New Results on the SSIE with an Operator of the form FΔ ⊂ Ɛ + Fx′ Involving the Spaces of Strongly Summable and Convergent Sequences Using the Cesàro Method

Bruno De Malafosse

doi:10.3390/axioms10030157

Abstract

Given any sequence a=(an)n≥1 of positive real numbers and any set E of complex sequences, we can use Ea to represent the set of all sequences y=(yn)n≥1 such that y/a=(yn/an)n≥1∈E. In this paper, we use the spaces w∞, w0 and w of strongly bounded, summable to zero and summable sequences, which are the sets of all sequences y such that n−1∑k=1nykn is bounded and tends to zero, and such that y−le∈w0, for some scalarl. These sets were used in the statistical convergence. Then we deal with the solvability of each of the SSIE FΔ⊂Ɛ+Fx′, where Ɛ is a linear space of sequences, F=c0, c, ℓ∞, w0, w or w∞, and F′=c0, c or ℓ∞. For instance, the solvability of the SSIE wΔ⊂w0+sxc relies on determining the set of all sequences x=xnn≥1∈U+ that satisfy the following statement. For every sequence y that satisfies the condition limn→∞n−1∑k=1nyk−yk−1−l=0, there are two sequences u and v, with y=u+v such that limn→∞n−1∑k=1nuk=0 and limn→∞vn/xn=L for some scalars l and L.

Highlights

We write ω for the set of all complex sequences y =k≥1, `∞, c and c0 for the sets of p all bounded, convergent and null sequences, respectively; andp = y ∈ ω : ∑∞ k =1 | y k | < ∞for 1 ≤ p < ∞
Instance, the solvability of the spaces inclusion equations (SSIE) w∆ ⊂ w0 + s x relies on determining the set of all sequences x =n≥1 ∈ U + that satisfy the following statement
We studied the SSIE c ⊂ Dr ∗ EC1 + s x with E ∈ {c, s1 } and s1 ⊂ Dr ∗ (s1 )C1 + s x, where C1 is the Cesàro operator defined by (C1 )n y = n−1 ∑nk=1 yk for all y, and we dealt with the solvability of the SSE associated with the previous SSIE and (c) defined by Dr ∗ EC1 + s x = c with E ∈ {c0, c, s1 } and Dr ∗ EC1 + s x = s1 with E ∈ {c, s1 }