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

Abstract

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 ∀ x 0 ∈ C , x n + 1 = α n x n + ( 1 − α n ) T x n , n ⩾ 0 . It is proved that if the control sequence { α n } is chosen so that κ < α n < 1 and ∑ n = 0 ∞ ( α n − κ ) ( 1 − α n ) = ∞ , then lim n → ∞ ‖ x n − T x n ‖ = d ( 0 , R ( A ) ¯ ) , where A = I − T and 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 { x n } 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].