Convergence of a Hybrid Iterative Scheme for Fixed Points of Nonexpansive Maps, Solutions of Equilibrium, and Variational Inequalities Problems

Let K be a close, convex, and nonempty subset of a real q-uniformly smooth Banach space E, which is also uniformly converx. For some k > 0, let T-i : K -> E i is an element of N and A : K -> E be family of nonexpansive maps and k-inverse strongly accretive map, respectively. Let G : KxK -> R be a bifunction satisfying some condition. Let P-k be a nonexpansive of E onto K. For some fixed real numbers delta is an element of(0,1), lambda is an element of(0,(qj/d(q))(1)/((q-1))), and arbitrary but fixed vectors x(1),u is an element of E, let {x(n)} and {y(n)} be sequences generated by G(y(n),eta) +(1/r) (eta-y(n), j(q)(y(n)-x(n)) > >= 0, for all eta is an element of K, x(n+1) = alpha(n)u+(1 - delta) (1 - alpha(n))x(n) + delta Sigma(i >= 1) sigma(in) TiPK(y(n) - lambda Ay(n)), n >= 1, where r is an element of (0,1) is fixed, and {alpha(n},) {sigma(i,n)} subset of (0,1) are sequences satisfying appropriate conditions. If F : = [boolean AND F-infinity(i=1)(T-i)] boolean AND VI(K,A) boolean AND EP(G) not equal theta, under some mild conditions, we prove that the sequences {x(n)} and {y(n)} converge strongly to some element in F.