Banach Sequence Spaces Research Articles

Existence of solutions of infinite system of nonlinear sequential fractional differential equations

In a recent paper (Filomat 32:4577–4586, 2018) the authors have investigated the existence and uniqueness of a solution for a nonlinear sequential fractional differential equation. To present an analytical improvement for Fazli–Nieto’s results with some conditions removed based on a new technique is the main objective of this paper. In addition, we introduce an infinite system of nonlinear sequential fractional differential equations and discuss the existence of a solution for them in the classical Banach sequence spaces c_{0} and ell_{p} by applying the Darbo fixed point theorem. Moreover, the proposed method is applied to several examples to show the clarity and effectiveness.

Ramsey theory without pigeonhole principle and the adversarial Ramsey principle

We develop a general framework for infinite-dimensional Ramsey theory with and without pigeonhole principle, inspired by Gowers’ Ramsey-type theorem for block sequences in Banach spaces and by its exact version proved by Rosendal. In this framework, we prove the adversarial Ramsey principle for Borel sets, a result conjectured by Rosendal that generalizes at the same time his version of Gowers’ theorem and Borel determinacy of games on integers.

Correction to: Fine spectra of the discrete generalized Cesàro operator on Banach sequence spaces

In the original article, in Theorem 7.4 (in all statements from (1) to (9)): the symbol C0 should be replaced by bv.

Fine spectra of the discrete generalized Cesàro operator on Banach sequence spaces

This paper concerns the spectrum and the fine spectrum of the discrete generalized Cesaro operator $$C_{t}$$ , where $$0\le t<1$$ , on Banach sequence spaces close to $$\ell ^{1}$$ and $$\ell ^{\infty }$$ . We derive some compactness results for the operator $$C_{t}$$ to describe the spectrum. Our technique involves standard operator theory and summability theory.

On a class of infinite system of third-order differential equations in lp via measure of noncompactness

In this paper, with the help of measure of noncompactness together with Darbo-type fixed point theorem, we focus on the infinite system of third-order differential equations u???i + au??i + bu?i + cui = fi(t, u1(t), u2(t),...) where fi ? C(R x R?,R) is ?-periodic with respect to the first coordinate and a,b,c ? R are constants. The aim of this paper is to obtain the results with respect to the existence of ?-periodic solutions of the aforementioned system in the Banach sequence space lp (1 ? p &lt; ?) utilizing the respective Green?s function. Furthermore, some examples are provided to support our main results.

Solvability for infinite systems of fractional differential equations in Banach sequence spaces lp and c0

This paper is devoted to an infinite system of nonlinear fractional differential equations in the Banach spaces c0 and lp with p ? 1. Existence results are obtained, by using the theory of measure of noncompactness and a new generalization of Darbo?s fixed point theorem. Some examples are also included to show the efficiency of our results.

Solvability of infinite system of nonlinear singular integral equations in the C(Itimes I, c) space and modified semi-analytic method to find a closed-form of solution

‎In this article‎, ‎we discuss about solvability of infinite systems of singular integral equations with two variables in the Banach sequence space $C(I times I‎, ‎c)$ by applying measure of noncompactness‎ and Meir-Keeler condensing operators‎. By presenting an example, we have illustrated our results‎. ‎For validity of the results we introduce a modified semi-analytic method in the case of two variables to make an iteration algorithm to find a closed-form of solution for the above problem‎. ‎The numerical results show that the produced sequence for approximating the solution of example is in the $c$ space with a high accuracy‎.

Disjoint hypercyclic weighted pseudoshift operators generated by different shifts

Let I be a countably infinite index set, and let X be a Banach sequence space over I. In this article, we characterize the disjoint hypercyclic and supercyclic weighted pseudoshift operators on X in terms of the weights, the OP-basis, and the shift mappings on I. Also, the shifts on weighted Lp spaces of a directed tree and the operator weighted shifts on ℓ2(Z,K) are investigated as special cases.

Delay Differential Equations in Sequence Spaces

The standard delay equations are newly studied in the context of classical separable Banach Sequence Spaces. As a classical solution is shown to exist, the associated semigroup and its infinitesimal generator are found, and some important properties of those operators are proven, including some spectral properties. As an application, it is shown how can these results be used to characterize the constrained null-controllability.

A remark on a paper of P. B. Djakov and M. S. Ramanujan

Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We show that if there exists a continuous linear unbounded operator between l-K\{o}the spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding K\{o}the matrices when every continuous linear operator between l-K\{o}the spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-K\{o}the spaces, under a splitting condition, causes the existence of a common basic subspace.

Polynomial Stability and Polynomial Instability for Skew-Evolution Semiflows

In this paper, we give characterizations for a polynomial stability of skew-evolution semiflows, using Banach spaces of sequences or functions. We obtain variants of some theorems due to Datko, for the case of a polynomial stability. In parallel, similar characterizations for a polynomial instability are also carried out.

Solvability for an infinite system of fractional order boundary value problems

We investigate the solvability for an infinite system of fractional order boundary value problems of differential equations in Banach sequence spaces l∞ and c. Our approach depends on Darbo’s fixed point theorem in conjunction with new measures of noncompactness in spaces C1(J,l∞) and C1(J,c).

