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1

Remarks on some variants of minimal point theorem and Ekeland variational principle with applications

Meghea Irina, Stamin Cristina Stefania

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

354--379

EN

This paper presents some variants of minimal point theorem together with corresponding variants of Ekeland variational principle. In the second part of this article, there is a discussion on Ekeland variational principle and minimal point theorem relative to it in uniform spaces. The aim of these series of important results is to highlight relations between them, some improvements and specific applications.

2

On split feasibility problem for finite families of equilibrium and fixed point problems in Banach spaces

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

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

658--675

EN

In this article, motivated by the works of Ali Akbar and Elahe Shahrosvand [Split equality common null point problem for Bregman quasi-nonexpansive mappings, Filomat 32 (2018), no. 11, 3917–3932], Eskandani et al. [A hybrid extragradient method for solving pseudomonotone equilibrium problem using Bregman distance, J. Fixed Point Theory Appl. 20 (2018), 132], B. Ali and M. H. Harbau [Convergence theorems for Bregman K-mappings and mixed equilibrium problems in reflexive Banach spaces, J. Funct. Spaces (2016) Article ID 5161682, 18 pages], and some other related results in the literature, we introduce a hybrid extragradient iterative algorithm that employs a Bregman distance approach for approximating a split feasibility problem for a finite family of equilibrium problems involving pseudomonotone bifunctions and fixed point problems for a finite family of Bregman quasi-asymptotically nonexpansive mappings using the concept of Bregman K-mapping in reflexive Banach spaces. Using our iterative algorithm, we state and prove a strong convergence result for approximating a common solution to the aforementioned problems. Furthermore, we give an application of our main result to variational inequalities and also report a numerical example to illustrate the convergence of our method. The result presented in this article extends and complements many related results in the literature.

3

On solving pseudomonotone equilibrium problems via two new extragradient-type methods under convex constraints

Khunpanuk Chainarong, Pakkaranang Nuttapol, Pholasa Nattawut

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

297--314

EN

The primary objective of this study is to develop two new proximal-type algorithms for solving equilibrium problems in real Hilbert space. Both new algorithms are analogous to the well-known two-step extragradient algorithm for solving the variational inequality problem in Hilbert spaces. The proposed iterative algorithms use a new step size rule based on local bifunction information instead of the line search technique. Two weak convergence theorems for both algorithms are well-established by letting mild conditions. The main results are used to solve the fixed point and variational inequality problems. Finally, we present several computational experiments to demonstrate the efficiency and effectiveness of the proposed algorithms.

4

Inertial iterative method with self-adaptive step size for finite family of split monotone variational inclusion and fixed point problems in Banach spaces

Ogwo Grace N., Alakoya Timilehin O., Mewomo Oluwatosin T.

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

193--216

EN

In this paper, we propose and study a new inertial iterative algorithm with self-adaptive step size for approximating a common solution of finite family of split monotone variational inclusion problems and fixed point problem of a nonexpansive mapping between a Banach space and a Hilbert space. This method combines the inertial technique with viscosity method and self-adaptive step size for solving the common solution problem. We prove a strong convergence result for the proposed method under some mild conditions. Moreover, we apply our result to study the split feasibility problem and split minimization problem. Finally, we provide some numerical experiments to demonstrate the efficiency of our method in comparison with some well-known methods in the literature. Our method does not require prior knowledge or estimate of the operator norm, which makes it easily implementable unlike so many other methods in the literature, which require prior knowledge of the operator norm for their implementation.

5

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.

6

Two strongly convergent self-adaptive iterative schemes for solving pseudo-monotone equilibrium problems with applications

Pakkaranang Nuttapol, Rehman Habib ur, Kumam Wiyada

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

280--298

EN

The aim of this paper is to propose two new modified extragradient methods to solve the pseudo-monotone equilibrium problem in a real Hilbert space with the Lipschitz-type condition. The iterative schemes use a new step size rule that is updated on each iteration based on the value of previous iterations. By using mild conditions on a bi-function, two strong convergence theorems are established. The applications of proposed results are studied to solve variational inequalities and fixed point problems in the setting of real Hilbert spaces. Many numerical experiments have been provided in order to show the algorithmic performance of the proposed methods and compare them with the existing ones.

7

Inertial shrinking projection algorithm with self-adaptive step size for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings

Olona Musa A., Alakoya Timilehin O., Owolabi Abd-semii O.-E., Mewomo Oluwatosin T.

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

47--67

EN

In this paper, we introduce a shrinking projection method of an inertial type with self-adaptive step size for finding a common element of the set of solutions of a split generalized equilibrium problem and the set of common fixed points of a countable family of nonexpansive multivalued mappings in real Hilbert spaces. The self-adaptive step size incorporated helps to overcome the difficulty of having to compute the operator norm, while the inertial term accelerates the rate of convergence of the proposed algorithm. Under standard and mild conditions, we prove a strong convergence theorem for the problems under consideration and obtain some consequent results. Finally, we apply our result to solve split mixed variational inequality and split minimization problems, and we present numerical examples to illustrate the efficiency of our algorithm in comparison with other existing algorithms. Our results complement and generalize several other results in this direction in the current literature.

8

An iterative algorithm for the system of split mixed equilibrium problem

Karahan Ibrahim, Jolaoso Lateef Olakunle

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

309--324

EN

In this article, a new problem that is called system of split mixed equilibrium problems is introduced. This problem is more general than many other equilibrium problems such as problems of system of equilibrium, system of split equilibrium, split mixed equilibrium, and system of split variational inequality. A new iterative algorithm is proposed, and it is shown that it satisfies the weak convergence conditions for nonexpansive mappings in real Hilbert spaces. Also, an application to system of split variational inequality problems and a numeric example are given to show the efficiency of the results. Finally, we compare its rate of convergence other algorithms and show that the proposed method converges faster.

