Ishikawa and Mann iteration methods for strongly accretive operators

LetE be a smooth Banach space. Suppose T:E →E is a strongly accretive map. It is proved that each of the two well known fixed point iteration methods (the Mann and Ishikawa iteration methods), under suitable conditions, converges strongly to a solution of the equationTx =f.     

Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings

In the present paper, we prove that the modified implicit iteration sequence for a finite family of asymptotically quasi-nonexpansive mappings converges strongly to a common fixed point of the family in a uniformly convex Banach space, requiring one member T in the family to be semi-compact. Our results extend and improve some recent results.     

Approximating Solutions of Maximal Monotone Operators in Hilbert Spaces

Let H be a real Hilbert space and let T: H→2^H be a maximal monotone operator. In this paper, we first introduce two algorithms of approximating solutions of maximal monotone operators. One of them is to generate a strongly convergent sequence with limit vT⁻¹0. The other is to discuss the weak convergence of the proximal point algorithm. Next, using these results, we consider the problem of finding a minimizer of a convex function. Our methods are motivated by Halpern's iteration and Mann's iteration.     

Ishikawa iterative process with errors for nonlinear equations of generalized monotone type in Banach spaces

The concept of the operators of generalized monotone type is introduced and iterative approximation methods for a fixed point of such operators by the Ishikawa and Mann iteration schemes {xn} and {yn} with errors is studied. Let X be a real Banach space and T : D ? X → 2^D be a multi‐valued operator of generalized monotone type with fixed points. A new general lemma on the convergence of real sequences is proved and used to show that {xn} converges strongly to a unique fixed point of T in D. This result is applied to the iterative approximation method for solutions of nonlinear equations with generalized strongly accretive operators. Our results generalize many of know results. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Viscosity approximation methods for nonexpansive multimaps in Banach spaces

We prove strong convergence of the viscosity approximation process for nonexpansive nonself multimaps. Furthermore, an explicit iteration process which converges strongly to a fixed point of multimap T is constructed. It is worth mentioning that, unlike other authors, we do not impose the condition ＂Tz = {z}＂ on the map T.     

(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     

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     

Fixed point iteration processes for asymptotic pointwise nonexpansive mappings in Banach spaces

Let X be a uniformly convex Banach space with the Opial property. Let T:C→C be an asymptotic pointwise nonexpansive mapping, where C is bounded, closed and convex subset of X. In this paper, we prove that the generalized Mann and Ishikawa processes converge weakly to a fixed point of T. In addition, we prove that for compact asymptotic pointwise nonexpansive mappings acting in uniformly convex Banach spaces, both processes converge strongly to a fixed point.     

Approximation Methods for a Common Fixed Point of a Finite Family of Nonexpansive Mappings

Let K be a nonempty closed and convex subset of a real Banach space E. Let T: K → E be a continuous pseudocontractive mapping and f:K → E a contraction, both satisfying weakly inward condition. Then for t ? (0, 1), there exists a sequence {y _t } ? K satisfying the following condition: y _t  = (1 ? t)f(y _t ) + tT(y _t ). Suppose further that {y _t } is bounded or F(T) ≠  and E is a reflexive Banach space having weakly continuous duality mapping J _? for some gauge ?. Then it is proved that {y _t } converges strongly to a fixed point of T, which is also a solution of certain variational inequality. Moreover, an explicit iteration process that converges strongly to a common fixed point of a finite family of nonexpansive mappings and hence to a solution of a certain variational inequality is constructed.     

Convergence analysis of projected fixed‐point iteration on a low‐rank matrix manifold

In this paper, we analyze the convergence of a projected fixed‐point iteration on a Riemannian manifold of matrices with fixed rank. As a retraction method, we use the projector splitting scheme. We prove that the convergence rate of the projector splitting scheme is bounded by the convergence rate of standard fixed‐point iteration without rank constraints multiplied by the function of initial approximation. We also provide counterexample to the case when conditions of the theorem do not hold. Finally, we support our theoretical results with numerical experiments.     

