The viscosity approximation method for multivalued G -nonexpansive mappings in Hadamard spaces endowed with graphs
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Journal of Fixed Point Theory and Applications
The viscosity approximation method for multivalued G-nonexpansive mappings in Hadamard spaces endowed with graphs Bancha Panyanak Abstract. Using the viscosity approximation method introduced by Moudafi (J Math Anal Appl 241:46–55, 2000), we can obtain strong convergence theorems for monotone increasing G-nonexpansive mappings in Hadamard spaces endowed with graphs. We also give sufficient conditions for the existence of solutions of the variational inequality problem in this setting. Our results generalize and improve many results in the literature. Mathematics Subject Classification. 47H09, 47H10. Keywords. Viscosity approximation method, monotone increasing Gnonexpansive mapping, Hadamard space, directed graph, variational inequality problem.
1. Introduction Let A be a nonempty subset of a metric space (X, d). A mapping g : A → A is said to be a contraction if there exists a constant k ∈ [0, 1), such that: d(g(x), g(y)) ≤ kd(x, y), for all x, y ∈ A.
(1.1)
If (1.1) is valid when k = 1, then g is said to be nonexpansive. A point x in A is called a fixed point of g if x = g(x). Let (H, ·, ·) be a Hilbert space and C a nonempty closed convex subset of H. Let f : C → C be a contraction and T : C → C a nonexpansive mapping with a nonempty fixed point set F (T ). The variational inequality problem is to find a point z in F (T ), such that: f (z) − z, z − q ≥ 0, for all q ∈ F (T ).
(1.2)
One of the powerful tools for solving such problems is the so-called viscosity approximation method which was introduced by Moudafi [28]. Precisely, he proved the following result. 0123456789().: V,-vol
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B. Panyanak
Theorem 1.1. Let H, C, f, T be as above. Let {xn } be the sequence in C defined by x1 ∈ C and: εn 1 xn+1 := f (xn ) ⊕ T (xn ), for all n ∈ N, 1 + εn 1 + εn ∞ where {εn } is a sequence in (0, 1), such that limn→∞ εn = 0, n=1 εn = ∞ 1 = 1. Then, {xn } converges strongly to the unique and limn→∞ ε1n − εn+1 solution of (1.2). A geodesic space X is called an Hadamard space if it is a complete metric space which satisfies the (CN) inequality. The precise definition is given in the next section. It is well known that every Hilbert space is an Hadamard space (see, e.g., [6]). Thus, it is natural to study the extensions of the known results in Hilbert spaces to the general setting of Hadamard spaces. Fixed point theory in Hadamard spaces was first studied by Kirk [24]. He showed that every nonexpansive mapping defined on a bounded closed convex subset of an Hadamard space always has a fixed point. Wangkeeree and Prechasilp [38] guaranteed the existence of solutions of the variational inequality problem in an Hadamard space. Later on, Panyanak and Suantai [31] extended Wangkeeree and Prechasilp’s result to the case of multivalued nonexpansive mappings. On the other hand, Jachymski [20] combined the concepts of fixed point theory and graph theory to prove a generalization of the Banach contraction principle in a complete metric space endowed with a graph. Beg et al. [4]
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