Iterative construction of fixed points of strictly pseudocontractive mappings

Let (E, ‖ ? ‖) be a smooth Banach space over the real field and A a nonempty closed bounded convex subset of E. Suppose T : A → A is a uniformly continuous strictly pseudocontractive selfmapping of A. Then, if [math001]satisfies [math001]the iteration process [math001] and [math001] converges strongly to the unique fixed point x of T. This is an improvement of a result of C.E. Chidume who established strong convergence of (x ⁿ to x in case E is L ^p or l ^p with [math001] making essential use of the inepuality [math001] which is kown to hold in these spaces for all x and y     

Iterative construction of fixed points of nonself-mappings in Banach spaces

The purpose of this paper is to study the iterative methods for constructing fixed points of nonself-mappings in Banach spaces. The concept of the class of asymptotically _QG$Q_G$-weakly contractive nonself-mappings is introduced and a new iterative algorithm for finding fixed points of this class of mappings is studied. Several strong convergence results on this algorithm are established under different conditions.     

Iterative construction of fixed points for multivalued operators of the monotone type

The fixed points of set valued operators satisfying a condition of the monotonicity type in Hilbert space are approximated by simple recursive averaging processes.     

Iterative approximation of fixed points of nonexpansive mappings

Let K be a nonempty closed convex subset of a real Banach space E which has a uniformly Gâteaux differentiable norm and be a nonexpansive mapping with F(T):={x∈K:Tx=x}≠∅. For a fixed δ∈(0,1), define by Sx:=(1−δ)x+δTx, ∀x∈K. Assume that {zt} converges strongly to a fixed point z of T as t→0, where zt is the unique element of K which satisfies zt=tu+(1−t)Tzt for arbitrary u∈K. Let {αn} be a real sequence in (0,1) which satisfies the following conditions: ; . For arbitrary x₀∈K, let the sequence {xn} be defined iteratively by

xn+1=αnu+(1−αn)Sxn.     

Iterative methods for interval inclusion of fixed points

The paper discusses a technique for handling numerical, iterative processes that combines the efficiency of ordinary floating-point iterations with the accuracy control that may be obtained by iterations in interval arithmetic. As illustration the technique is used for the solution of fixed point problems in one and several variables.     

Iterative approximation of fixed points of Lipschitz pseudocontractive maps

Let be a -uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g., ). Let be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset of and let be arbitrary. Then the iteration sequence defined by , converges strongly to a fixed point of , provided that and have certain properties. If is a Hilbert space, then converges strongly to the unique fixed point of closest to .

     

Iterative methods for the computation of fixed points of demicontractive mappings

This paper surveys some of the main convergence properties of the Mann-type iteration for the demicontractive mappings. Some variants of the Mann iteration that ensure the strong convergence, like the (CQ) algorithm and a variant for the asymptotically demicontractive mappings are also considered. The usual framework of our study is a (real) Hilbert space and only to a certain extent some particular Banach spaces. Historical aspects are pointed out and some applications for the convex feasibility problem are discussed.     

(渐近)非扩张映象的不动点的迭代逼近

Let E be a uniformly convex Banach space which satisfies Opial‘s condition or has aFrechet differentiable norm,and C be a bounded closed convex subset of E. If T： C→C is(asymptotically)nonexpansive,then the modified Ishikawa iteration process defined by     

Banach空间中φ-半压缩型算子不动点的迭代逼近

在一般的Banach空间中讨论了Φ—强拟增生算子方程的解和Φ—半压缩型算子不动点的迭代逼近问题，算子无Lipschitz假设或有界性要求，证明简捷，得到的结果统一、改进和推广了文[1—10]中的相应结果。     

Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces

The iteration scheme for families of nonexpansive mappings, essentially due to Halpern [Bull. Amer. Math. Soc. 73 (1967) 957-961], is established in a Banach space. The main theorem extends a recent result of O'Hara et al. [Nonlinear Anal. 54 (2003) 1417-1426] to a Banach space setting. For the same iteration scheme, with finitely many mappings, a complementary result to a result of Jung and Kim [Bull. Korean Math. Soc. 34 (1997) 93-102] (also Bauschke [J. Math. Anal. Appl. 202 (1996) 150-159]) is obtained by imposing other condition on the sequence of parameters. Our results also improve results in [C. R. Acad. Sci. Sér A-B Paris 284 (1977) 1357-1359; J. Math. Anal. Appl. 211 (1997) 71-83; Arch. Math. 59 (1992) 486-491] in framework of a Hilbert space.     

Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces

Let X be a real Banach space with a normalized duality mapping uniformly norm-to-weak^? continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping JΦ with gauge ?. Let f be an α-contraction and {Tn} a sequence of nonexpansive mappings, we study the strong convergence of explicit iterative schemes

(1)     

Iterative approximation to common fixed points of a countable family of nonexpansive mappings

We proved several strong convergence results by using the conception of a uniformly asymptotically regular sequence {T _n } of nonexpansive mappings in a reflexive Banach space which admits a weakly continuous duality mapping J _?(l ^p (1?p?t)?=?t ^p?1. The results presented develop and complement the corresponding ones by Song, Y. and Chen, R., 2007 [Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces. Nonlinear Analysis, 66, 591–603], Song, Y., Chen, R. and Zhou, H., 2007 [Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces. Nonlinear Analysis, 66, 1016–1024] and O'Hara, J.G., Pillay, P. and Xu, H.K., 2006 [Iterative approaches to convex feasibility problem in Banach Space. Nonlinear Analysis, 64, 2022–2042], O'Hara, J.G., Pillay, P. and Xu, H.K., 2003 [Iterative approaches to fineding nearest common fixed point of nonexpansive mappings in Hilbert spaces. Nonlinear Analysis, 54, 1417–1426] and Jung, J.S., 2005 [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications, 302, 509–520] and many other existing literatures.     

Iterative processes for common fixed points of two different families of mappings with applications

In this paper, an implicit iterative process is investigated for common fixed points of two different families of nonlinear mappings. Theorems of strong and weak convergence are established in real Hilbert spaces. As an application of the iterative process, inclusion problems are considered.     

Banach空间中可数非扩张映像族公共不动点迭代算法

提出一种新的迭代算法用于求解实一致光滑Banach空间上可数非扩张映像族的公共不动点.在一定条件下证明了迭代算法产生的序列强收敛到一个公共不动点,并且此不动点也是一个变分不等式的解.此结果改进和推广了已有的相关结果.     

Iterative approximation for a zero of accretive operator and fixed points problems in Banach space

In this paper we introduced an iteration scheme for viscosity approximation for a zero of accretive operator and fixed points problems in a reflexive Banach space with weakly continuous duality mapping. A new iterative sequence is introduced and strong convergence of the algorithm x_n is proved. The results improve and extend the results of Hu and Liu [L. Hu and L. Liu, A new iterative algorithm for common solutions of a finite family of accretive operators, Nonlinear Anal. 70 (2009) 2344-2351] and some others.     

Iterative methods for solving equilibrium problems, variational inequalities and fixed points of nonexpansive semigroups

In this work, strong convergence theorems by the viscosity approximation method associated with Meir–Keeler contractions are established for solving fixed point problems of a nonexpansive semigroup, a system of equilibrium problems and variational inequality problems in a real Hilbert space. Further, applications related to commutative semigroup are obtained.