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1

Sharp sufficient condition for the convergence of greedy expansions with errors in coefficient computation

Valiullin Artur R., Valiullin Albert R., Solodov Alexei P.

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

254--264

EN

Generalized approximate weak greedy algorithms (gAWGAs) were introduced by Galatenko and Livshits as a generalization of approximate weak greedy algorithms, which, in turn, generalize weak greedy algorithm and thus pure greedy algorithm. We consider a narrower case of gAWGA in which only a sequence of absolute errors {ξn}∞n=1 is nonzero. In this case sufficient condition for a convergence of a gAWGA expansion to an expanded element obtained by Galatenko and Livshits can be written as ∑∞n=1ξ2n<∞ . In the present article, we relax this condition and show that the convergence is guaranteed for ξn=o(1√n) . This result is sharp because the convergence may fail to hold for ξn≍1√n .

2

Generalized common fixed point theorem for generalized hybrid mappings in Hilbert spaces

Kondo Atsumasa

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

752--759

EN

In this article, we prove a common fixed point theorem for commutative nonlinear mappings that jointly satisfy a certain condition. From the main theorem, a common fixed point theorem for commutative generalized hybrid mappings is derived as a special case. Our novel approach significantly expands the applicable range of mappings for well-known fixed point theorems to be effective. Examples are presented to explicitly illustrate this contribution.

3

Coupled fixed point theorems under new coupled implicit relation in Hilbert spaces

Kim Kyung Soo

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

81--89

EN

The aim of this paper is to study existence and uniqueness of coupled fixed point for a family of self-mappings satisfying a new coupled implicit relation in a Hilbert space. We also prove well-posedness of a coupled fixed point problem.

4

An extragradient inertial algorithm for solving split fixed-point problems of demicontractive mappings, with equilibrium and variational inequality problems

Okeke Chibueze C., Ugwunnadi Godwin C., Jolaoso Lateef O.

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

506--527

EN

The purpose of this article is to study and analyse a new extragradient-type algorithm with an inertial extrapolation step for solving split fixed-point problems for demicontractive mapping, equilibrium problem, and pseudomonotone variational inequality problem in real Hilbert spaces. One of the advantages of the proposed algorithm is that a strong convergence result is achieved without a prior estimate of the Lipschitz constant of the cost operator, which is very difficult to find. In addition, the stepsize is generated at each iteration by some simple computations, which allows it to be easily implemented without the prior knowledge of the Lipschitz constant of the cost operator. Some numerical experiments are reported to show the performance and behaviour of the sequence generated by our algorithm. The obtained results in this article extend and improve many related recent results in this direction in the literature.

5

Strong convergence inertial projection algorithm with self-adaptive step size rule for pseudomonotone variational inequalities in Hilbert spaces

Wairojjana Nopparat, Pakkaranang Nuttapol, Pholasa Nattawut

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

110--128

EN

In this paper, we introduce a new algorithm for solving pseudomonotone variational inequalities with a Lipschitz-type condition in a real Hilbert space. The algorithm is constructed around two algorithms: the subgradient extragradient algorithm and the inertial algorithm. The proposed algorithm uses a new step size rule based on local operator information rather than its Lipschitz constant or any other line search scheme and functions without any knowledge of the Lipschitz constant of an operator. The strong convergence of the algorithm is provided. To determine the computational performance of our algorithm, some numerical results are presented.

6

Generalized split null point of sum of monotone operators in Hilbert spaces

Mebawondu Akindele A., Abass Hammed A., Oyewole Olawale K., Aremu Kazeem O., Narain Ojen K.

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

359--376

EN

In this paper, we introduce a new type of a generalized split monotone variational inclusion (GSMVI) problem in the framework of real Hilbert spaces. By incorporating an inertial extrapolation method and an Halpern iterative technique, we establish a strong convergence result for approximating a solution of GSMVI and fixed point problems of certain nonlinear mappings in the framework of real Hilbert spaces. Many existing results are derived as corollaries to our main result. Furthermore, we present a numerical example to support our main result and propose an open problem for interested researchers in this area. The result obtained in this paper improves and generalizes many existing results in the literature.

7

Structure of n-quasi left m-invertible and related classes of operators

Duggal Bhagwati Prashad, Kim In Hyun

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

249--268

EN

Given Hilbert space operators T, S ∈ B(H), let Δ and δ ∈ B(B (H)) denote the elementary operators ΔT,S(X) = (LT RS − I) (X) = TXS - X and δT,S(X) = (LT – RS)(X) = TX - XS. Let d = Δ or δ. Assuming T commutes with S∗, and choosing X to be the positive operator S∗nSn for some positive integer n, this paper exploits properties of elementary operators to study the structure of n-quasi [m, d]-operators dm T,S (X) = 0 to bring together, and improve upon, extant results for a number of classes of operators, such as n-quasi left m-invertible operators, n-quasi m-isometric operators, n-quasi m-self-adjoint operators and n-quasi (m, C) symmetric operators (for some conjugation C of H). It is proved that Sn is the perturbation by a nilpotent of the direct sum of an operator Sn1 = (…)n satisfying dmT1S1(I1) = 0 , T1 = (…) , with the 0 operator; if S is also left invertible, then Sn is similar to an operator B such that dmB∗,B(I) = 0. For power bounded S and T such that ST∗ - T∗S = 0 and ΔTS(S∗nSn) = 0, S is polaroid (i.e., isolated points of the spectrum are poles). The product property, and the perturbation by a commuting nilpotent property, of operators T, S satisfying dmT,S (I) = 0, given certain commutativity properties, transfers to operators satisfying S∗ndmT,S (I)Sn = 0.

