Sunny Nonexpansive Retraction Research Articles

Existence of sunny nonexpansive retractions and approximation of fixed points of a representation of nonexpansive mappings

This paper presents an implicit scheme for a representation of nonexpansive mappings on a closed convex subset of a smooth uniformly convex Banach space with respect to a left-regular sequence of means defined on a subset of l∞(S). The main results are to establish an existence theorem of a sunny nonexpansive retraction and to create an algorithm for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces.

Iterative manner involving sunny nonexpansive retractions for nonlinear operators from the perspective of convex programming as applicable to differential problems, image restoration and signal recovery

<abstract><p>In this paper, using sunny nonexpansive retractions which are different from the metric projection in Banach spaces, we develop the $ CR $-iteration algorithm in view of two quasi-nonexpansive nonself mappings and also give the convergence analysis for the proposed method in the setting of uniformly convex Banach spaces. Furthermore, our results can be applied for the purpose of finding common zeros of accretive operators, convexly constrained least square problems and convex minimization problems. Regarding application, some numerical experiments involving real-world problems are provided, with focus on differential problems, image restoration problems and signal recovery problems.</p></abstract>

New iterative methods for nonlinear operators as concerns convex programming applicable in differential problems, image deblurring, and signal recovering problems

In this paper, using sunny nonexpansive retractions, which are different from the metric projection in Banach spaces, a new type of study regarding the iterative methods in view of two quasi‐nonexpansive nonself mappings is presented. We also give the convergence analysis for the proposed method in the background of uniformly convex Banach spaces. Moreover, we apply our results to find solutions of common zeros of accretive operators, convexly constrained least square problems, and convex minimization problems. Furthermore, we also discuss novel applications of these methods to differential problems, image deblurring, and signal recovering problems.

The finitary content of sunny nonexpansive retractions

We use techniques of proof mining to extract a uniform rate of metastability (in the sense of Tao) for the strong convergence of approximants to fixed points of uniformly continuous pseudocontractive mappings in Banach spaces which are uniformly convex and uniformly smooth, i.e. a slightly restricted form of the classical result of Reich. This is made possible by the existence of a modulus of uniqueness specific to uniformly convex Banach spaces and by the arithmetization of the use of the limit superior. The metastable convergence can thus be proved in a system which has the same provably total functions as first-order arithmetic and therefore one may interpret the resulting proof in Gödel’s system [Formula: see text] of higher-type functionals. The witness so obtained is then majorized (in the sense of Howard) in order to produce the final bound, which is shown to be definable in the subsystem [Formula: see text]. This piece of information is further used to obtain rates of metastability to results which were previously only analyzed from the point of view of proof mining in the context of Hilbert spaces, i.e. the convergence of the iterative schemas of Halpern and Bruck.

On the quantitative asymptotic behavior of strongly nonexpansive mappings in banach and geodesic spaces

We give explicit rates of asymptotic regularity for iterations of strongly nonexpansive mappings T in general Banach spaces as well as rates of metastability (in the sense of Tao) in the context of uniformly convex Banach spaces when T is odd. This, in particular, applies to linear norm-one projections as well as to sunny nonexpansive retractions. The asymptotic regularity results even hold for strongly quasi-nonexpansive mappings (in the sense of Bruck), the addition of error terms and very general metric settings. In particular, we get the first quantitative results on iterations (with errors) of compositions of metric projections in CAT(ĸ)-spaces (ĸ > 0). Under an additional compactness assumption we obtain, moreover, a rate of metastability for the strong convergence of such iterations.

A viscosity approximation method for weakly relatively nonexpansive mappings by the sunny nonexpansive retractions in Banach spaces

In this paper, we introduce a new viscosity approximation method by using the shrinking projection algorithm to approximate a common fixed point of a countable family of nonlinear mappings in a Banach space. Under quite mild assumptions, we establish the strong convergence of the sequence generated by the proposed algorithm and provide an affirmative answer to an open problem posed by Maingé (Comput. Math. Appl. 59:74-79, 2010) for quasi-nonexpansive mappings. In contrast with related processes, our method does not require any demiclosedness principle condition imposed on the involved operators belonging to the wide class of quasi-nonexpansive operators. As an application, we also introduce an iterative algorithm for finding a common element of the set of common fixed points of an infinite family of quasi-nonexpansive mappings and the set of solutions of a mixed equilibrium problem in a real Banach space. We prove a strong convergence theorem by using the proposed algorithm under some suitable conditions. Our results improve and generalize many known results in the current literature.

