Abstract
Motivated and inspired by the growing contribution with respect to iterative approximations from some researchers in the literature, we design and investigate two types of brand-new semi-implicit viscosity iterative approximation methods for finding the fixed points of nonexpansive operators associated with contraction operators in complete {operatorname{CAT}(0)} spaces and for solving related variational inequality problems. Under some suitable assumptions, strong convergence theorems of the sequences generated by the approximation iterative methods are devised, and a numerical example and some applications to related variational inequality problems are included to verify the effectiveness and practical utility of the convergence theorems. Our main results presented in this paper do not only improve, extend and refine some corresponding consequences in the literature, but also show that the additional variational inequalities, general variational inequality systems and equilibrium problems can be solved via approximation of the iterative sequences. Finally, we provide an open question for future research.
Highlights
In this paper, we consider the following two kinds of new semi-implicit viscosity approximation methods of iterative forms (in short, (TVIM-I) and (TVIM-II), respectively) for nonexpansive operator T associated with contraction operator in CAT(0) space X: ⎧ ⎨vn = anf ⊕ (1 an)T,⎩un+1 = bnun ⊕ (1 – bn)vn, ∀n ≥ 1
If T : E → E is a nonexpansive operator with F(T) = ∅, f : E → E is a contraction with coefficient k ∈ [0, 1), and (1.3) holds, for any given u1 ∈ E, the sequence {un} generated by (TVIM-I) converges strongly to q ∈ F(T) such that q = PF(T)f (q), which is a unique solution of the variational inequality (1.7)
We investigate the strong convergence of the iterative approximation method (TVIM-II)
Summary
We consider the following two kinds of new semi-implicit viscosity approximation methods of iterative forms (in short, (TVIM-I) and (TVIM-II), respectively) for nonexpansive operator T associated with contraction operator in CAT(0) space X:. Xiong and Lan Journal of Inequalities and Applications (2020) 2020:145 and an )T Where u1 ∈ E ⊆ X is an arbitrary given element, f : E → E is a contraction operator and number sequences {an}, {bn} ⊆ (0, 1) satisfy the following conditions:. ⎩0 < lim infn→∞ bn ≤ lim supn→∞ bn < 1. Remark 1.1 (i) The iterative procedures (TVIM-I) (1.1) and (TVIM-II) (1.2) with the implicit midpoint rule are well-defined. Defining an operator G1 : E → E by G1(v) =
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