8

A strong convergence theorem for a zero of the sum of a finite family of maximally monotone mappings

Wega Getahun B., Zegeye Habtu, Boikanyo Oganeditse A.

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

152--166

EN

The purpose of this article is to study the method of approximation for zeros of the sum of a finite family of maximally monotone mappings and prove strong convergence of the proposed approximation method under suitable conditions. The method of proof is of independent interest. In addition, we give some applications to the minimization problems and provide a numerical example which supports our main result. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

9

Geometry Interpretation of Differentiability of Rock Types in Hilbert‘s Space

Feriancikova Katarina, Lazarova Edita, Krulakova Maria, Ivanicova Lucia, Lesso Igor

Inżynieria Mineralna

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2019

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R. 21, nr 1/2

49--54

EN

Signal of vibrations accompanying the rotary drilling of three rock types (andesite, limestone and granite) by diamond core-drill bits was processed and evaluated in order to track the signal characteristics of tested rock types. Mathematical procedures of Hilbert’s abstract space were applied to express the differences between the rock types based on vibration signal. Experiments were performed using the laboratory drilling rig designed and constructed at the Institute of Geotechnics SAS providing automated continuous monitoring of key process parameters (thrust force, rotation speed, torque, advance rate, etc.). Nominal regime of thrust force 5000 N and rotation speed 1000 rpm was used in the experiments along with monitoring with sampling frequency 17 kHz. The vibration signal was recorded by accelerometers in three orthogonal directions: axial in the drilling directions and two radial directions in horizontal and vertical planes. For the purposes of evaluation, only the vibrations in axial direction were assessed as their signal exhibits the highest entropy. A method providing the expression of mutual differences between the vibrations formed during the drilling of different rock types was developed, which enables to set the differences in abstract space to the planar visualization.

PL

Sygnały drgań pochodzących z wierceniu obrotowego trzech rodzajów skał (andezyt, wapień i granit) za pomocą diamentowych wierteł rdzeniowych został przetworzony i oceniony w celu śledzenia charakterystyk sygnałowych badanych rodzajów skał. Zastosowano matematyczne procedury przestrzeni Hilberta, aby wyrazić różnice między rodzajami skał w oparciu o sygnał wibracyjny. Eksperymenty przeprowadzono na laboratoryjnej platformie wiertniczej zaprojektowanej i skonstruowanej w Instytucie Geotechniki SAS, zapewniającej zautomatyzowane ciągłe monitorowanie kluczowych parametrów procesu (siły ciągu, prędkości obrotowej, momentu obrotowego, prędkości posuwu itp.). W doświadczeniach zastosowano nominalną wartość siły nacisku 5000 N i prędkości obrotowej 1000 rpm wraz z monitorowaniem częstotliwości 17 kHz. Sygnał drgań został zarejestrowany przez akcelerometry w trzech kierunkach ortogonalnych: osiowym w kierunkach wiercenia i dwóch promieniowych w płaszczyznach poziomej i pionowej. Do celów oceny oceniono jedynie drgania w kierunku osiowym, ponieważ ich sygnał wykazuje najwyższą entropię. Opracowano metodę wyrażania wzajemnych różnic między drganiami powstającymi podczas wiercenia różnych rodzajów skał, która umożliwia przeniesienie różnic z przestrzeni Hilberta na wizualizację dwuwymiarową.

10

Reproducing Kernel Hilbert Space and Coalescence Hidden-variable Fractal Interpolation Functions

Prasad Srijanani Anurag

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

467--474

EN

Reproducing Kernel Hilbert Spaces (RKHS) and their kernel are important tools which have been found to be incredibly useful in many areas like machine learning, complex analysis, probability theory, group representation theory and the theory of integral operator. In the present paper, the space of Coalescence Hidden-variable Fractal Interpolation Functions (CHFIFs) is demonstrated to be an RKHS and its associated kernel is derived. This extends the possibility of using this new kernel function, which is partly self-affine and partly non-self-affine, in diverse fields wherein the structure is not always self-affine.

11

A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces

Jolaoso Lateef Olakunle, Taiwo Adeolu, Alakoya Timilehin Opeyemi, Mewomo Oluwatosin Temitope

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

183--203

EN

We consider a new subgradient extragradient iterative algorithm with inertial extrapolation for approximating a common solution of variational inequality problems and fixed point problems of a multivalued demicontractive mapping in a real Hilbert space. We established a strong convergence theorem for our proposed algorithm under some suitable conditions and without prior knowledge of the Lipschitz constant of the underlying operator. We present numerical examples to show that our proposed algorithm performs better than some recent existing algorithms in the literature.

