United Nations Educational, Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS CONVERGENCE OF HYBRID STEEPEST DESCENT METHOD FOR VARIATIONAL INEQUALITIES IN BANACH SPACES

Let E be a real q−uniformly smooth Banach space with constant dq, q ≥ 2. Let T : E → E and G : E → E be a nonexpansive map and an η−strongly accretive map which is also κ− Lipschitzian, respectively. Let {λn} be a real sequence in [0, 1] satisfying some appropriate conditions. For δ ∈ (0, ( qη dqκ )q−1), define a sequence {xn} iteratively in E by x0 ∈ E, xn+1 = T n+1xn = Txn − δλn+1G(Txn), n ≥ 0. Then, {xn} converges strongly to the unique solution x ∗ of the variational inequality problem V I(G,K) (search for x ∈ K such that 〈Gx, jq(y − x )〉 ≥ 0 ∀ y ∈ K), where K := Fix(T ) = {x ∈ E : Tx = x} 6 = ∅. A convergence theorem related to finite family of nonexpansive maps is also proved. MIRAMARE – TRIESTE July 2007 [email protected] [email protected] [email protected]