Hybrid viscosity-like approximation methods for nonexpansive mappings in Hilbert spaces

Consider on a real Hilbert space HH a nonexpansive mapping TT with a fixed point, a contraction ff with coefficient 0<α<10<α<1, and two strongly positive linear bounded operators A,BA,B with coefficients γ̄∈(0,1) and β>0β>0, respectively. Let 0<γα<β0<γα<β. We introduce a general iterative algorithm defined by xn+1≔(I−λn+1A)Txn+λn+1[Txn−μn+1(BTxn−γf(xn))],∀n≥1, with μn→μ(n→∞)μn→μ(n→∞), and prove the strong convergence of the iterative algorithm to a fixed point x̃∈Fix(T)≕C which is the unique solution of the variational inequality (for short, V I(A−I+μ(B−γf),C)): 〈[A−I+μ(B−γf)]x̃,x−x̃〉≥0,∀x∈C. On the other hand, assume CC is the intersection of the fixed-point sets of a finite number of nonexpansive mappings on HH. We devise another iterative algorithm which generates a sequence {xn}{xn} from an arbitrary initial point x0∈Hx0∈H. The sequence {xn}{xn} is proven to converge strongly to an element of CC which is the unique solution x∗x∗ of the V I(A−I+μ(B−γf),C). Applications to constrained generalized pseudoinverses are included.