Approximation of Nearest Common Fixed Point of Nonexpansive Mappings in Hilbert Spaces

The purpose of this paper is to study the convergence problem of the iteration scheme xn＋l = λn＋1y ＋ （1 - λn＋1）Tn＋1xn for a family of infinitely many nonexpansive mappings T1, T2,... in a Hilbert space. It is proved that under suitable conditions this iteration scheme converges strongly to the nearest common fixed point of this family of nonexpansive mappings. The results presented in this paper extend and improve some recent results.     

没有Lipschitz假设的ψ-半压缩映象的不动点与ψ-强拟增生算子方程解的逼近

设X为实一致光滑Banach空间,T:D(T)∈X→X为ψ-半压缩映象且在它的不动点q处是局部有界的.本文证明了Mann迭代与Ishikawa迭代过程强收敛于T的唯一不动点g.几个相关的结果处理ψ-强拟增生算子方程的解的迭代构造.本文所得到的结果扩展并推广了Xu和Roach,Zhou和Jia等人的相应结果.     

Fixed point Ishikawa iterations

We first establish sufficient conditions on the coefficients of the Ishikawa iteration process to guarantee that, if the Ishikawa iterates of a continuous selfmap T, of the unit interval, converge, then they converge to a fixed point of T. Second we obtain sufficient conditions to guarantee that the iterations converge.     

Banach空间中Reich-Takahashi迭代法的强收敛定理

设E是具有一致正规结构的实Banach空间,其范数是一致Gateaux可微的;设D是E的非空有界闭凸子集,T:D→D是渐近非扩张映象.本文证明了,在一些适当的条件下,由修正的Reich-Takahashi迭代法(1.2)式所定义的序列{xn}强收敛到渐近非扩张映象的不动点,其中x0是D中一任给点,{αn},{β}是区间[0,1]中满足某些限制的实数列.     

Stability of Mann’s iterates under metric regularity

We consider a Mann-like iteration for solving the inclusion xT(x) where is a set-valued mapping, defined from a Banach space X into itself, which is metrically regular near a point in its graph. We study the behavior of the iterates generated by our method and prove that they inherit the regularity properties of the mapping T. First we consider the case when the mapping T is metrically regular, then the case when it is strongly metrically regular. Finally, we present an inexact version of our method and we study its convergence when the mapping T is strongly metrically subregular.     

Random products of contractions in metric and Banach Spaces

Suppose (X, d) is a metric space and {T₀,…, T_N} is a family of quasinonexpansive self-mappings on X. We give conditions sufficient to guarantee that every possible iteration of mappings drawn from {T₀,…, T_N} converges. As a consequence, if C₀,…, C_N are closed convex subsets of a Hilbert space with nonempty intersection, one of which is compact, and the proximity mappings are iterated in any order (provided only that each is used infinitely often), then the resulting sequence converges strongly to a point of the common intersection.     

Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings

In this paper, a necessary and sufficient conditions for the strong convergence to a common fixed point of a finite family of continuous pseudocontractive mappings are proved in an arbitrary real Banach space using an implicit iteration scheme recently introduced by Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Fuct. Anal. Optim. 22 (2001) 767-773] in condition αn∈(0,1], and also strong and weak convergence theorem of a finite family of strictly pseudocontractive mappings of Browder-Petryshyn type is obtained. The results presented extend and improve the corresponding results of M.O. Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73-81].     

The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps

The convergence of modified Mann iteration is equivalent to the convergence of modified Ishikawa iterations, when T is an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive map.     

Termination criteria for inexact fixed‐point schemes

We analyze inexact fixed‐point iterations where the generating function contains an inexact solve of an equation system to answer the question of how tolerances for the inner solves influence the iteration error of the outer fixed‐point iteration. Important applications are the Picard iteration and partitioned fluid‐structure interaction. For the analysis, the iteration is modeled as a perturbed fixed‐point iteration, and existing analysis is extended to the nested case x = F ( S ( x )). We prove that if the iteration converges, it converges to the exact solution irrespective of the tolerance in the inner systems, provided that a nonstandard relative termination criterion is employed, whereas standard relative and absolute criteria do not have this property. Numerical results demonstrate the effectiveness of the approach with the nonstandard termination criterion. Copyright © 2015 John Wiley & Sons, Ltd.