Unbounded solutions to abstract boundary value problems of fractional differential equations on a half line

In this paper, we investigate the existence of solutions to abstract boundary value problems of fractional differential equations on . The choice of the Banach space allows the solutions to be unbounded. Our approach mainly depends on the technique of Hausdorff measure of noncompactness in conjunction with the criterion of compactness in . Finally, an example in the Banach sequence space ℓ1 is given to illustrate our results.

Order spectrum of the Cesàro operator in Banach lattice sequence spaces

The discrete Cesaro operator C acts continuously in various classical Banach sequence spaces within $$ {\mathbb {C}}^{{\mathbb {N}}}.$$ For the coordinatewise order, many such sequence spaces X are also complex Banach lattices [eg. $$c_0, \ell ^p $$ for $$ 1 < p \le \infty , $$ and $$ {{\text {ces}}}(p)$$ for $$ p \in \{ 0 \} \cup ( 1, \infty )$$]. In such Banach lattice sequence spaces, C is always a positive operator. Hence, its order spectrum is well defined within the Banach algebra of all regular operators on X. The purpose of this note is to show, for every X belonging to the above list of Banach lattice sequence spaces, that the order spectrum $$ \sigma _\mathrm{o} (C)$$ of Ccoincides with its usual spectrum $$ \sigma ( C)$$ when C is considered as a continuous linear operator on the Banach space X.

Solvability of infinite systems of second-order differential equations with boundary conditions in ℓP

In this paper, governed by the fundamental solutions we introduce the Green’s function of the second-order differential equations in general form with respect to boundary conditions and deal with the solvability of the infinite system of second-order differential equations with p, q ∈ C([0, T ], ℝ) and the boundary conditions u i (0) = u i (T ) = 0. We remark that the subjected system has not been previously considered and this investigation complements several results in the literature. Using the ideas of Hausdorff measure of noncompactness and Meir-Keleer condensing operator we seek the sufficient conditions to justify the existence of solutions for the aforementioned system in Banach sequence space ℓ p (1 ≤ p < ∞). Finally, an example is given to ascertain the efficiency of the results.

Disjoint Supercyclic Weighted Pseudo-Shifts on Banach Sequence Spaces

In this article, we present several equivalent conditions ensuring the disjoint supercyclicity of finite weighted pseudo-shifts acting on an arbitrary Banach sequence space. The disjoint supercyclic properties of weighted translations on locally compact discrete groups, the direct sums of finite classical weighted backward shifts, and the bilateral backward operator weighted shifts can be viewed as special cases of our main results. Furthermore, we exhibit an interesting fact that any finite bilateral weighted backward shifts on the space l2 (ℤ) never satisfy the d-Supercyclicity Criterion by a simple proof.

Solvability of infinite systems of third-order differential equations in $$c_{0}$$ by Meir–Keeler condensing operators

Throughout this work, using the technique of measure of noncompactness together with Meir–Keeler condensing operators, we study the solvability of the following infinite system of third-order differential equations in the Banach sequence space $$c_0 $$ as a closed subspace of $$ \ell ^{\infty } $$ : $$\begin{aligned} u_{i}^{\prime \prime \prime }+au_{i}^{\prime \prime }+bu_{i}^{\prime }+cu_{i} =f_{i}(t,u_{1}(t),u_{2}(t),\ldots ) \end{aligned}$$ where $$ f_{i}\in C(\mathbb {R}\times \mathbb {R}^{\infty }, \mathbb {R}) $$ is $$ \omega $$ -periodic with respect to the first coordinate and $$ a,b,c \in \mathbb {R} $$ are constant. Our approach depends on the Green’s function corresponding to the aforesaid system and deduce some conclusions relevant to the existence of $$ \omega $$ -periodic solutions in Banach sequence space $$c_{0} $$ . In addition, some examples are supplied to illustrate the usefulness of the outcome.

Positive solutions for an infinite system of fractional order boundary value problems

In this paper, we investigate the existence of positive solutions for an infinite system of fractional order boundary value problems in a Banach sequence space c. Our analysis relies on the Krasnosel’skii fixed point theorem in a cone in conjunction with the criterion of relative compactness in C^{1}(J, mathbf{c}). Finally, an example is given to illustrate our abstract results.

A Class of Hereditarily $ell_p(c_0)$ Banach spaces

We extend the class of Banach sequence spaces constructed by Ledari, as presented in ''A class of hereditarily $ell_1$ Banach spaces without Schur property'' and obtain a new class of hereditarily $ell_p(c_0)$ Banach spaces for $1leq p<infty$. Some other properties of this spaces are studied.

Solvability of infinite systems of nonlinear integral equations in two variables by using semi-analytic method

In this article, we generalize and investigate existence of solution for infinite systems of nonlinear integral equations with two variables in a given Banach sequence space BC(R+ x R+,c) using Meir-Keeler condensing and noncompactness. Validity of results are shown with the help of an illustrative example. We also introduce a coupled semi-analytic method in the case of two variables in order to construct an iteration algorithm to find a numerical solution for above-mentioned problem. The numerical results show that the produced sequence for approximating the solution in the examples is in the Banach sequence space BC(R+ x R+,c) itself.