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

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 157
Author(s):  
Bruno de Malafosse
Keyword(s):  
Linear Space ◽  
Statistical Convergence ◽  
Real Numbers ◽  
Positive Real ◽  
Complex Sequences ◽  
Cesàro Method ◽  
Convergent Sequences

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⊂E+Fx′, where E 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.

scholarly journals Extension of Some Results on the (SSIE) and the (SSE) of the Form F ⊂ ℇ + F′ x and ℇ + Fx = F

2017 ◽  
Vol 59 (1) ◽  
pp. 107-123
Author(s):  
Bruno de Malafosse
Keyword(s):  
Linear Space ◽  
Real Numbers ◽  
Positive Real ◽  
Complex Sequences

Abstract 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 ℇ is a linear space of sequences and F′ is either c0, or ℓ∞ and we solve the (SSIE) c0 ⊂ ℇ + sx for ℇ ⊂ (sα)∆ 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 (ℓrp)∆ + 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].

scholarly journals Matrix transformations and application to perturbed problems of some sequence spaces equations with operators

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 5123-5130
Author(s):  
Malafosse de ◽  
Ali Fares ◽  
Ali Ayad
Keyword(s):  
Linear Space ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Real Numbers ◽  
Positive Real ◽  
Complex Sequences ◽  
Perturbed Equation

Given any sequence z = (zn)n?1 of positive real numbers 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; in particular, cz = s(c) z denotes the set of all sequences y such that y/z converges. Starting with the equation Fx = Fb we deal with some perturbed equation of the form ? + Fx = Fb, where ? is a linear space of sequences. In this way we solve the previous equation where ? =(Ea)T and (E,F) ? {(l?,c), (c0,l?), (c0,c), (lp,c), (lp,l?), (w0,l?)} with p ? 1, and T is a triangle.

scholarly journals Statistical convergence of double sequences on probabilistic normed spaces defined by $[ V, \lambda, \mu ]$-summability

2014 ◽  
Vol 33 (2) ◽  
pp. 59-67
Author(s):  
Pankaj Kumar ◽  
S. S. Bhatia ◽  
Vijay Kumar
Keyword(s):  
Statistical Convergence ◽  
Convergence Criteria ◽  
Double Sequences ◽  
Normed Spaces ◽  
Real Numbers ◽  
Positive Real ◽  
Probabilistic Normed Spaces

In this paper, we aim to generalize the notion of statistical convergence for double sequences on probabilistic normed spaces with the help of two nondecreasing sequences of positive real numbers $\lambda=(\lambda_{n})$ and $\mu = (\mu_{n})$  such that each tending to zero, also $\lambda_{n+1}\leq \lambda_{n}+1, \lambda_{1}=1,$ and $\mu_{n+1}\leq \mu_{n}+1, \mu_{1}=1.$ We also define generalized statistically Cauchy double sequences on PN space and establish the Cauchy convergence criteria in these spaces.

scholarly journals Calculations on Matrix Transformations Involving an Infinite Tridiagonal Matrix

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 218
Author(s):  
Ali Fares ◽  
Ali Ayad ◽  
Bruno de Malafosse
Keyword(s):  
Tridiagonal Matrix ◽  
Matrix Transformations ◽  
Real Numbers ◽  
Positive Real ◽  
Complex Sequences

Given any sequence z=znn≥1 of positive real numbers and any set E of complex sequences, we write Ez for the set of all sequences y=ynn≥1 such that y/z=yn/znn≥1∈E; in particular, sz0 denotes the set of all sequences y such that y/z tends to zero. Here, we consider the infinite tridiagonal matrix Br,s,t˜, obtained from the triangle Br,s,t, by deleting its first row. Then we determine the sets of all positive sequences a=ann≥1 such that EaBr,s,t˜⊂Ea, where E=ℓ∞, c0, or c. These results extend some recent results.

STRONGLY (VΛ,A,P) - SUMMABLE SEQUENCE SPACES DEFINED BY A MODULUS

2007 ◽  
Vol 12 (4) ◽  
pp. 419-424 ◽  
Author(s):  
Tunay Bilgin ◽  
Yilmaz Altun
Keyword(s):  
Key Words ◽  
Statistical Convergence ◽  
Sequence Spaces ◽  
Modulus Function ◽  
Complex Numbers ◽  
Infinite Matrix ◽  
Real Numbers ◽  
Positive Real ◽  
Summable Sequence ◽  
De La Vallée Poussin

We introduce the strongly (Vλ,A,p) ‐ summable sequences and give the relation between the spaces of strongly (Vλ,A,p) ‐ summable sequences and strongly (Vλ,A,p) ‐ summable sequences with respect to a modulus function when A = (α ik ) is an infinite matrix of complex numbers and ρ = (pi) is a sequence of positive real numbers. Also we give natural relationship between strongly (Vλ, A,p) ‐ convergence with respect to a modulus function and strongly Sλ (A) ‐ statistical convergence. Key words: De la Vallee‐Poussin mean, modulus function, statistical convergence.

