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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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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.

3

An iterative approximation of common solutions of split generalized vector mixed equilibrium problem and some certain optimization problems

Mewomo Oluwatosin T., Oyewole Olawale K.

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

335--358

EN

In this paper, we study the problem of finding a common solution of split generalized vector mixed equlibrium problem (SGVMEP), fixed point problem (FPP) and variational inequality problem (VIP). We propose an inertial-type iterative algorithm, which uses a projection onto a feasible set and a linesearch, which can be easily calculated. We prove a strong convergence of the sequence generated by the proposed algorithm to a common solution of SGVMEP, fixed point of a quasi- ϕ -nonexpansive mapping and VIP for a general class of monotone mapping in 2-uniformly convex and uniformly smooth Banach space E1 and a smooth, strictly convex and reflexive Banach space E2 . Some numerical examples are presented to illustrate the performance of our method. Our result improves some existing results in the literature.

4

Convergence theorems for generalized hemicontractive mapping in p-uniformly convex metric space

Ugwunnadi Godwin C., Izuchukwu Chinedu, Mewomo Oluwatosin T.

Journal of Applied Analysis

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2020

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Vol. 26, nr 2

221--229

EN

In this paper, we introduce and study an Ishikawa-type iteration process for the class of generalized hemicontractive mappings in p-uniformly convex metric spaces, and prove both Delta-convergence and strong convergence theorems for approximating a fixed point of generalized hemicontractive mapping in complete p-uniformly convex metric spaces. We give a surprising example of this class of mapping that is not a hemicontractive mapping. Our results complement, extend and generalize numerous other recent results in CAT(0) spaces.

5

Convergence theorem for a finite family of asymptotically demicontractive multi-valued mappings in CAT(0) spaces

Ugwunnadi Godwin C., Mewomo Oluwatosin T., Izuchukwu Chinedu

Journal of Applied Analysis

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2020

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Vol. 26, nr 1

117--130

EN

In this paper, we introduce the class of asymptotically demicontractive multivalued mappings and establish a strong convergence theorem of the modified Mann iteration to a common fixed point of a finite family of asymptotically demicontractive multivalued mappings in a complete CAT(0) space. We also give a numerical example of our iterative method to show its applicability.

6

On θ-generalized demimetric mappings and monotone operators in Hadamard spaces

Ogwo Grace N., Izuchukwu Chinedu, Aremu Kazeem O., Mewomo Oluwatosin T.

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

95--111

EN

Our main interest in this article is to introduce and study the class of θ-generalized demimetric mappings in Hadamard spaces. Also, a Halpern-type proximal point algorithm comprising this class of mappings and resolvents of monotone operators is proposed, and we prove that it converges strongly to a fixed point of a θ-generalized demimetric mapping and a common zero of a finite family of monotone operators in a Hadamard space. Furthermore, we apply the obtained results to solve a finite family of convex minimization problems, variational inequality problems and convex feasibility problems in Hadamard spaces.