SPLIT COMMON FIXED POINT PROBLEMS AND HIERARCHICAL VARIATIONAL INEQUALITY PROBLEMS IN HILBERT SPACES

Let Hi and H-2 be real Hilbert spaces and let alpha > 0. Let U be an alpha-inverse strongly monotone mapping of H-2 into H-2. Let B be a maximal monotone operator on H-1. Let S be a demimetric mapping of H-1 into H-1. Let k is an element of(0,1) and let g be a k-contraction of H-1 into itself. Let V be a (gamma) over bar -strongly monotone and L-Lipschitzian continuous operator of H-1 into H-1 with (gamma) over bar > 0 and L > 0. Take mu, gamma is an element of R as follows: 0 < mu < 2 (gamma) over bar /L-2,L- 0 < gamma < (gamma) over bar- L-2 mu/2 /k Let A : H-1 -> H-2 be a bounded linear operator such that parallel to A parallel to not equal 0. Suppose F(S) boolean AND B(-1)0 boolean AND A(-1)(u(-1)0) not equal empty set, where F(S) is the set of fixed points of S and B(-1)0 and U(-1)0 are the sets of zero points of B and U, respectively. In this paper, we prove a strong convergence theorem for finding a point z(0) of F(S) boolean AND B(-1)0 boolean AND A(-1)(u(-1)0), where z(0) is a unique fixed point of PF(s)boolean AND B-10A-1(u-10)(I-V+gamma g). This point z(0) is an element of F(S) n B(-1)0 n A(-1)(U(-1)0) is also a unique solution of the hierarchical variational inequality ((V - gamma g)z(0), q - z(0)) >= 0, V q is an element of F(S) n B(-1)0 n A(-1)(U(-1)0). Using this result, we obtain new and well-known strong convergence theorems in Hilbert spaces.