Abstract
In this paper, we study a schematic approximation of solutions of a split null point problem for a finite family of maximal monotone operators in real Hilbert spaces. We propose an iterative algorithm that does not depend on the operator norm which solves the split null point problem and also solves a generalized mixed equilibrium problem. We prove a strong convergence of the proposed algorithm to a common solution of the two problems. We display some numerical examples to illustrate our method. Our result improves some existing results in the literature.
Highlights
Let C be a nonempty, closed and convex subset of a real Hilbert space H
If A is a monotone operator, we can define, for each r > 0, a nonexpansive single-valued mapping JrA : R( I + rA) → D ( A) by JrA := ( I + rA)−1 which is generally known as the resolvent of A
This paper considered the approximation of common solutions of a split null point problem for a finite family of maximal monotone operators and generalized mixed equilibrium problem in real Hilbert spaces
Summary
Let C be a nonempty, closed and convex subset of a real Hilbert space H. For solving (8), Byrne et al [42] proposed the following iterative algorithm: For r > 0 and an arbitrary x0 ∈ H1 , A xn+1 = Jr 1 ( xn − γL∗ ( I − JrA2 ) Lxn ), where γ ∈ (0, 2/|| L||2 ). They prove a weak convergence of (9) to a solution of (8). We consider the problem of finding the common solution of the GMEP (2) and the SNPP for a finite family of intersection of maximal monotone operator in the frame work of real Hilbert spaces. We prove a strong convergence theorem of the proposed algorithm to the common solution of problem given by (11)
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