Fixed point solutions for variational inequalities in image restoration over q-uniformly smooth Banach spaces

In this paper, we introduce new implicit and explicit iterative methods for finding a common fixed point set of an infinite family of strict pseudo-contractions by the sunny nonexpansive retractions in a real q-uniformly and uniformly convex Banach space which admits a weakly sequentially continuous generalized duality mapping. Then we prove the strong convergence under mild conditions of the purposed iterative scheme to a common fixed point of an infinite family of strict pseudo-contractions which is a solution of some variational inequalities. Furthermore, we apply our results to study some strong convergence theorems in and spaces with . Our results mainly improve and extend the results announced by Ceng et al. (Comput. Math. Appl. 61:2447-2455, 2011) and many authors from Hilbert spaces to Banach spaces. Finally, we give some numerical examples for support our main theorem in the end of the paper. MSC:47H09, 47H10, 47H17, 47J25, 49J40.

Nonlinear Ergodic Theorem for Positively Homogeneous Nonexpansive Mappings in Banach Spaces

Recently, two retractions (projections), which are different from the metric projection and the sunny nonexpansive retraction in a Banach space, were found. In this article, using nonlinear analytic methods and new retractions, we prove a nonlinear ergodic theorem for positively homogeneous and nonexpansive mappings in a uniformly convex Banach space. The limit points are characterized by using new retractions.

CONVERGENCE OF AN ITERATIVE ALGORITHM FOR SYSTEMS OF GENERALIZED VARIATIONAL INEQUALITIES

In this paper, we introduce and consider a new system of generalized variational inequalities involving five different operators. Using the sunny nonexpansive retraction technique we suggest and analyze some new explicit iterative methods for this system of variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.

A two-step iterative process for common fixed points in Banach spaces

Suppose that K is a nonempty closed convex subset of a uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T1,T2:K→E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with two sequences {kn(i)}⊂[1,∞) satisfying ∑n=1∞(kn(i)−1)<∞(i=1,2) and F≔F(T1)∩F(T2)={x∈K:T1x=T2x=x}≠0̸, respectively. For any given x1∈K, suppose that {xn} is a sequence generated iteratively by {xn+1=(1−αn)(PT1)nxn+αn(PT2)nyn,yn=(1−βn)xn+βn(PT1)nxn,n∈N, where {αn} and {βn} are sequences in [a,1−a] for some a∈(0,1). Strong convergence of {xn} to a common fixed point of T1 and T2 is obtained by assuming the complete continuity of only one of the two mappings and then under the condition (A′) which is weaker than compactness of the domain. Moreover, weak convergence is obtained under Opial’s condition. Our results generalize a number of corresponding results in the contemporary literature. Another interesting fact is that our results obtained even for only one mapping are new.

Modified Noor′s Extragradient Method for Solving Generalized Variational Inequalities in Banach Spaces

Motivated and inspired by Korpelevich′s and Noor′s extragradient methods, we suggest an extragradient method by using the sunny nonexpansive retraction which has strong convergence for solving the generalized variational inequalities in Banach spaces.

New Iterative Manner Involving Sunny Nonexpansive Retractions for Pseudocontractive Mappings

Iterative methods for pseudocontractions have been studied by many authors in the literature. In the present paper, we firstly propose a new iterative method involving sunny nonexpansive retractions for pseudocontractions in Banach spaces. Consequently, we show that the suggested algorithm converges strongly to a fixed point of the pseudocontractive mapping which also solves some variational inequality.

An Alternative Regularization Method forEquilibrium Problems and Fixed Point ofNonexpansive Mappings

We introduce a new regularization iterative algorithm for equilibrium and fixed point problems of nonexpansive mapping. Then, we prove a strong convergence theorem for nonexpansive mappings to solve a unique solution of the variational inequality and the unique sunny nonexpansive retraction. Our results extend beyond the results of S. Takahashi and W. Takahashi (2007), and many others.

Two-step projection methods for a system of variational inequality problems in Banach spaces

Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E and let ? C be a sunny nonexpansive retraction from E onto C. Let the mappings $${T, S: C \to E}$$ be ? 1-strongly accretive, μ 1-Lipschitz continuous and ? 2-strongly accretive, μ 2-Lipschitz continuous, respectively. For arbitrarily chosen initial point $${x^0 \in C}$$ , compute the sequences {x k } and {y k } such that $${\begin{array}{ll} \quad y^k = \Pi_C[x^k-\eta S(x^k)],\\ x^{k+1} = (1-\alpha^k)x^k+\alpha^k\Pi_C[y^k-\rho T(y^k)],\quad k\geq 0, \end{array}}$$ where {? k } is a sequence in [0,1] and ?, ? are two positive constants. Under some mild conditions, we prove that the sequences {x k } and {y k } converge to x* and y*, respectively, where (x*, y*) is a solution of the following system of variational inequality problems in Banach spaces: $${\left\{\begin{array}{l}\langle \rho T(y^*)+x^*-y^*,j(x-x^*)\rangle\geq 0, \quad\forall x \in C,\\\langle \eta S(x^*)+y^*-x^*,j(x-y^*)\rangle\geq 0,\quad\forall x \in C.\end{array}\right.}$$ Our results extend the main results in Verma (Appl Math Lett 18:1286---1292, 2005) from Hilbert spaces to Banach spaces. We also obtain some corollaries which include some results in the literature as special cases.

