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1

Operators with memory in Schramm spaces

Wróbel Małgorzata

Journal of Applied Mathematics and Computational Mechanics

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2023

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Vol. 22, nr 3

69--80

EN

We show that every operator with memory acting between Banach spaces CΦBV(I) of continuous functions of bounded variation in the sense of Schramm defined on a compact interval I of a real axis, is a Nemytskij composition operator with the continuous generating function. Moreover, some consequences for uniformly bounded operators with memory will be given. As a by-product, we obtain that a Banach space CΦBV(I) has the uniform Matkowski property.

2

Weakly locally uniformly rotund norm which is not locally uniformly rotund

Malec Marek

Journal of Mathematics and Applications

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2021

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Vol. 44

71--73

EN

The aim of this paper is to provide a proof of the fact that a weakly locally uniformly rotund norm does not have to be locally uniformly rotund. This result is well-known for experts in Geometry of Banach Spaces. However, since the justification of this result is omitted in the literature, we believe that the present note may be helpful for students or novices in the theory.

3

Corrigendum to the paper “Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings”

Gabeleh Moosa

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

9--10

EN

The purpose of this short note is to present a correction of the proof of the main result given in the paper “Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings, ”Demonstr. Math. 53 (2020), 38-43.

4

On the non-hypercyclicity of scalar type spectral operators and collections of their exponentials

Markin Marat V.

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

352--359

EN

Generalizing the case of a normal operator in a complex Hilbert space, we give a straightforward proof of the non-hypercyclicity of a scalar type spectral operator A in a complex Banach space as well as of the collection {etA}t≥0 of its exponentials, which, under a certain condition on the spectrum of the operator A, coincides with the C0-semigroup generated by A. The spectrum of A lying on the imaginary axis, we also show that non-hypercyclic is the strongly continuous group {etA}t∈R of bounded linear operators generated by A. From the general results, we infer that, in the complex Hilbert space L2(R), the anti-self-adjoint differentiation operator A≔ddx with the domain D(A)≔W12(R) is non-hypercyclic and so is the left-translation strongly continuous unitary operator group generated by A.

5

Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings

Gabeleh Moosa, Künzi Hans-Peter A.

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

38--43

EN

In this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [Proximal normal structure and nonexpansive mappings, Studia Math. 171 (2005), 283–293] immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper [Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.

6

Nonlinear fractional differential equations with non-instantaneous impulses in Banach spaces

Benchohra M., Slimane M.

Journal of Mathematics and Applications

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2018

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Vol. 41

39--51

EN

This paper is devoted to study the existence of solutions for a class of initial value problems for non-instantaneous impulsive fractional differential equations involving the Caputo fractional derivative in a Banach space. The arguments are based upon Monch's fixed point theorem and the technique of measures of noncompactness.

7

Noncyclic Meir-Keeler contractions and best proximity pair theorems

Gabeleh M., Markin J.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

171--181

EN

In this article, we consider the class of noncyclic Meir-Keeler contractions and study the existence and convergence of best proximity pairs for such mappings in the setting of complete CAT(0) spaces. We also discuss asymptotic pointwise noncyclic Meir-Keeler contractions in the framework of uniformly convex Banach spaces and generalize a main result of Chen [Chen C. M., A note on asymptotic pointwise weaker Meir-Keeler type contractions, Appl. Math. Lett., 2012, 25, 1267-1269]. Examples are given to support our main results.

8

Approximate property of a functional equation with a general involution

Bae J.-H., Park W.-G.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

304--308

EN

In this paper, we prove the Hyers-Ulam stability of the functional equation f(x + y, z + w) + f(x + σ (y), z + τ(w)) = 2f(x, z) + 2f(y, w), where σ, τ are involutions.

9

An abstract nonlocal functional-differential second order evolution problem

Byszewski L.

Technical Transactions

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2017

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Vol. 1(114)

147--154

EN

The aim of the paper is to prove two theorems on the existence and uniqueness of mild and classical solutions of a semilinear functional-differential evolution second order equation together with nonlocal conditions. The theory of strongly continuous cosine families of linear operators in a Banach space is applied. The paper is based on publications [1–9] and is a generalization of paper [6].

PL

W artykule udowodniono dwa twierdzenia o istnieniu i jednoznaczności całkowych i klasycznych rozwiązań semiliniowego funkcjonalno-różniczkowego zagadnienia ewolucyjnego rzędu drugiego z warunkami nielokalnymi. W tym celu zastosowano teorię rodziny cosinus liniowych operatorów w przestrzeni Banacha. Artykuł bazuje na publikacjach [1–9] i jest pewnym uogólnieniem publikacji [6].

10

Weak solutions of fractional order differential equations via Volterra-Stieltjes integral operator

El-Sayed A. M. A., El-Sayed W. G., Abd El-Mowla A. A. H.

Journal of Mathematics and Applications

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2017

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Vol. 40

85--96

EN

The fractional derivative of the Riemann-Liouville and Caputo types played an important role in the development of the theory of fractional derivatives, integrals and for its applications in pure mathematics ([18], [21]). In this paper, we study the existence of weak solutions for fractional differential equations of Riemann-Liouville and Caputo types. We depend on converting of the mentioned equations to the form of functional integral equations of Volterra-Stieltjes type in reflexive Banach spaces.

