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
∀x0 ∈ C, xn+1 = αnxn +(1−αn)T xn, n 0.
It is proved that if the control sequence {αn} is chosen so that κ < αn < 1 and
∞
n=0(αn − κ)(1 − αn) = ∞, then
limn→∞ xn − T xn = 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 {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].
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