9

Viscosity approximation methods for nonexpansive multi-valued nonself mappings and equilibrium problems

Cholamjiak P., Cholamjiak W., Suantai S.

Demonstratio Mathematica

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2014

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Vol. 47, nr 2

382--395

EN

In this paper, strong convergence theorems by the viscosity approximation method for nonexpansive multi-valued nonself mappings and equilibrium problems are established under some suitable conditions in a Hilbert space. The obtained results extend and improve the corresponding results existed in the literature.

10

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].

11

Analiza dynamiki maszyn roboczych z wykorzystaniem macierzowych metod mechaniki

Grabski J.

Zeszyty Naukowe. Rozprawy Naukowe / Politechnika Łódzka

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1998

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Z. 252

3-177

PL

W pracy przedstawione są metody analizy, a także modele układów, pozwalające na rozwiązywanie wielu praktycznie ważnych zadań z dziedziny dynamiki maszyn roboczych. Zapis równań więzów, zasady prac przygotowanych oraz metod generowania równań ruchu takich jak równania Lagrange'a I-go i Il-go rodzaju, Maggi'ego, Boltzmanna-Hamela przy pomocy rachunku macierzowego pozwolił na budowę prostych algorytmów i programów do obliczeń numerycznych. Macierzowy zapis oraz zaproponowane modyfikacje klasycznych metod mechaniki analitycznej umożliwiły zarówno znaczną, automatyzację procesu generowania równań, jak również poszukiwania rozwiązań wielu zadań mechaniki przy pomocy komputera. Przy rozpatrywaniu zagadnień równowagi zwraca się uwagę na sytuacje, w których pojawiają się problemy z jednoznacznością rozwiązań otrzymywanych dla modelu matematycznego analizowanego układu oraz sposoby eliminowania tego rodzaju trudności. Opracowane algorytmy generowania równań równowagi oraz równań ruchu zostały wykorzystane w obliczeniach szeregu modeli maszyn roboczych. Zamieszczone w tej pracy rozwiązania obejmują zagadnienia dotychczas niedostatecznie poznane. Są to przede wszystkim analiza przestrzennych modeli maszyn roboczych, badanie ruchu ładunku w postaci bryły sztywnej (model pozwalający na rozpatrywanie ruchu ogólnego, w miejsce zakładanego zwykle ruchu postępowego). Przeprowadzone obliczenia, których wyniki zamieszczone są w pracy, pokazują zarówno wpływ sprężystości podpór na obciążenia liny utrzymującej nosiwo, jak również zmiany obciążeń w zależności od założonego modelu liny (rozciągliwa i nierozciagliwa), a także różnice w trajektoriach spowodowane modelowaniem ładunku przy pomocy punktu materialnego oraz przy pomocy trójwymiarowej bryły. Proponowany model ładunku pozwala na uwzględnienie wpływu nierównomiernego rozmieszczenie masy, obciążenia wywołanego naporem powietrza, położenia punktu zaczepienia, początkowej prędkości kątowej. Zastosowane modele ładunku i podłoża umożliwiają, analizę procesu podnoszenia ładunku z chropowatego podłoża, z uwzględnieniem fazy napinania liny, poślizgu bądź obrotu ładunku wokół krawędzi czy naroża, ewentualnych zderzeń ładunku z podłożem i wahań ładunku. Przyjęta metoda badania wrażliwości, polegająca na różniczkowaniu równań ruchu względem badanych parametrów, pozwala na analizę własności układu w wybranym obszarze zmiany parametrów.

EN

The methods of analysis, as well as the models of systems allowing one to solve many important practical problems in the field of engineering machine dynamics are presented. The matrix notations used in constraint equations, principles of virtual work and methods for generation of motion equations, such as Langrangels equations of the first and second kind, Maggi's and Boltzmann-Hamel's equations, permit the formulation of simple algorithms and programs for numerical calculations. The matrix notations and the suggested modifications of the classical analytical mechanics methods enable one to automate significantly the generation procedure of equations, as well as to find solutions to many mechanical problems by means of computers. While considering the problems of equilibrium, attention is paid to the issues in which problems with uniqueness of the solutions obtained for the mathematical model of the system under analysis occur and to the ways of eliminating this kind of problems. The formulated algorithms for the generation of equilibrium equations and motion equations have been employed in computation of many models of engineering machines. The solutions included in the present paper concern the problems not known enough yet. First of all, these problems comprise: an analysis of 3D engineering machine models, investigations of the motion of the load in the form of a rigid body (the model which allows one to analyse a general motion instead of the usually assumed translatory motion). The calculations whose results are presented here show both an influence of the elasticity of supports on the load of the rope handling the material, and changes in the load with respect to the assumed rope model (extensible and inextensible), as well as differences in trajectories caused by load modelling by means of a particle or a 3D solid. The proposed load model makes it possible to take into account an influence of the nonuniform mass distribution, the loading caused by air pressure, the location of the point of load holding, and the initial angular velocity of load. The load and foundation models used enable an analysis of the load hoisting process from rough ground, including a phase of rope stretching, slide or rotation of the load around its edge or corner, possible impacts of the load against ground and load swinging. The assumed method for sensitivity investigations, consisting in differentiation of the motion equations with respect to the parameters under investigation, allows for an analysis of the system properties in the selected region of changes in the parameters.