A Common Fixed Point Theorem Using an Iterative Method

We prove the existence of common fixed points for two selfmaps T and f of a convex metric space X via the convergence of modified Mann iteration where T is a nonlinear f -weakly contractive selfmap of X and range of f is complete. An invariant approximation result is also proved.

International Journal of Engineering Sciences & Research Technology, 2014

In this paper we prove a fixed point theorem for nonexpansive mapping using a well known result of Ky Fan’s best approximation theorem in Hilbert space setting. AMS Subject Classification: 47H10, 54H25

2009

Let X be a real uniformly convex Banach space and C a closed convex nonempty subset of X. Let {Ti}ri1 be a finite family of nonexpansive self-mappings of C. For a given x1 ∈ C, let {xn} and {xin}, i 1, 2,..., r, be sequences defined x0n xn, x1n a1n1 T1x 0 n 1 − a1n1 x

2000

We suggest and analyze an iterative algorithm for a finite family of nonexpansive mappings T 1 ,T 2 ,...,T r . Further, we prove that the proposed iterative algorithm converges strongly to a common fixed point of T 1 ,T 2 ,...,T r .

Applied Mathematics and Computation, 2006

Let E be either a strictly convex and reflexive Banach spaces with a uniformly Gâteaux differentiable norm or a reflexive Banach spaces with a weakly sequentially continuous duality mapping, and K be a nonempty closed convex subset of E. For a family of finite many nonexpansive mappings {T l } (l = 1,2,. . . , N) and fixed contractive mapping f : K ! K, define iteratively a sequence {x n } as follows:

Journal of Applied Mathematics and Computing, 2011

We modify an iteration process of Agarwal et al. (J. Nonlinear Convex Anal. 8(1):61-79, 2007) to the case of two mappings and prove some weak and strong convergence theorems for two asymptotically nonexpansive mappings. We also point out that this process cannot be used for three mappings in its existing form even for nonexpansive mappings. We have to impose an extra condition to get convergence. We give an example to show that there do exist two nonexpansive mappings satisfying that condition.

Fixed Point Theory and Applications, 2005

Let K be a nonempty closed convex subset of a reflexive real Banach space E which has a uniformly Gâteaux differentiable norm. Assume that K is a sunny nonexpansive retract of E with Q as the sunny nonexpansive retraction. Let T i : K → E, i = 1,...,r, be a family of nonexpansive mappings which are weakly inward. Assume that every nonempty closed bounded convex subset of K has the fixed point property for nonexpansive mappings. A strong convergence theorem is proved for a common fixed point of a family of nonexpansive mappings provided that T i , i = 1,2,...,r, satisfy some mild conditions. Publication Date

Indian Journal of Mathematics

Carpathian Journal of Mathematics

Using the technique of enrichment of contractive type mappings by Krasnoselskij averaging, presented here for the first time, we introduce and study the class of enriched nonexpansive mappings in Hilbert spaces. In order to approximate the fixed points of enriched nonexpansive mappings we use the Krasnoselskij iteration for which we prove strong and weak convergence theorems. Examples to illustrate the richness of the new class of contractive mappings are also given. Our results in this paper extend some classical convergence theorems established by Browder and Petryshyn in [Browder, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228] from the case of nonexpansive mappings to that of enriched nonexpansive mappings,thus including many other important related results from literature as particular cases.

Journal of Computational and Applied Mathematics, 2009

In this paper, we introduce a modified Ishikawa iterative process for approximating a fixed point of nonexpansive mappings in Hilbert spaces. We establish some strong convergence theorems of the general iteration scheme under some mild conditions. The results improve and extend the corresponding results of many others.

Computers & Mathematics with Applications, 2011

Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let T i : C → H, i = 1, 2, . . . , N, be a finite family of generalized asymptotically nonexpansive mappings. It is our purpose, in this paper to prove strong convergence of Mann's type method to a common fixed point of {T i : i = 1, 2, . . . , N} provided that the interior of common fixed points is nonempty. No compactness assumption is imposed either on T or on C . As a consequence, it is proved that Mann's method converges for a fixed point of nonexpansive mapping provided that interior of F (T ) ̸ = ∅. The results obtained in this paper improve most of the results that have been proved for this class of nonlinear mappings.

Fixed Point Theory and Applications, 2007

Suppose that K is a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T 1 ,T 2 : K → E be two weakly inward and asymptotically nonexpansive mappings with respect to P with sequences {K n }, {l n } ⊂ [1,∞), lim n→∞ k n = 1, lim n→∞ l n = 1, F(T 1) ∩ F(T 2) = {x ∈ K : T 1 x = T 2 x = x} = ∅, respectively. Suppose that {x n } is a sequence in K generated iteratively by x 1 ∈ K, x n+1 = α n x n + β n (PT 1) n x n + γ n (PT 2) n x n , for all n ≥ 1, where {α n }, {β n }, and {γ n } are three real sequences in [ ,1 − ] for some > 0 which satisfy condition α n + β n + γ n = 1. Then, we have the following. (1) If one of T 1 and T 2 is completely continuous or demicompact and ∞ n=1 (k n − 1) < ∞, ∞ n=1 (l n − 1) < ∞, then the strong convergence of {x n } to some q ∈ F(T 1) ∩ F(T 2) is established. (2) If E is a real uniformly convex Banach space satisfying Opial's condition or whose norm is Fréchet differentiable, then the weak convergence of {x n } to some q ∈ F(T 1) ∩ F(T 2) is proved.

Based on the very recent work by Censor-Segal [4] and inspired by Xu [6], Zhao-Yang [10] and Bauschke-Combettes [2], in this paper, we study the algorithm proposed by Moudafi [11] for the class of quasinon expansive operators to solve the split common fixedpoint problem (SCFP) in the framework of Hilbert space. Furthermore we proved the strong convergence for the (SCFP), which extend and improve the result of Moudafi [11] from a weak convergence to a strong convergence.

Nonlinear Analysis: Theory, Methods &amp; Applications, 2010

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 0 < µ < 2η/k 2 , 0 < γ < µ(η − µk 2 2)/α = τ /α. We proved that the sequence {x n } generated by the iterative method x n+1 = α n γ f (x n) + (I − µα n F)Tx n converges strongly to a fixed pointx ∈ F ix (T), which solves the variational inequality (γ f − µF)x, x −x ≤ 0, for x ∈ F ix (T).