### Approximating common solutions of variational inequalities by iterative algorithms with applications

Xiaolong Qin, Sun Young Cho, Yeol Je Cho
2011 Glasnik Matematicki - Serija III
In this paper, we introduce an iterative scheme for a general variational inequality. Strong convergence theorems of common solutions of two variational inequalities are established in a uniformly convex and 2-uniformly smooth Banach space. As applications, we, still in Banach spaces, consider the convex feasibility problem. 2010 Mathematics Subject Classification. 47H05, 47H09, 47J25. Recall that the classical variational inequality, denoted by V I(C, A), is to find u ∈ C such that It is well
more » ... nown that for given z ∈ H and u ∈ C satisfy the inequality if and only if u = P C z, where P C denotes the metric projection from H onto C. From the above, we see that u ∈ C is a solution to the problem (1.1) if and only if u satisfies the following equation: where ρ > 0 is a constant. This implies that the problem (1.1) and the problem (1.2) are equivalent. This alternative formula is very important from the numerical analysis point of view. Many authors studied iterative methods for the problem (1.1) provided that A has some monotonicity. Recently, Aoyama, Iiduka and Takahashi () introduced and analyzed a general variational inequality which can be viewed as a Banach version of the variational inequality (1.1). Let C be a nonempty closed convex subset of a Banach space E and E * the dual space of E. Let ·, · denote the pairing between E and E * . For q > 1, the generalized duality mapping J q : E → 2 E * is defined by for all x ∈ E. In particular, J = J 2 is called the normalized duality mapping. It is known that J q (x) = x q−2 J(x) for all x ∈ E. If E is a Hilbert space, then J = I, the identity mapping. Further, we have the following properties of the generalized duality mapping J q : (1) J q (x) = x q−2 J 2 (x) for all x ∈ E with x = 0; (2) J q (tx) = t q−1 J q (x) for all x ∈ E and t ∈ [0, ∞); (3) J q (−x) = −J q (x) for all x ∈ E. Let U E = {x ∈ E : x = 1}. A Banach space E is said to uniformly convex if, for any ǫ ∈ (0, 2], there exists δ > 0 such that, for any x, y ∈ U E ,