11

On regulated functions

Banaś J., Kot M.

Journal of Mathematics and Applications

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2017

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Vol. 40

21--36

EN

The main purpose of this review article is to present the concept of a regulated function and to indicate the connection of the class of regulated functions with other significant classes of functions. In particular, we give a characterization of regulated functions in terms of step functions and we show that the linear space of regulated functions forms a Banach space under the classical supremum norm.

12

Measures of noncompactness in a Banach algebra and their applications

Dudek S.

Journal of Mathematics and Applications

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2017

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Vol. 40

69--84

EN

In this paper we study the existence of solutions of a nonlinear quadratic integral equation of fractional order. This equation is considered in the Banach space of real functions defined, continuous and bounded on the real half axis. Additionally, using the technique of measures of noncompactness we obtain some characterization of considered integral equation. We provide also an example illustrating the applicability of our approach.

13

Multivariate and abstract approximation theory for Banach space valued functions

Anastassiou G. A.

Demonstratio Mathematica

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2017

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Vol. 50, nr 1

208--222

EN

Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued Fréchet differentiable functions to the unit operator, as well as other basic approximations including those under convexity. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators for which we study their approximation properties. We derive pointwise and uniform estimates, which imply the approximation of these operators to the unit assuming Fréchet differentiability of functions, and then we continue with basic approximations. At the end we study the special case where the approximated function fulfills a convexity condition resulting into sharp estimates. We give applications to Bernstein operators.

14

A note on range-kernel uncomplementation

Kubrusly C., Duggal B. P.

Demonstratio Mathematica

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2017

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Vol. 50, nr 1

252--260

EN

This note exhibits a Banach-space operator such that neither the range nor the kernel is complemented both for the operator and its adjoint.

15

A strong convergence result for systems of nonlinear operator equations involving total asymptotically nonexpansive mappings in uniformly convex Banach spaces

Abdelhakim A. A.

Silesian Journal of Pure and Applied Mathematics

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2016

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Vol. 6, iss. 1

27--48

EN

We prove the strong convergence of an implicit iterative procedure to a solution of a system of nonlinear operator equations involving total asymptotically nonexpansive operators in uniformly convex Banach spaces.

16

Some new existence results and stability concepts for fractional partial random differential equations

Abbas S., Benchohra M., Darwish M. A.

Journal of Mathematics and Applications

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2016

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Vol. 39

5--22

EN

In the present paper we provide some existence results and Ulam’s type stability concepts for the Darboux problem of partial fractional random differential equations in Banach spaces, by applying the measure of noncompactness and a random fixed point theorem with stochastic domain.

17

Schauder’s fixed-point theorem in approximate controllability problems

Babiarz A., Klamka J., Niezabitowski M.

International Journal of Applied Mathematics and Computer Science

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2016

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Vol. 26, no. 2

263--275

EN

The main objective of this article is to present the state of the art concerning approximate controllability of dynamic systems in infinite-dimensional spaces. The presented investigation focuses on obtaining sufficient conditions for approximate controllability of various types of dynamic systems using Schauder’s fixed-point theorem. We describe the results of approximate controllability for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, impulsive neutral functional evolution integro-differential systems, stochastic impulsive systems with control-dependent coefficients, nonlinear impulsive differential systems, and evolution systems with nonlocal conditions and semilinear evolution equation.

18

Intuitionistic fuzzy almost Cauchy–Jensen mappings

Gordji M. E., Abbaszadeh S.

Demonstratio Mathematica

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2016

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Vol. 49, nr 1

18--25

EN

In this paper, we first investigate the Hyers–Ulam stability of the generalized Cauchy–Jensen functional equation of p-variable (…) in an intuitionistic fuzzy Banach space. Then, we conclude the results for Cauchy–Jensen functional equation of p-variable (…). Next, we discuss the intuitionistic fuzzy continuity of Cauchy–Jensen mappings.

19

Banach fixed-point theorem in semilinear controllability problems – a survey

Klamka J., Babiarz A., Niezabitowski M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2016

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Vol. 64, nr 1

21--35

EN

The main aim of this article is to review the existing state of art concerning the complete controllability of semilinear dynamical systems. The study focus on obtaining the sufficient conditions for the complete controllability for various systems using the Banach fixedpoint theorem. We describe the results for stochastic semilinear functional integro-differential system, stochastic partial differential equations with finite delays, semilinear functional equations, a stochastic semilinear system, a impulsive stochastic integro-differential system, semilinear stochastic impulsive systems, an impulsive neutral functional evolution integro-differential system and a nonlinear stochastic neutral impulsive system. Finally, two examples are presented.

20

The random of lacunary statistical on χ2 over p-metric spaces defined by Musielak

Subramanian N., Babu R., Thirunavukkarasu P.

Journal of Mathematics and Applications

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2015

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Vol. 38

133--150

EN

Mursaleen introduced the concepts of statistical convergence in random 2-normed spaces. Recently Mohiuddine and Aiyup defined the notion of lacunary statistical convergence and lacunary statistical Cauchy in random 2-normed spaces. In this paper, we define and study the notion of lacunary statistical convergence and lacunary of statistical Cauchy sequences in random on χ2 over p- metric spaces dfined by Musielak and prove some theorems which generalizes Mohiuddine and Aiyup results.