Abstract
Let C be a closed and convex subset of a real Hilbert space H. Let T be a 2-generalized hybrid mapping of C into itself, let A be an α-inverse strongly-monotone mapping of C into H, and let B and F be maximal monotone operators on and respectively. The purpose of this paper is to introduce a general iterative scheme for finding a point of which is a unique solution of a hierarchical variational inequality, where is the set of fixed points of T, and are the sets of zero points of and F, respectively. A strong convergence theorem is established under appropriate conditions imposed on the parameters. Further, we consider the problem for finding a common element of the set of solutions of a mathematical model related to mixed equilibrium problems and the set of fixed points of a 2-generalized hybrid mapping in a real Hilbert space.
Highlights
Let H be a Hilbert space, and let C be a nonempty closed convex subset of H
Lin and Takahashi [ ] obtained the strong convergence theorem for finding a point p ∈ (A + B)– ∩ F– which is a unique solution of a hierarchical variational inequality, where A is an α-inverse strongly-monotone mapping of C into H, and B and F are maximal monotone operators on D(B) ⊂ C and D(F) ⊂ C, respectively
We introduce a new general iterative scheme for finding a common element of F(T) ∩ (A + B)– ∩ F– which is a unique solution of a hierarchical variational inequality, where F(T) is the set of fixed points of T, (A + B)– and F– are the sets of zero points of A + B and F, respectively
Summary
Let H be a Hilbert space, and let C be a nonempty closed convex subset of H. A mapping T : C → C is said to be generalized hybrid if there are α, β ∈ R such that α Tx – Ty + ( – α) x – Ty ≤ β Tx – y + ( – β) x – y for all x, y ∈ C. An (α, β)-generalized hybrid mapping is nonexpansive for α = and β = , nonspreading for α. A mapping T is called -generalized hybrid if there exist α , α , β , β ∈ R such that α T x – Ty + α Tx – Ty + ( – α – α ) x – Ty. Recall that a linear bounded operator B is strongly positive if there is a constant γ > with the property. Hojo et al [ ] proved the strong convergence theorem of Halpern type [ ] for -generalized hybrid mappings in a Hilbert space as follows.
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