On the proximal point method in Hadamard spaces

Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces

“…whereγ is the derivative of the curve γ : R → X, define by γ(t) := (t, e t ) for each t ∈ R. Then (X, d) is a complete p-uniformly convex metric space with p = 2 and parameter c = 2. Let f := ||.|| 2 2 : X → R. Then, f is proper, convex and lower semicontinuous in (X, d) (see [8,Example 7.1]). Now, define T 1 , T 2 : X → X by T 1 (x 1 , x 2 ) = (x 1 , e x1 ) and T 2 (x 1 , x 2 ) = (−x 1 , e −x1 ) for all x = (x 1 , x 2 ) ∈ X.…”

“…They proved the convergence of (8) to a fixed point of the contractive-like operator and also established some data dependence results for their proposed multi-step iterative scheme. In the same vein, Basarir and Sahin [4] studied (8) in the framework of CAT(0) spaces with k-strictly pseudocontractive mappings and they established the convergent results under some suitable conditions. Many authors have also used the multi-step algorithms to approximate fixed points of nonlinear mappings in CAT(0) spaces (see [32,5,36,37] and the references therein).…”

“…He established the ∆-convergence of (2) with the assumptions that f has a minimizer in X and ∞ n=1 λ n = ∞. Other authors have also studied the PPA in the framework of Hadamard spaces (see for example [8,20]). MPs are closely related to other optimization problems such as the monotone inclusion problems [21], equilibrium problems [23] (see also [19]), variational inequality problems [20] and many more.…”

“…MPs are closely related to other optimization problems such as the monotone inclusion problems [21], equilibrium problems [23] (see also [19]), variational inequality problems [20] and many more. Thus, finding solutions of MPs using the PPA in Hilbert, Banach and Hadamard spaces still remains an active area of research in nonlinear and convex analysis and optimization see [1,2,8,15,16,31,38,39,40,43] and other references therein. In an attempt to introduced and generalized the PPA from these spaces to p-uniformly convex metric spaces, Choi and Ji [9] introduced the notion of p-resolvent operator in a p-uniformly convex metric space as a generalization of the Moreau-Yosida resolvent in a CAT(0) space as follows:…”

Monotone relations in Hadamard spaces

“…In [9], Khatibzadeh and Ranjbar, investigated some properties of monotone operators and their resolvents and also proximal point algorithm in Hadamard spaces. Chaipunya and Kumam [6] studied the general proximal point method for finding a zero point of a maximal monotone set-valued vector field defined on Hadamard spaces and valued in its linear dual. They proved the relation between the maximality and Minty's surjectivity condition.…”