Laurentiu Leustean, Adriana Nicolae
In this note we point out that results on the asymptotic behaviour of an alternative iterative method are corollaries of corresponding results on the well-known Halpern iteration.
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A class of extended Ishikowa Iterative processes is proposed and studied, which involves many kinds of Mann and Ishikawa iterative processes. The main conclusion of the present work extends and generalizes some recent results of this research line.
Alternative iterative methods for a nonexpansive mapping in a Banach space are proposed and proved to be convergent to a common solution to a fixed point problem and a variational inequality. We give rates of asymptotic regularity for such iterations using proof-theoretic techniques. Some applications of the convergence results are presented.
In this paper by using $W_{n}$-mapping, we introduce a composite iterative method for finding a common fixed point for infinite family of nonexpansive mappings and a solution of a certain variational inequality. Furthermore, the strong convergence of the proposed iterative method is established. Finally, some simulation examples are presented. Our results improve and extend the previous results.
We introduce and investigate an iterative scheme for approximating common fixed point of a family of Bregman relatively-nonexpansive mappings in real reflexive Banach spaces. We prove strong convergence theorem of the sequence generated by our scheme under some appropriate conditions. Furthermore, our scheme and results unify some known results obtained in this direction.
Let $H$ be an infinite dimensional, reflexive, separable Hilbert space and $NA(H)$ the class of all norm-attainble operators on $H.$ In this note, we study an implicit scheme for a canonical representation of nonexpansive contractions in norm-attainable classes.
It is our aim in this article to correct the wrong impression created in the paper of A. Rafiq titled: "Convergence theorems for uniformly L-Lipschitzian Asymptotically Nonexpansive mappings" which appeared in Acta Universitatis Apulensis no. 20/2009, 183-192.
In this study, we introduce a new iterative processes to approximate common fixed points of an infinite family of quasi-nonexpansive mappings and obtain a strongly convergent iterative sequence to the common fixed points of these mappings in a uniformly convex Banach space. Also we prove that this process to approximate zeros of an infinite family of accretive operators and we obtain a strong convergence result for these operators. Our results improve and generalize many known results in the current literature.
In this paper, we introduce a new iterative method to find a common solution of a generalized mixed equilibrium problem, a variational inequality problem and a hierarchical fixed point problem for a demicontinuous nearly nonexpansive mapping. We prove that the proposed method converges strongly to a common solution of above problems under the suitable conditions. It is also noted that the main theorem is proved without usual demiclosedness condition. Also, under the appropriate assumptions on the control sequences and operators, our iterative method can be reduced to recent methods. So, the re…
In this paper, we introduce new implicit and explicit iterative schemes which converge strongly to a unique solution of variational inequality problems for strongly accretive operators over a common fixed point set of finite family of nonexpansive mappings in $q$-uniformly smooth real Banach spaces. As an application, we introduce an iteration process which converges strongly to a solution of the variational inequality which is a common fixed point of finite family of strictly pseudocontractive mappings. Our theorems extend, generalize, improve and unify the corresponding results of Xu \cite{2…
In this paper, we first introduce an iterative process in modular function spaces and then extend the idea of a λ-firmly nonexpansive mapping from Banach spaces to modular function spaces. We call such mappings as (λ,ρ)-firmly nonexpansive mappings. We incorporate the two ideas to approximate fixed points of (λ,ρ)-firmly nonexpansive mappings using the above mentioned iterative process in modular function spaces. We give an example to validate our results.
By using the Ishikawa iterative algorithm, we approximate the fixed points and the best proximity points of a relatively non expansive mapping. Also, we use the von Neumann sequence to prove the convergence result in a Hilbert space setting. A comparison table is prepared using a numerical example which shows that the Ishikawa iterative algorithm is faster than some known iterative algorithms such as Picard and Mann iteration.
The concept of asymptotically nonexpansive mappings is an important generalization of the class of nonexpansive mappings. Implicit midpoint procedures are extremely fundamental for solving equations involving nonlinear operators. This paper studies the convergence analysis of the class of asymptotically nonexpansive mappings by the implicit midpoint iterative procedures. The necessary conditions for the convergence of the class of asymptotically nonexpansive mappings are established, by using a well-known iterative algorithm which plays important roles in the computation of fixed points of non…
In this paper, we study a new iterative method for finding the fixed point of a weak Bregman relatively nonexpansive mapping and the set of solutions of generalized mixed equilibrium problems in Banach spaces.