Lifting convergent sequences with networks

1971 ◽  
Vol 70 (1) ◽  
pp. 27-30
Author(s):  
J. H. Webb
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Real Numbers ◽  
Positive Real ◽  
Positive Integers ◽  
Locally Convex ◽  
Convergent Sequences

Definition (Moukoko Priso(2)). A locally convex spaceE[T] is said to have a strict absorbent network of type Σ if there exists in E a familyof absolutely convex absorbent sets such that(1) if {nk} is a sequence of positive integers andfor each k, then the seriesconverges in E[T](2) for each sequence {nk} there is a sequence {λk} of positive real numbers such that, ifand 0 ≤ μk ≤ λkfor each k, then(i) converges in E[T], and(ii) for each p.

scholarly journals A new perspective on paranormed Riesz sequence space of non-absolute type

2015 ◽  
Vol 3 (4) ◽  
pp. 150 ◽  
Author(s):  
Murat Candan
Keyword(s):  
Sequence Space ◽  
Topological Properties ◽  
Real Numbers ◽  
Positive Real ◽  
Band Matrix ◽  
Current Article ◽  
Matrix Class ◽  
Alpha Beta ◽  
New Perspective ◽  
Convergent Sequences

<p>The current article mainly dwells on introducing Riesz sequence space \(r^{q}(\widetilde{B}_{u}^{p})\) which generalized the prior studies of Candan and Güneş [28], Candan and Kılınç [30]  and consists of all sequences whose \(R_{u}^{q}\widetilde{B}\)-transforms are in the space \(\ell(p)\), where \(\widetilde{B}=B(r_{n},s_{n})\) stands for double sequential band matrix \((r_{n})^{\infty}_{n=0}\) and \((s_{n})^{\infty}_{n=0}\) are given convergent sequences of positive real numbers. Some topological properties of the new brand sequence space have been investigated as well as \(\alpha\)- \(\beta\)-and \(\gamma\)-duals. Additionally, we have also constructed the basis of \(r^{q}(\widetilde{B}_{u}^{p})\). Eventually, we characterize a matrix class on the sequence space. These results are more general and more comprehensive than the corresponding results in the literature.</p>

scholarly journals On (λ,µ)-statistical convergence of double sequences on intuitionistic fuzzy normed spaces

Filomat ◽  
2011 ◽  
Vol 25 (2) ◽  
pp. 109-120 ◽  
Author(s):  
Vijay Kumar ◽  
M. Mursaleen
Keyword(s):  
Statistical Convergence ◽  
Double Sequences ◽  
Normed Spaces ◽  
Real Numbers ◽  
Positive Real ◽  
Intuitionistic Fuzzy ◽  
Intuitionistic Fuzzy Normed Spaces

In this paper, we define (?, ?)- statistical convergence and (?, ?)-statistical Cauchy double sequences on intuitionistic fuzzy normed spaces (IFNS in short), where ? = (?n ) and ? = (?m) be two non-decreasing sequences of positive real numbers such that each tending to ? and ?n+1 ? ?n + 1, ?1 = 1; ?m+1 ? ?m + 1, ?1 = 1. We display example that shows our method of convergence is more general for double sequences in intuitionistic fuzzy normed spaces.

On New Classes of Sequence Spaces Inclusion Equations Involving the Sets C0, C, lP, (1 ≤ P ≤ ∞), W0 and W∞

2017 ◽  
Vol 84 (3-4) ◽  
pp. 211 ◽  
Author(s):  
Bruno de Malafosse
Keyword(s):  
Sequence Spaces ◽  
Real Numbers ◽  
Positive Real ◽  
Complex Sequences

<p>Given any sequence a = (a<sub>n</sub>)<sub>n≥1</sub> of positive real numbers and any set <em>E</em> of complex sequences, we write E<sub>a</sub> for the set of all sequences y = (y<sub>n</sub>)<sub>n≥1</sub> such that y/a = (y<sub>n</sub>/a<sub>n</sub>)<sub>n≥1</sub> ∈ E; in particular, c<sub>a</sub> denotes the set of all sequences y such that y/a converges. Let Φ = {c<sub>0</sub>, c, l<sub>∞</sub>, l<sub>p</sub>, w<sub>0</sub>, w<sub>∞</sub>},(p≥1).. In this paper we apply a result stated in [9] and we deal with the class of (SSIE) of the form F ⊂ E<sub>a</sub>+F'<sub>x</sub> where F∈{c<sub>0,</sub>l<sub>p</sub>, w<sub>0</sub>, w<sub>∞</sub>} and E, F' ∈ Φ. We then obtain the solvability of the corresponding (SSIE) in the particular case when a = (r<sup>n</sup>)<sub>n</sub> and we deal with the case when F = F'. Finally we solve the equation E<sub>r</sub> + (l<sub>p</sub>)<sub>x</sub> = l<sub>p</sub> with E = c<sub>0</sub>, c, s<sub>1</sub>, or l<sub>p</sub> (p≥1). These results extend those stated in [10].</p>

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

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