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1
Content available remote 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.
2
Content available remote 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.
3
Content available remote 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.
4
Content available remote 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.
5
Content available remote 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.
6
Content available remote 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.
7
Content available remote On the proximal point algorithm and demimetric mappings in CAT(0) spaces
Aremu K. O., Izuchukwu C., Ugwunnadi G. C., Mewomo O. T.
Demonstratio Mathematica
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2018
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Vol. 51, nr 1
277--294
EN
In this paper, we introduce and study the class of demimetric mappings in CAT(0) spaces. We then propose a modified proximal point algorithm for approximating a common solution of a finite family of minimization problems and fixed point problems in CAT(0) spaces. Furthermore, we establish strong convergence of the proposed algorithm to a common solution of a finite family of minimization problems and fixed point problems for a finite family of demimetric mappings in complete CAT(0) spaces. A numerical example which illustrates the applicability of our proposed algorithm is also given. Our results improve and extend some recent results in the literature.
8
Content available remote 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.
9
Content available remote Well-posedness of the fixed point problem for ø-max-contractions
Akkouchi M.
Fasciculi Mathematici
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2010
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Nr 45
5-12
EN
We study the well-posedness of the fixed point problem for self-mappings of a metric space which are ø-max-contractions (see [6]).
10
Content available remote Well-posedness of fixed point problem for mappings satisfying an implicit relation
Akkouchi M., Popa V.
Demonstratio Mathematica
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2010
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Vol. 43, nr 4
923-929
EN
The notion of well-posedness of a fixed point problem has generated much interest to a several mathematicians, for example, F. S. De Blassi and J. Myjak (1989), S. Reich and A. J. Zaslavski (2001), B. K. Lahiri and P. Das (2005) and V. Popa (2006 and 2008). The aim of this paper is to prove for mappings satisfying some implicit relations in orbitally complete metric spaces, that fixed point problem is well-posed.
11
Content available Construction of algebraic-analytic discrete approximations for linear and nonlinear hyperbolic equations in R2. Pt. 1
Luśtyk M., Prytula M.
Opuscula Mathematica
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2007
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Vol. 27, no. 1
25-36
EN
An algebraic-analytic method for constructing discrete approximations of linear hyperbolic equations based on a generalized d'Alembert formula of the Lytvyn and Riemann expressions for Cauchy data is proposed. The problem is reduced to some special case of the fixed point problem.
12
Content available remote Well-posedness and porosity of a certain class of operators
Lahiri B., Das P.
Demonstratio Mathematica
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2005
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Vol. 38, nr 1
169--176
EN
We prove that several fixed point problems are well-posed and study the porosity behaviour of a certain class of operators.
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