12

Ergodic and fixed point theorems for sequences and nonlinear mappings in a Hilbert space

Rouhani B. D.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

27--36

EN

In this paper, we introduce the notion of 2-generalized hybrid sequences, extending the notion of nonexpansive and hybrid sequences introduced and studied in our previous work [Djafari Rouhani B., Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. thesis, Yale University, 1981; and other published in J. Math. Anal. Appl., 1990, 2002, and 2014; Nonlinear Anal., 1997, 2002, and 2004], and prove ergodic and convergence theorems for such sequences in a Hilbert space H. Subsequently, we apply our results to prove new fixed point theorems for 2-generalized hybrid mappings, first introduced in [Maruyama T., Takahashi W., Yao M., Fixed point and mean ergodic theorems for new nonlinear mappings in Hilbert spaces, J. Nonlinear Convex Anal., 2011, 12, 185-197] and further studied in [Lin L.-J., Takahashi W., Attractive point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces, Taiwanese J. Math., 2012, 16, 1763-1779], defined on arbitrary nonempty subsets of H.

13

A unified algorithm for solving split generalized mixed equilibrium problem, and for finding fixed point of nonspreading mapping in Hilbert spaces

Jolaoso L. O., Oyewole K. O., Okeke C. C., Mewomo O. T.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

211--232

EN

The purpose of this paper is to study a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces. We introduce a new iterative algorithm and prove its strong convergence for approximating a common solution of a split generalized mixed equilibrium problem and a fixed point problem for nonspreading mappings in real Hilbert spaces. Our algorithm is developed by combining a modified accelerated Mann algorithm and a viscosity approximation method to obtain a new faster iterative algorithm for finding a common solution of these problems in real Hilbert spaces. Also, our algorithm does not require any prior knowledge of the bounded linear operator norm. We further give a numerical example to show the efficiency and consistency of our algorithm. Our result improves and compliments many recent results previously obtained in this direction in the literature.

14

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.

15

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.

16

Ryszard Grząślewicz (1953–2005)

Hudzik H., Mielczarek G., Płuciennik R.

Wiadomości Matematyczne

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2014

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T. 50, Nr 1

85--99

PL

Z wielkim wzruszeniem przyjęliśmy zaszczyt napisania wspomnienia o niezwykle zdolnym matematyku i naszym wielkim przyjacielu, profesorze Ryszardzie Grząślewiczu. Postaramy się przybliżyć sylwetkę człowieka, który niemal całe swoje dorosłe życie związał z Politechniką Wrocławską.

17

Porosity and the bounded linear regularity property

Reich S., Zaslavski A. J.

Journal of Applied Analysis

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2014

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Vol. 20, nr 1

1--6

EN

H. H. Bauschke and J. M. Borwein showed that in the space of all tuples of bounded, closed, and convex subsets of a Hilbert space with a nonempty intersection, a typical tuple has the bounded linear regularity property. This property is important because it leads to the convergence of infinite products of the corresponding nearest point projections to a point in the intersection. In the present paper we show that the subset of all tuples possessing the bounded linear regularity property has a porous complement. Moreover, our result is established in all normed spaces and for tuples of closed and convex sets, which are not necessarily bounded.

18

On derivations of operator algebras with involution

Širovnik N., Vukman J.

Demonstratio Mathematica

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2014

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Vol. 47, nr 4

784--790

EN

The purpose of this paper is to prove the following result. Let X be a complex Hilbert space, let L(X) be an algebra of all bounded linear operators on X and let A(X) (…) L(X) be a standard operator algebra, which is closed under the adjoint operation. Suppose there exists a linear mapping D : A(X) → L(X) satisfying the relation 2D(AA*A) = D(AA*)A + AA*D(A) + D(A)A*A + AD(A*A) for all A (…) A(X). In this case, D is of the form D(A) = [A,B] for all A (…) A(X) and some fixed B (…) L(X), which means that D is a derivation.

19

Modified Mann iterative algorithms by hybrid projection methods for nonexpansive semigroups and mixed equilibrium problems

Katchang P., Kumam P.

Journal of Applied Analysis

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2012

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Vol. 18, nr 2

259-273

EN

In this paper, we propose a modified Mann iterative algorithm by two hybrid projection methods for finding a common element of the set of fixed points of nonexpansive semigroups and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Then, we obtain interesting and new strong convergence theorems for the sequences generated by these processes by using the hybrid projection methods in the mathematical programming. The results presented in this paper extend and improve the corresponding one by Nakajo and Takahashi [J. Math. Anal. Appl. 279 (2003), 372-379].

20

Extending Lipschitz mappings continuously

Kopecka E.

Journal of Applied Analysis

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2012

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Vol. 18, nr 2

167-177

EN

We consider short mappings from a bounded subset of a Euclidean space into that space, that is, mappings which do not increase distances between points. By Kirszbraun's theorem any such mapping can be extended to the entire space to be short again. In general, the extension is not unique. We show that there are single-valued extension operators continuous in the supremum norm. The multivalued extension operator is lower semicontinuous.