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1

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.

2

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.

3

Direct estimate for some operators of Durrmeyer type in exponential weighted space

Krech G., Wachnicki E.

Demonstratio Mathematica

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2014

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Vol. 47, nr 2

336--349

EN

In the present paper, we investigate the convergence and the approximation order of some Durrmeyer type operators in exponential weighted space. Furthermore, we obtain the Voronovskaya type theorem for these operators.

4

Convergence of an implicit iteration process with errors for two asymptotically nonexpansive mappings

Temir S.

Demonstratio Mathematica

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2013

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Vol. 46, nr 4

781--793

EN

The purpose of this paper is to introduce an implicit iterative process with errors for approximating common fixed point of two finite families of asymptotically nonexpansive mappings in the framework of Banach space. The results presented in this paper extend and generalize the corresponding results of Qin et al. [Convergence analysis of implicit iterative algorithms for asymptotically nonexpansive mappings, Appl. Math. Comp. 210 (2009), 542–550], Thakur [Weak and strong convergence of composite implicititeration process, Appl. Math. Comp. 190 (2007), 965–973] and some others.

5

Convergence of implicit iterative process for a finite family of I-nonexpansive mappings

Temir S., Aykanat G.

Demonstratio Mathematica

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2009

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Vol. 42, nr 1

175-184

EN

We prove that an implicit iterative process conyerges weakly and strongly to a common fixed point of a finite family of I-nonexpansive mappings in a Banach space. The results presented in this paper extend and improve the corresponding results of [1, 3, 11, 12]

6

A convergence theorem for some mean value fixed point iteration procedures

Berinde V.

Demonstratio Mathematica

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2005

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Vol. 38, nr 1

177--184

EN

A general convergence theorem for the Ishikawa fixed point iteration procedure in a large class of quasi-contractive type operators is given. As particular cases, it contains convergence theorems for Picard, Krasnoselskij and Mann iterations, theorems which extend and generalize several results in the literature.