Iterative Approximation of Common Fixed Points of Two Nonself Asymptotically Nonexpansive Mappings

Suppose that K is nonempty closed convex subset of a uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction and F : = F(T1)∩F(T2) = {x ∈ K : T1x = T2x = x} ≠ ∅. Let T1, T2 : K → E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with two sequences satisfying (i = 1,2), respectively. For any given x1 ∈ K, suppose that {xn} is a sequence generated iteratively by , , n ∈ ℕ, where {αn} and {βn} are sequences in [a, 1 − a] for some a ∈ (0,1). Under some suitable conditions, the strong and weak convergence theorems of {xn} to a common fixed point of T1 and T2 are obtained.

Weak and strong convergence theorems of three step iteration process with errors for nonself-asymptotically nonexpansive mappings

Let E be a real uniformly convex and smooth Banach space with P as a sunny nonexpansive retraction, K be a nonempty closed convex subset of E . Let { T i } i = 1 3 : K → E be three of weakly inward and asymptotically nonexpansive mappings with respect to P . It is proved that three step iteration process with errors converges weakly and strongly to a common fixed point of { T i } i = 1 3 under certain conditions. It presents some new results in this paper.

Path Convergence and Approximation of Common Zeroes of a Finite Family of m‐Accretive Mappings in Banach Spaces

Let E be a real Banach space which is uniformly smooth and uniformly convex. Let K be a nonempty, closed, and convex sunny nonexpansive retract of E, where Q is the sunny nonexpansive retraction. If E admits weakly sequentially continuous duality mapping j, path convergence is proved for a nonexpansive mapping T : K → K. As an application, we prove strong convergence theorem for common zeroes of a finite family of m‐accretive mappings of K to E. As a consequence, an iterative scheme is constructed to converge to a common fixed point (assuming existence) of a finite family of pseudocontractive mappings from K to E under certain mild conditions.

VISCOSITY APPROXIMATIONS FOR NONEXPANSIVE NONSELF-MAPPINGS IN BANACH SPACES

Abstract. Strong convergence theorem of the explicit viscosity itera-tive scheme involving the sunny nonexpansive retraction for nonexpansivenonself-mappings is established in a reexive and strictly convex Banachspaces having a weakly sequentially continuous duality mapping. Themain result improves the corresponding result of [19] to the more generalclass of mappings together with certain di erent control conditions. 1. IntroductionLet Ebe a real Banach space and Cbe a nonempty closed convex subsetof E. Recall that a mapping f: C!Cis a contraction on Cif there exists aconstant k2(0;1) such that kf(x) f(y)kkkx yk; x; y2C:We use C to denote the collection of mappings fverifying the above inequality. That is, C = ff : C!Cjf is a contraction with constant kg:Let T : C!Cbea nonexpansive mapping (recall that a mapping T: C!Cis nonexpansive ifkTx Tykkx yk; x; y2C) and F(T) denote the set of xed points ofT; that is, F(T) = fx2C: x= Txg:In 1967, Halpern [5] rstly introduced the following explicit iterative scheme(1.1) in Hilbert space,x

COMMON FIXED POINTS OF A PAIR OF NON-EXPANSIVE MAPPINGS WITH APPLICATIONS TO CONVEX FEASIBILITY PROBLEMS

AbstractLet C be a non-empty closed convex subset of a reflexive and strictly convex Banach space E which also has a weakly continuous duality map Jφ(x) with the gauge φ. Let S and T be non-expansive mappings from C into itself such that F = F(S) ∩ F(T) ≠ ∅. Let {αn} and {βn} be sequences in (0, 1). Let {xn} be a sequence defined by where u ∈ C is a given point. Assume that the following restrictions imposed on the control sequences are satisfied: Then the sequence {xn} converges strongly to x* ∈ F, where x* = Q(u) and Q: C → F is the unique sunny non-expansive retraction from C onto F.

Convergence theorems on viscosity approximation methods for a finite family of nonexpansive non-self-mappings

Let E be a Banach space, C a nonempty closed convex subset of E , f : C → C a contraction, and T i : C → E nonexpansive mappings with nonempty F ≔ ⋂ i = 1 N F i x ( T i ) , where N ≥ 1 is an integer and F i x ( T i ) is the set of fixed points of T i . Assume that C is a sunny nonexpansive retract of E with Q as the sunny nonexpansive retraction. It is proved that an iterative scheme x n + 1 = λ n + 1 f ( x n ) + ( 1 − λ n + 1 ) Q T n + 1 x n ( n ≥ 0 ) converges strongly to a solution in F of a certain variational inequality provided E is strictly convex, reflexive and has a weakly sequentially continuous duality mapping and provided the sequence { λ n } satisfies certain conditions and the sequence { x n } satisfies weak asymptotic regularity. An application of the main result is also given.