A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings

Abstract

In this paper, we present a new modified self-adaptive inertial subgradient extragradient algorithm in which the two projections are made onto some half spaces. Moreover, under mild conditions, we obtain a strong convergence of the sequence generated by our proposed algorithm for approximating a common solution of variational inequality problem and common fixed point of a finite family of demicontractive mappings in a real Hilbert space. The main advantages of our algorithm are: strong convergence result obtained without prior knowledge of the Lipschitz constant of the related monotone operator, the two projections made onto some half-spaces and the inertial technique which speeds up rate of convergence. Finally, we present an application and a numerical example to illustrate the usefulness and applicability of our algorithm.

Iterative method for solving split common fixed point problem of asymptotically demicontractive mappings in Hilbert spaces

“…It is well-known that P C x is characterized by the inequality x − P C x, y − P C x ≤ 0, ∀y ∈ C. Moreover, P C is nonexpansive and F (P C ) = C. Since P 2 C = P C , it follows by induction that P n C = P C , for all n ∈ N. For details on P C , see [1,29,30]. Remark 2.…”

Inertial Mann-Krasnoselskii Algorithm With Self Adaptive Stepsize for Split Variational Inclusion Problem and Paramonotone Equilibria

“…The inertial type algorithms which originated from the heavy ball method of the two order time dynamical system can be regarded as a means of speeding up the convergence rates of iterative schemes. Some recent results on inertial algorithms can be found in [17,19].…”

Krasnoselski–Mann-type inertial method for solving split generalized mixed equilibrium and hierarchical fixed point problems

“…In particular, the term θ n (x nx n-1 ) which is called the inertial extrapolation term was proposed as a remarkable tool for speeding up the convergence properties of iterative methods, and the inertial type algorithm has been studied and modified in various forms by many authors (see [1,4,6,37,42,51]). For example, the inertial forward-backward splitting methods [35], the inertial Douglas-Rachford splitting methods [8] and the inertial proximal methods [40], the inertial extragradient methods [27], the inertial subgradient extragradient methods [43], the inertial shrinking projection methods [41].…”