In this work, a new concept of nonself total asymptotically nonexpansive mapping is introduced and an iterative process is considered for two nonself totally asymptotically nonexpansive mappings. Weak and strong convergence theorems for computing common fixed points of two nonself total asymptotically nonexpansive mappings are established in the framework of Banach spaces. Finally, we give an example which show that our theorems are applicable.
In this paper, we study a new iterative method for a common fixed point of a finite family of Bregman strongly nonexpansive mappings in the frame work of reflexive real Banach spaces. Moreover, we prove the strong convergence theorem for finding common fixed points with the solutions of a mixed equilibrium problem.
For a nonempty convex subset $C$ of a Hadamard space $X$, it is proved that $u=P_Cx$ if and only if $\langle\overrightarrow{xu},\overrightarrow{uy}\rangle \geqslant0$ for all $y\in C$. As an application of this characterization, we prove strong convergence of two iterative algorithms with perturbations for nonexpansive mappings.
We introduce an iterative process for finding common fixed point of finite family of quasi-Bregman nonexpansive mappings which is a unique solution of some equilibrium problem.
The demiclosedness principle is one of the key tools in nonlinear analysis and fixed point theory. In this note, this principle is extended and made more flexible by two mutually orthogonal affine subspaces. Versions for finitely many (firmly) nonexpansive operators are presented. As an application, a simple proof of the weak convergence of the Douglas-Rachford splitting algorithm is provided.
In this paper, we introduce a new class of Bregman generalized $α$-nonexpansive mappings in terms of Bregman distances, and investigate the Ishikawa and Noor iterations for these mappings. We establish weak and strong convergence theorems of Ishikawa and Noor iterative schemes for Bregman generalized $α$-nonexpansive mappings in Banach spaces. Furthermore, we propose an example of our generated mapping and some numerical examples which support our main theorem. Our results are new and improve the recent ones in the literature.
In this paper, we study $Δ$- convergence of iterations for a sequence of strongly quasi-nonexpansive mappings as well as the strong convergence of the Halpern type regularization of them in Hadamard spaces. Then, we give some their applications in iterative methods, convex and pseudo-convex minimization(proximal point algorithm), fixed point theory and equilibrium problems. The results extend several new results in the literature and some of them seem new even in Hilbert spaces.
Strong convergence of a new iterative process based on the Shrinking projection method to a common element of the set of common fixed points of an infinite family of relatively quasi-nonexpansive multivalued mappings and the solution set of an equilibrium problem in a Banach space is established. Our results improved and extend the corresponding results announced by many others.
Remarks on b-enriched nonexpansive mappings
该研究针对文献中提出的富集非扩张映射概念展开分析，指出其与经典非扩张映射概念重合，需重新审视，为度量不动点理论的相关文献研究提供参考并指明后续澄清方向。
Weak and strong convergence of an implicit iterative process with errors for a finite family of asymptotically quasi $I-$nonexpansive mappings in Banach space
本文证明了Banach空间中有限族渐近拟$I_j-$非扩张映射和渐近拟非扩张映射带误差隐式迭代过程弱收敛和强收敛到公共不动点。
In this paper, we introduce a new problem called the split feasibility and fixed point equality problems (SFFPEP) and propose a new iterative algorithm for solving the problem (SFFPEP) for the class of quasi-nonexpansive mappings in Hilbert spaces. Furthermore, we study the convergence of the proposed algorithm. At the end, we give numerical example that illustrate our theoretical result. The SFFPEP is a generalization of the split feasibility problem (SFP), split feasibility and fixed point problems (SFFPP) and split equality fixed point problem (SEFPP).
This paper provides a fixed point theorem and iterative construction of a common fixed point for a general class of nonlinear mappings in the setup of uniformly convex hyperbolic spaces. We translate a multi-step iteration, essentially due to Chidume and Ofoedu [4] in such setting for the approximation of common fixed points of a finite family of total asymptotically nonexpansive mappings. As a consequence, we establish strong and del-convergence results which extend and generalize various corresponding results announced in the current literature.
