Relaxed viscosity approximation methods for fixed point problems and variational inequality problems (Q955639)

Let \(X\) be a strictly convex and reflexive Banach space with a uniformly Gâteaux differentiable norm and \(C\) a nonempty closed convex subset of \(X\). Let \(\{T_n\}_{n=1}^\infty\) be a sequence of nonexpansive self-mappings on \(C\) with the nonempty common fixed point set \(F\) and \(f:C\to C\) a given contractive map. The authors present an iterative scheme which converges strongly to some \(p\in F\) which is the unique solution to the variational inequality \[ \langle(I - f)p, J(p-x^*)\rangle\leq 0, \quad \forall x^* \in F, \] where \(J\) is the duality map of \(X\).