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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

2022

Vol. 55, nr 1

193--216

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.

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

2021

Vol. 54, nr 1

47--67

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.