Inertial-Type Algorithm for Solving Split Common Fixed Point Problems in Banach Spaces

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

“…The inertial method involves a two-step iteration in which the next iterate is defined by making use of the previous two iterates and this generally increase the rate of convergence of iterative algorithms. The inertial type algorithm has been studied and modified in various forms by many authors (see [2,3,4,9,47,51,52]). For example, an inertial hybrid proximal-extragradient algorithm which is a combination of hybrid proximal extragradient and inertial type algorithm for a maximal monotone operator was proposed by Bot and Csetnek [9].…”

A Strong Convergence Theorem for Split Null Point Problem and Generalized Mixed Equilibrium Problem in Real Hilbert Spaces

“…Observe that u n can be rewritten as u n = K F r n (x n − r n g(x n )) for each n. Fix p ∈ Ω. Since p = K F r n (p − r n p), g is β-inverse strongly monotone and r n ∈ (0, 2β), for any n ∈ R, we have from (21) and Lemma 2 (ii) that…”

“…For an arbitrary x 0 ∈ H 1 , let {x n } ⊂ H 1 be a sequence defined iteratively by (21) satisfying the conditions of Lemma 6. Then {x n } converges strongly to p ∈ Ω, where p = P Ω ∇ f (p).…”

“…The GMEP includes as special cases, minimization problem, variational inequality problem, fixed point problem, nash equilibrium etc. GMEP (2) and these special cases have been studied in Hilbert, Banach, Hadamard and p-uniformly convex metric spaces , see [11][12][13][14][15][16][17][18][19][20][21].…”

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

“…In general, the convergence rate of Krasnoselski-Mann iterative method and hybrid iterative method is slow. 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].…”