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Solvability of new (SSIE) involving the continuous and residual spectra of the generalized difference operator B (r, s) on c0

Malafosse Bruno de

Fasciculi Mathematici

2023

nr 66

61--75

Let U+ be the set of all positive sequences. Then, given any sequence z = (zn)n≥1 ∈ U+ and any set E of complex sequences, we write Ez for the set of all sequences y = (yn)n≥1 such that y/z = (yn/zn)n≥1 ∈ E. We use the notation sz = (ℓ∞)z. In this paper, for given r, s≠ 0 and for every λ ∈ ℂ, we determine the set of all positive sequences x = (xn)n≥1 that satisfy the (SSIE) with an operator (c0)B(r,s)−λI ⊂ Ɛ + sx, where Ɛ ⊂ sθ for some θ ∈ U+ is a linear space of sequences, in each of the cases, (1) |λ - r| > |s|, or λ = r, (2) |λ - r| = |s| and (3) |λ - r| < |s| and λ ≠ r. These cases are associated with the continuous and residual spectra σc (B (r, s), c0) and σr (B (r, s), c0), of B (r, s) on c0, determined by Altay and Başar in [2]. We apply these results to the solvability of the (SSIE) (c0)B(r,s)−λI ⊂ s(c)R +sx for all λ ∈ ℂ and R > 0. Then we deal with the (SSIE) (c0)Δ−λI ⊂ bvp + sx and (c0)B(r,s)−λI ⊂ ERɑ + sx, for E = c0, c, or ℓ∞, where Rɑ, ɑ ∈ U+, is the Rhaly matrix. These results extend those stated in [21].

Solvability of certain sequence spaces inclusion equations with operators

Malafosse B. de

Demonstratio Mathematica

2013

Vol. 46, nr 2

299--314

In this paper, we deal with sequence spaces inclusion equations (SSIE), which are determined by an inclusion where each term is a sum or a sum of products of sets of the form a(T) and f(x)(T) where f maps U+ to itself, and (...), the sequence x is the unknown and T is a given triangle. Here, we determine the set of all sequences x with positive entries such that (…) and (…) where (...). We are led to study, among other things, the inclusion equations (…) and (…) where (…) is the operator of first differences defined by (…) for (…) with (…). The first (SSIE) leads to determine the set of all sequences x such that (…) and (…) implies (…). These results generalize some of the results given in [1].