On the proximal point algorithm and demimetric mappings in CAT(0) spaces

Abstract

In this paper, we introduce and study the class of demimetric mappings in CAT(0) spaces.We then propose a modified proximal point algorithm for approximating a common solution of a finite family of minimization problems and fixed point problems in CAT(0) spaces. Furthermore,we establish strong convergence of the proposed algorithm to a common solution of a finite family of minimization problems and fixed point problems for a finite family of demimetric mappings in complete CAT(0) spaces. A numerical example which illustrates the applicability of our proposed algorithm is also given. Our results improve and extend some recent results in the literature.

On Mixed Equilibrium Problems in Hadamard Spaces

“…In this case, x * is called a minimizer of Ψ and argmin y∈C Ψ(y) denotes the set of minimizers of Ψ. MPs are very useful in optimization theory and convex and nonlinear analysis. One of the most popular and effective methods for solving MPs is the proximal point algorithm (PPA) which was introduced in Hilbert space by Martinet [1] in 1970 and was further extensively studied in the same space by Rockafellar [2] in 1976. e PPA and its generalizations have also been studied extensively for solving MP (1) and related optimization problems in Banach spaces and Hadamard manifolds (see [3][4][5][6][7] and the references therein), as well as in Hadamard and p-uniformly convex metric spaces (see [8][9][10][11][12][13] and the references therein).…”

A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces

“…The mapping P C is called the metric projection from H onto C. It is well known that P C has the following characteristics: 2 , for every x, y ∈ H, (ii) for x ∈ H and z ∈ C, z = P C x if and only if ⟨x − z, z − y⟩ ≥ 0, for all y ∈ C, (10) (iii) for x ∈ H and y ∈ C,…”

“…This field is experiencing an explosive growth in both theory and applications. Several iterative methods have been developed for solving the VIP and its related optimization problems, see [1][2][3][4][5] and references therein. It is well known that the VIP is equivalent to the following fixed point problem (see [6])…”

On θ-generalized demimetric mappings and monotone operators in Hadamard spaces

“…Let C be a nonempty subset of a real Hilbert space H Demimetric mappings are of central importance in optimization since they contain many common types of operators emanating from optimization. For instance, the class of k-demimetric mappings with ∈ (−∞ ) k , 1 is known to cover the class of θ-generalized hybrid mappings, the metric projections and the resolvents of maximal monotone operators (which are known as useful tools for solving optimization problems) in Hilbert spaces (see [1,2] and references therein). Thus, many authors have studied this class of mappings in both Hilbert and Banach spaces (see [1,[3][4][5]).…”

“…Thus, many authors have studied this class of mappings in both Hilbert and Banach spaces (see [1,[3][4][5]). This was recently extended to Hadamard spaces by Aremu et al [2]. They defined demimetric mappings in a Hadamard space as follows: let C be a nonempty subset of a CAT(0) space X.…”