On a Mann type implicit iteration process for continuous pseudo-contractive mappings

Let K$K$ be a nonempty closed convex subset of a Banach space E$E$, T:K→K$T:K\to K$ a continuous pseudo-contractive mapping. Suppose that {_αn}$\left\{{\alpha }_n\right\}$ is a real sequence in [0,1]$[0,1]$ satisfying appropriate conditions; then for arbitrary _x0∈K$x_0\in K$, the Mann type implicit iteration process {_xn}$\left\{x_n\right\}$ given by _xn=_αnxn−1+(1−_αn)T_xn,n≥0$x_n={\alpha }_n{x}_{n-1}+(1-{\alpha }_n)T{x}_n,n\ge 0$, strongly and weakly converges to a fixed point of T$T$, respectively.     

The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces

A recent trend in the iterative methods for constructing fixed points of nonlinear mappings is to use the viscosity approximation technique. The advantage of this technique is that one can find a particular solution to the associated problems, and in most cases this particular solution solves some variational inequality. In this paper, we try to extend this technique to find a particular common fixed point of a finite family of asymptotically nonexpansive mappings in a Banach space which is reflexive and has a weakly continuous duality map. Both implicit and explicit viscosity approximation schemes are proposed and their strong convergence to a solution to a variational inequality is proved.     

Further investigation on iteration processes for pseudocontractive mappings with application

A new iterative method for approximating fixed points of bounded and continuous pseudocontractive mapping is proposed and a strong convergence theorem is obtained. As an application, we prove that a slight modification of our new scheme could be employed for approximating zeros of bounded and continuous accretive operators. Our theorems extend and unify most of the results that have been proved for this class of mappings.     

Convergence of iterative algorithms for multivalued mappings in Banach spaces

We show strong and weak convergence for Mann iteration of multivalued nonexpansive mappings T$T$ in a Banach space. Furthermore, we give a strong convergence of the modified Mann iteration which is independent of the convergence of the implicit anchor-like continuous path _zt∈tu+(1−t)T_zt$z_t\in tu+(1-t)T{z}_t$.     

Strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups

Strong convergence theorems are obtained for a finite family of asymptotically nonexpansive mappings and semigroups by the modified Mann method.     

A new iteration process for finite families of generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings

A new iteration process is introduced and proved to converge strongly to a common fixed point for a finite family of generalized Lipschitz nonlinear mappings in a real reflexive Banach space E$E$ with a uniformly Gâteaux differentiable norm if at least one member of the family is pseudo-contractive. It is also proved that a slight modification of the process converges to a common zero for a finite family of generalized Lipschitz accretive operators defined on E$E$. Results for nonexpansive families are obtained as easy corollaries. Finally, the new iteration process and the method of proof are of independent interest.     

Convergence of Ishikawa’s iteration method for pseudocontractive mappings

Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let T_i:C→C,i=1,2,…,N, be a finite family of Lipschitz pseudocontractive mappings. It is our purpose, in this paper, to prove strong convergence of Ishikawa’s method to a common fixed point of a finite family of Lipschitz pseudocontractive mappings provided that the interior of the common fixed points is nonempty. No compactness assumption is imposed either on T or on C. Moreover, computation of the closed convex set C_n for each n≥1 is not required. The results obtained in this paper improve on most of the results that have been proved for this class of nonlinear mappings.     

A general iterative algorithm for nonexpansive mappings in Hilbert spaces

Let H be a real Hilbert space. Suppose that T is a nonexpansive mapping on H with a fixed point, f is a contraction on H with coefficient 0<α<1, and F:H→H is a k-Lipschitzian and η-strongly monotone operator with k>0,η>0. Let . We proved that the sequence {x_n} generated by the iterative method x_n+1=α_nγf(x_n)+(I−μα_nF)Tx_n converges strongly to a fixed point , which solves the variational inequality , for x∈F_ix(T).     

Fixed point solutions of generalized equilibrium problems for nonexpansive mappings

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a generalized equilibrium problem in a real Hilbert space. Then, strong convergence of the scheme to a common element of the two sets is proved. As an application, problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem is solved. Moreover, solution is given to the problem of finding a common element of fixed points set of nonexpansive mappings and the set of solutions of a variational inequality problem.     

