Abstract
In this paper, a new iterative algorithm is proposed for finding a common solution to a constrained convex minimization problem, a quasi-variational inclusion problem and the fixed point problem of a strictly pseudo-contractive mapping in a real Hilbert space. It is proved that the sequence generated by the proposed algorithm converges strongly to a common solution of the three above described problems. By applying this result to some special cases, several interesting results can be obtained.
Highlights
1 Introduction Variational inequalities, introduced by Hartman and Stampacchia [ ] in the early sixties, are one of the most interesting and intensively studied classes of mathematical problems. They are a very powerful tool of the current mathematical technology and have been extended to study a considerable amount of problems arising in mechanics, physics, optimization and control, nonlinear programming, transportation equilibrium and engineering sciences
Throughout this paper, we assume that H is a real Hilbert space with the inner product ·, · and the induced norm ·, and let C be a nonempty closed convex subset of H
In order to find a common element of the solution set of quasi-variational inclusion problem ( . ) and the fixed point set of k-strictly pseudo-contractive mapping ( . ), which is a solution of the following constrained convex minimization problem: min f (x), ( . )
Summary
Variational inequalities, introduced by Hartman and Stampacchia [ ] in the early sixties, are one of the most interesting and intensively studied classes of mathematical problems. ) and the fixed point set of k-strictly pseudo-contractive mapping A monotone multi-valued mapping M is called maximal if for any (x, f ) ∈ H × H, x – y, f – g ≥ for every (y, g) ∈ G(M) implies f ∈ Mx. Remark .
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