A General Iterative Approach to Variational Inequality Problems and Optimization Problems
© Jong Soo Jung. 2011
Received: 4 October 2010
Accepted: 14 November 2010
Published: 30 November 2010
We introduce a new general iterative scheme for finding a common element of the set of solutions of variational inequality problem for an inverse-strongly monotone mapping and the set of fixed points of a nonexpansive mapping in a Hilbert space and then establish strong convergence of the sequence generated by the proposed iterative scheme to a common element of the above two sets under suitable control conditions, which is a solution of a certain optimization problem. Applications of the main result are also given.
Let be a real Hilbert space with inner product and induced norm . Let be a nonempty closed convex subset of and be self-mapping on . We denote by the set of fixed points of and by the metric projection of onto .
We denote the set of solutions of the variational inequality problem (1.1) by . The variational inequality problem has been extensively studied in the literature; see [1–5] and the references therein.
where is a contraction with constant , , , and . He proved that the sequence generated by (1.3) strongly converges strongly to a point in , which is the unique solution of a certain variational inequality.
where , are infinitely many closed convex subsets of such that , , is a real number, is a strongly positive bounded linear operator on (i.e., there is a constant such that , for all ), and is a potential function for (i.e., for all ). For this kind of optimization problems, see, for example, Deutsch and Yamada , Jung , and Xu [12, 13] when and for a given point in .
In this paper, motivated by the above-mentioned results, we introduce a new general composite iterative scheme for finding a common point of the set of solutions of the variational inequality problem (1.1) for an inverse-strongly monotone mapping and the set of fixed points of a nonexapansive mapping and then prove that the sequence generated by the proposed iterative scheme converges strongly to a common point of the above two sets, which is a solution of a certain optimization problem. Applications of the main result are also discussed. Our results improve and complement the corresponding results of Chen et al. , Iiduka and Takahashi , Jung , and others.
2. Preliminaries and Lemmas
So, if , then is a nonexpansive mapping of into . The following result for the existence of solutions of the variational inequality problem for inverse strongly-monotone mappings was given in Takahashi and Toyoda .
We need the following lemmas for the proof of our main results.
Lemma 2.3 (Xu ).
Lemma 2.4 (Marino and Xu ).
3. Main Results
In this section, we present a new general composite iterative scheme for inverse-strongly monotone mappings and a nonexpansive mapping.
Now we divide the proof into several steps.
As a direct consequence of Theorem 3.1, we have the following results.
Using Theorem 3.1, we found a strong convergence theorem for finding a common fixed point of a nonexpansive mapping and a strictly pseudo-contractive mapping.
Using Theorem 3.1, we also obtain the following result.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0017007).
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