Convergence analysis of a Halpern type algorithm for accretive operators

In this paper, we study the convergence of a Halpern type proximal point algorithm for accretive operators in Banach spaces. Our results fill the gap in the work of Zhang and Song (2012) [1] and, consequently, all the results there can be corrected accordingly.     

Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings

In this paper, we prove strong convergence theorems to a zero of monotone mapping and a fixed point of relatively weak nonexpansive mapping. Moreover, strong convergence theorems to a point which is a fixed point of relatively weak nonexpansive mapping and a solution of a certain variational problem are proved under appropriate conditions.     

Implicit iteration process for a finite family of asymptotically hemi-contractive mappings in Banach spaces

The purpose of this paper is to study the strong convergence of a process of implicit iteration to a common fixed point for a finite family of asymptotically hemi-contractive mappings. Our results extend a recent result of M.O. Osilike and B.G. Akuchu [Common fixed points of a finite family of asymptotically pseudocontractive maps, Fixed Point Theory Appl. 2 (2004) 81–88] from Hilbert spaces to p$p$-uniformly convex Banach spaces with p>1$p>1$.     

Strong convergence theorems for a finite family of nonexpansive mappings and semigroups via the hybrid method

Strong convergence theorems are obtained for a finite family of nonexpansive mappings and semigroups by the hybrid method.     

Hybrid pseudo-viscosity approximation schemes for systems of equilibrium problems and fixed point problems of infinite family and semigroup of non-expansive mappings

In this paper, we introduce hybrid pseudo-viscosity approximation schemes with strongly positive bounded linear operators for finding a common element of the set of solutions to a system of equilibrium problems, the set of fixed points of an infinite family and left amenable semigroup of non-expansive mappings in the frame work of Hilbert spaces. Our goal is to prove a result of strong convergence for hybrid pseudo-viscosity approximation schemes to approach a solution of systems of equilibrium problems which is also a common fixed point of an infinite family and left amenable semigroup of non-expansive mappings. The results presented in this paper can be treated as an extension and improvement of the corresponding results announced by Ceng et al. [L.C. Ceng, Q.H. Ansari, and J.C. Yao, Hybrid pseudo-viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many non-expansive mappings, Nonlinear Analysis 4 (2010) 743-754] and many others.     

Approximation of common fixed points for a countable family of relatively nonexpansive mappings in a Banach space and applications

In this paper, we prove a strong convergence theorem by the hybrid method for a countable family of relatively nonexpansive mappings in a Banach space. We also establish a new control condition for the sequence of mappings {T_n} which is weaker than the control condition in Lemma 3.1 of Aoyama et al. [K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007) 2350-2360]. Moreover, we apply our results for finding a common fixed point of two relatively nonexpansive mappings in a Banach space and an element of the set of solutions of an equilibrium problem in a Banach space, respectively. Our results are applicable to a wide class of mappings.     

Strong convergence theorems for finitely many nonexpansive mappings

Under the assumption that E$E$ is a reflexive Banach space whose norm is uniformly Gêteaux differentiable and which has a weakly continuous duality mapping _Jφ$J_{\phi }$ with gauge function φ$\phi$, Ceng–Cubiotti–Yao [Strong convergence theorems for finitely many nonexpansive mappings and applications, Nonlinear Analysis 67 (2007) 1464–1473] introduced a new iterative scheme for a finite commuting family of nonexpansive mappings, and proved strong convergence theorems about this iteration. In this paper, only under the hypothesis that E$E$ is a reflexive Banach space which has a weakly continuous duality mapping _Jφ$J_{\phi }$ with gauge function φ$\phi$, and several control conditions about the iterative coefficient are removed, we present a short and simple proof of the above theorem.     

Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups

The Mann iterations for nonexpansive mappings have in general only weak convergence in a Hilbert space. We modify an iterative method of Mann's type introduced by Nakajo and Takahashi [Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379] for nonexpansive mappings and prove strong convergence of our modified Mann's iteration processes for asymptotically nonexpansive mappings and semigroups.     

A note on the paper “A new iteration process for generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings”

In this note, we will modify several gaps in Chidume and Ofoedu [C.E. Chidume, E.U. Ofoedu, A new iteration process for generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings, Nonlinear Anal. (2006), in press (doi:10.1016/j.na.2006.05.012)].