Solvability of sequence spaces equations of the from (Ea)Δ + Fx = Fb

2015

scientific article

english

- BK space
- spaces of strongly bounded sequences
- sequence spaces equations
- sequence spaces
- equations with operator

EN
Given any sequence *a* = (*a _{n}*)

_{n≥1}of positive real numbers and any set

*E*of complex sequences, we write

*E*for the set of all sequences

_{a}*y*= (

*y*)

_{n}_{n≥1}such that

*y*/

*a*= (

*y*/

_{n}*a*)

_{n}_{n≥1}∈

*E*; in particular,

*s*

_{a}^{(c)}denotes the set of all sequences

*y*such that

*y*/

*a*converges. For any linear space

*F*of sequences, we have

*F*=

_{x}*F*if and only if

_{b}*x*/

*b*and

*b*/

*x*∈

*M*(

*F*,

*F*). The question is: what happens when we consider the perturbed equation

*E*+

*F*=

_{x}*F*where

_{b}*E*is a special linear space of sequences? In this paper we deal with the perturbed sequence spaces equations (SSE), defined by (

*E*)∆ +

_{a}*s*

_{x}^{(c)}=

*s*

_{b}^{(c)}where

*E*=

*c*

_{0}, or

*l*, (

_{p}*p*>1) and ∆ is the operator of the first difference defined by ∆

*=*

_{n}y*y*−

_{n}*y*−1 for all

_{n}*n*≥ 1 with the convention

*y*0 = 0. For

*E*=

*c*

_{0}the previous perturbed equation consists in determining the set of all positive sequences

*x*= (

*x*)

_{n}*that satisfy the next statement. The condition*

_{n}*y*/

_{n}*b*→

_{n}*L*

_{1}holds if and only if there are two sequences

*u*,

*v*with

*y*=

*u*+

*v*such that ∆

*/*

_{n}u*a*→ 0 and

_{n}*v*/

_{n}*x*n →

*L*

_{2}(

*n*→∞)for all

*y*and for some scalars

*L*

_{1}and

*L*

_{2}. Then we deal with the resolution of the equation (

*E*)∆ +

_{a}*s*

_{x}^{0}=

*s*

_{b}^{0}for

*E*=

*c*, or

*s*

_{1}, and give applications to particular classes of (SSE).

109 - 131

CC BY-NC-ND (attribution - noncommercial - no derivatives)

public

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^{}and Poznan Supercomputing and Networking Center

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