Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces

Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ-strict pseudo-contraction on C. Consider Mann's iteration algorithm given by∀x0∈C,xn+1=αnxn+(1−αn)Txn,n⩾0. It is proved that if the control sequence {αn}{αn} is chosen so that κ<αn<1κ<αn<1 and ∑n=0∞(αn−κ)(1−αn)=∞, then limn→∞‖xn−Txn‖=d(0,R(A)¯), where A=I−TA=I−T and d(0,D)d(0,D) denotes the distance between the origin and the subset set D of H. As a consequence of this result, we prove that if T has a fixed point in C  , then {xn}{xn} converges weakly to a fixed point of T. Also, we extend a result due to Reich to κ-strict pseudo-contractions in the Hilbert space setting. Further, by virtue of hybridization projections, we establish a strong convergence theorem for Lipschitz pseudo-contractions. The results presented in this paper improve or extend the corresponding results of Browder and Petryshyn [F.E. Browder, W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20 (1967) 197–228], Rhoades [B.E. Rhoades, Fixed point iterations using infinite matrices, Trans. Amer. Math. Soc. 196 (1974) 162–176] and of Marino and Xu [G. Marino, H.-K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (1) (2007) 336–346].