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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic behavior of solutions
of difference equations in Banach spaces


Authors: Cristóbal González and Antonio Jiménez-Melado
Journal: Proc. Amer. Math. Soc. 128 (2000), 1743-1749
MSC (1991): Primary 39A10; Secondary 47N99
Published electronically: February 3, 2000
MathSciNet review: 1695135
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the first order difference equation

\begin{displaymath}\Delta x_n = \sum _{i=0}^\infty a_n^i f(x_{n+i}), \end{displaymath}

and give necessary and sufficient conditions so that there exist solutions which are asymptotically constant. These results generalize those given earlier by Popenda and Schmeidel. As an application we give necessary and sufficient conditions for the second order difference equation

\begin{displaymath}\Delta (q_n \Delta x_n) + p_n f(x_n) =0 \end{displaymath}

to have asymptotically constant solutions.


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Additional Information

Cristóbal González
Affiliation: Departamento de Análisis Matemático, Universidad de Málaga, Fac. Ciencias, 29071 Málaga, Spain
Email: gonzalez@anamat.cie.uma.es

Antonio Jiménez-Melado
Affiliation: Departamento de Análisis Matemático, Universidad de Málaga, Fac. Ciencias, 29071 Málaga, Spain
Email: jimenez@anamat.cie.uma.es

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05490-3
PII: S 0002-9939(00)05490-3
Keywords: Difference equation, asymptotic behavior
Received by editor(s): July 21, 1998
Published electronically: February 3, 2000
Additional Notes: This research was partially supported by a grant from Ministerio de Educación y Cultura (Spain) PB97-1081, and from La Junta de Andalucía
Communicated by: Hal L. Smith
Article copyright: © Copyright 2000 American Mathematical Society