R. P. Agarwal, Difference equations and inequalities, 2nd ed., Monogr. Textbooks Pure Appl. Math. 228, Marcel Dekker, New York, 2000. MR 1740241 (2001f:39001)
  • [AZ90] J. Appell and P. P. Zabrejko, Nonlinear superposition operators, Cambridge University Press, Cambridge, 1990. MR 1066204 (91k:47168)
  • [BG80] J. Banaś and K. Goebel, Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics 60, Marcel Dekker, New York, 1980. MR 591679 (82f:47066)
  • [CS67] C. V. Coffman and J. J. Schäffer, Dichotomies for linear difference equations, Math. Ann. 172 (1967), 139-166. MR 0214946 (35:5791)
  • [Dar55] G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84-92. MR 0070164 (16:1140f)
  • [DG05] F. Dal and G. Sh. Guseinov, Properties of discrete composition operators, J. Difference Equ. Appl. 11 (2005), no. 1, 21-27. MR 2112803 (2005g:47119)
  • [EP07] K. Ey and C. Pötzsche, Asymptotic behavior of recursions via fixed point theory, J. Math. Anal. Appl. 337 (2008), no. 2, 1125-1141. MR 2386362 (2009b:39015)
  • [Gor71] S. P. Gordon, Stability and summability of solutions of difference equations, Math. Syst. Theory 5 (1971), 56-65. MR 0303141 (46:2279)
  • [Mad70] I. J. Maddox, Elements of Functional Analysis, Cambridge University Press, London-New York, 1970. MR 0390692 (52:11515)
  • [NP97] R. Naulin and M. Pinto, Stability of discrete dichotomies for linear difference systems, J. Difference Equ. Appl. 3 (1997), no. 2, 101-123. MR 1467605 (98g:39007)
  • [Pin98] M. Pinto, Weighted convergent and bounded solutions of difference systems, Comput. Math. Appl. 36 (1998), no. 10-12, 391-400. MR 1666156 (99j:39009)
  • [PS05] E. N. Petropoulou and P. D. Siafarikas, Existence of complex $ \ell_2$ solutions of linear delay systems of difference equations, J. Difference Equ. Appl. 11 (2005), no. 1, 49-62. MR 2112805 (2005h:39027)
  • [Sad67] B. N. Sadovski{\v{\i\/}}\kern.15em, On a fixed-point principle, Funct. Anal. Appl. 1 (1967), no. 2, 151-153. MR 0211302 (35:2184)
  • [Sas06] B. Sasu, Uniform dichotomy and exponential dichotomy of evolution families on the half-line, J. Math. Anal. Appl. 323 (2006), 1465-1478. MR 2260196 (2007j:34078)
  • [Wil84] A. Wilansky, Summability through functional analysis, North-Holland Mathematics Studies 85, North-Holland, Amsterdam, 1984. MR 738632 (85d:40006)

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Additional Information

Christian Pötzsche
Affiliation: Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, D-85748 Garching, Germany

Keywords: Recursion, iteration, nonautonomous difference equation, attractivity, $\ell ^p$-stability, admissibility, sequence space, functional-analytical method, measure of noncompactness
Received by editor(s): June 20, 2007
Received by editor(s) in revised form: August 27, 2007
Published electronically: May 13, 2009
Communicated by: Andreas Seeger
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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