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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.

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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.

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The convergence of Jungck-type iterative schemes for generalized contractive-like operators

Olaleru J. O., Akewe H.

Fasciculi Mathematici

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2010

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Nr 45

87-98

EN

We introduce the Jungck-multistep iteration and show that it converges strongly to the unique common fixed point of a pair of weakly compatible generalized contractive-like opera- tors defined on a Banach space. As corollaries, the results show that the Jungck-Mann, Jungck-Ishikawa and Jungck-Noor itera- tions can also be used to approximate the common fixed points of such maps. The results are improvements, generalizations and extensions of the work of Olatinwo and Imoru [13], Olatinwo [14-15]. Consequently, several results in literature are generalized.