Asymptotic behavior of solutions of difference equations in Banach spaces

González, Cristóbal; Jiménez-Melado, Antonio

doi:10.1090/S0002-9939-00-05490-3

Asymptotic behavior of solutions of difference equations in Banach spaces
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by Cristóbal González and Antonio Jiménez-Melado PDF

Proc. Amer. Math. Soc. 128 (2000), 1743-1749 Request permission

Abstract:

In this paper we consider the first order difference equation \[ \Delta x_n = \sum _{i=0}^\infty a_n^i f(x_{n+i}), \] 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 \[ \Delta (q_n \Delta x_n) + p_n f(x_n) =0 \] to have asymptotically constant solutions.

References

Andrzej Drozdowicz and Jerzy Popenda, Asymptotic behavior of the solutions of the second order difference equation, Proc. Amer. Math. Soc. 99 (1987), no. 1, 135–140. MR 866443, DOI 10.1090/S0002-9939-1987-0866443-0
P. K. Lin and Y. Sternfeld, Convex sets with the Lipschitz fixed point property are compact, Proc. Amer. Math. Soc. 93 (1985), no. 4, 633–639. MR 776193, DOI 10.1090/S0002-9939-1985-0776193-5
Jerzy Popenda and Ewa Schmeidel, On the asymptotic behavior of solutions of linear difference equations, Publ. Mat. 38 (1994), no. 1, 3–9. MR 1291948, DOI 10.5565/PUBLMAT_{3}8194_{0}1
Ewa Schmeidel, On the asymptotic behaviour of solutions of difference equations, Demonstratio Math. 30 (1997), no. 1, 193–197. MR 1446611