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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].

Extension of some results on the (SSIE) and the (SSE) of the from F ⊂ Ɛ+F’x and Ɛ+Fx = F

de Malafosse B.

Fasciculi Mathematici

2017

Nr 59

107--123

Given any sequence a = (an)n≥1 of positive real numbers and any set E of complex sequences, we write Ea for the set of all sequences y = (yn)n≥1 such that y/a = (yn/an)n≥1 ∈ E. In this paper we deal with the solvability of the (SSIE) of the form l∞ ⊂ Ɛ + F’x where E is a linear space of sequences and F’ is either c0, or l∞ and we solve the (SSIE) c0 ⊂ Ɛ + sx for Ɛ ⊂ (sα)Δ and α ∈ c0. Then we study the (SSIE) c ⊂ Ɛ + s( c) x and the (SSE) Ɛ +s( c ) x = c. Then we apply the previous results to the solvability of the (SSE) of the form (lpr)Δ ) + Fx = F for p ≥ 1 and F is any of the sets c0, c, or l∞. These results extend some of those given in [8] and [9].

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

Malafosse B. de

Fasciculi Mathematici

2015

Nr 55

109--131

Given any sequence a = (an)n≥1 of positive real numbers and any set E of complex sequences, we write Ea for the set of all sequences y = (yn)n≥1 such that y/a = (yn/an)>)n≥1 Є E; in particular, sa(c) denotes the set of all sequences y such that y/a converges. For any linear space F of sequences, we have Fx = Fb if and only if x/b and b/x Є M (F, F). The question is: what happens when we consider the perturbed equation Ɛ + Fx = Fb where Ɛ is a special linear space of sequences? In this paper we deal with the perturbed sequence spaces equations (SSE), defined by (Ea)Δ + sx(c) = sb(c) where E = c0, or lp, (p > 1) and Δ is the operator of the first difference defined by Δny = yn - yn-1 for all n ≥ 1 with the convention y>sub>0 = 0. For E = c0 the previous perturbed equation consists in determining the set of all positive sequences x = (xn)n that satisfy the next statement. The condition yn/bn → L1 holds if and only if there are two sequences u, v with y = u + v such that Δnu/an → 0 and vn/xn → L2 (n → ∞) for all y and for some scalars L1 and L2. Then we deal with the resolution of the equation (Ea)Δ + sx0 = sb>0 for E = c, or s1, and give applications to particular classes of (SSE).