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We propose a modified hybrid projection algorithm to approximate a common fixed point of a -strict pseudocontraction and of two sequences of nonexpansive mappings. We prove a strong convergence theorem of the proposed method and we obtain, as a particular case, approximation of solutions of systems of two equilibrium problems.

Let H be a Hilbert space and let C be a closed, convex and nonempty subset of H. If $T:C\to H$ is a non-self and non-expansive mapping, we can define a map $h:C\to\mathbb{R}$ by $h(x):=\inf\{\lambda\geq 0:\lambda x+(1-\lambda)Tx\in C\}$ . Then, for a fixed $x_{0}\in C$ and for $\alpha_{0}:=\max\{1/2, h(x_{0})\}$ , we define the Krasnoselskii-Mann algorithm $x_{n+1}=\alpha _{n}x_{n}+(1-\alpha_{n})Tx_{n}$ , where $\alpha_{n+1}=\max\{\alpha_{n},h(x_{n+1})\}$ . We will prove both weak and strong convergence results when C is a strictly convex set and T is an inward mapping.

Let H be a real Hilbert space. Consider on H a nonexpansive family T = {T(t) :0 ≤ t < ∞} with a common fixed point, a contraction f with the coefficient 0

We introduce two iterative algorithms for nonexpansive mappings in Hilbert spaces. We prove that the proposed algorithms strongly converge to a fixed point of a nonexpansive mapping .

We introduce a new iterative method for finding a common element of the set of solutions of a generalized equilibrium problem with a relaxed monotone mapping and the set of common fixed points of a countable family of nonexpansive mappings in a Hilbert space and then prove that the sequence converges strongly to a common element of the two sets. Using this result, we prove several new strong convergence theorems in fixed point problems, variational inequalities, and equilibrium problems.

First we present some interesting variants of Mann’s method. In the last section, we show that many existing results in the literature are concrete realizations of our general scheme under varying assumptions on the coefficients.

In this paper we use a matrix approach to approximate solutions of variational inequalities in Hilbert spaces. The methods studied combine new or well-known iterative methods (as the original Mann method) with regularized processes that involve regular matrices in the sense of Toeplitz. We obtain ergodic type results and convergence.