Abstract
We introduce new implicit and explicit iterative schemes for finding a common element of the set of fixed points of k-strictly pseudocontractive mapping and the set of zeros of the sum of two monotone operators in a Hilbert space. Then we establish strong convergence of the sequences generated by the proposed schemes to a common point of two sets, which is a solution of a certain variational inequality. Further, we find the unique solution of the quadratic minimization problem, where the constraint is the common set of two sets mentioned above. As applications, we consider iterative schemes for the Hartmann-Stampacchia variational inequality problem and the equilibrium problem coupled with fixed point problem.MSC:47H05, 47H09, 47H10, 47J05, 47J07, 47J25, 47J20, 49M05.
Highlights
Let H be a real Hilbert space with the inner product ·, · and the induced norm ·
Inspired and motivated by the above-mentioned recent works, in this paper, we introduce new implicit and explicit iterative schemes for finding a common element of the set of the solutions of the monotone inclusion problem (MIP) ( . ) with a set-valued maximal monotone operator B and an inverse-strongly monotone mapping A and the set of fixed points of a k-strictly pseudocontractive mapping T
We find the unique solution of the quadratic minimization problem: x = min x : x ∈ F(T) ∩ (A + B)
Summary
Let H be a real Hilbert space with the inner product ·, · and the induced norm ·. If T is a k-strictly pseudocontractive mapping on C, the fixed point set F(T) is closed convex, so that the projection PF(T) is well defined, and F(PCT) = F(T).
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