On the Rate of Asymptotic Regularity of Iterative Methods for Nonexpansive Mappings in CAT(0) Spaces and Hyperbolic Optimization
本文将Banach和Hilbert空间中的渐近正则性证明技术扩展到完备CAT(0)空间，并引入双曲优化的Halpern型优化器作为欧几里得HalpernSGD的非线性对应。
This paper deals with a modified iterative projection method for approximating a solution of the hierarchical fixed point problem for a sequene of nearly nonexpansive mappings with respect to a nonexpansive mapping. It is shown that under certain approximate assumptions on the operators and parameters, the modified iterative sequence converges strongly to a common element of the set of the common fixed points of nearly nonexpansive mappings.Also, this point solves some variational inequality. As a special case, this projection method can be used to find the minimum norm solution of the given v…
In this paper, first we introduce a new mapping for finding a common fixed point of an infinite family of nonexpansive mappings then we consider iterative method for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for $α$-inverse-strongly monotone mapping in a Hilbert space. We show that under suitable conditions, the sequence converges strongly to a common element of the above three sets. Our results presented in this paper improve and exten…
We introduce an abstract algorithm that aims to find the Bregman projection onto a closed convex set. As an application, the asymptotic behaviour of an iterative method for finding a fixed point of a quasi Bregman nonexpansive mapping with the fixed-point closedness property is analyzed. We also show that our result is applicable to Bregman subgradient projectors.
On unification of the strong convergence theorems for a finite family of total asymptotically nonexpansive mappings in Banach spaces
通过引入新的显式迭代格式统一有限族全渐近I-非扩张映射公共不动点的逼近方法，构造了非渐近非扩张的全渐近非扩张映射实例，并在一致凸Banach空间中证明了强收敛定理。
In this paper, we prove the existence of fixed points of mappings satisfying the condition (Da), a kind of generalized nonexpansive mappings, on a weakly compact convex subset in a Banach space satisfying Opial's condition. And we use Sahu([6]) and Thakur([10])'s iterative scheme to establish several convergence theorems in uniformly convex Banach spaces and give an example to show that this scheme converges faster than the scheme in [1]
This note discusses some aspects of the asymptotic behaviour of nonexpansive maps. Using metric functionals, we make a connection to the invariant subspace problem and prove a new result for nonexpansive maps of $\ell^{1}$. We also point out some inaccurate assertions appearing in the literature on this topic.
We provide a direct proof of a result regarding the asymptotic behavior of alternating nearest point projections onto two closed and convex sets in a Hilbert space. Our arguments are based on nonexpansive mapping theory.
In this paper, we generalize the existence result in [14] and prove convergence theorems of the iterative scheme in [12, 16] for monotone generalized alpa-nonexpansive mappings in uniformly convex partially ordered hyperbolic metric spaces. And we also give a numerical example to show that this scheme converges faster than the scheme in [14] and apply the result to the integral equation.
A Solution of variational inequality problem for a finite family of nonexpansive mappings in Hilbert spaces
本文针对Hilbert空间上有限族非扩张映射，提出显式迭代过程并证明其强收敛到公共不动点，且该不动点解决不动点集上的变分不等式。
In this paper, we introduce and study a new extragradient iterative process for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a variational inequality for an inverse strongly monotone mapping in a real Hilbert space. Also, we prove that under quite mild conditions the iterative sequence defined by our new extragradient method converges strongly to a solution of the fixed point problem for an infinite family of nonexpansive mappings and the classical variational inequality problem. In addition, utilizing this resul…
In this note, at first we prove that the existence of best proximity points for cyclic relatively nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic relatively nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that a main result of the paper "Proximal normal structure and relatively nonexpansive mappings", Studia Math., (171~(2005) 283--293) immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of it…
Because of Minty's classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator theory, and convex optimization. In this note, we show that if finitely many firmly nonexpansive mappings defined on a real Hilbert space are given and each of these mappings is asymptotically regular, which is equivalent to saying that they have or "almost have" fixed points, then the same is true for their composition. This significantly generalizes the resul…
The split common fixed point problems has found its applications in various branches of mathematics both pure and applied. It provides us a unified structure to study a large number of nonlinear mappings. Our interest here is to apply these mappings and propose some iterative methods for solving the split common fixed point problems and its variant forms, and we prove the convergence results of these algorithms. As a special case of the split common fixed problems, we consider the split common fixed point equality problems for the class of finite family of quasi-nonexpansive mappings. Furtherm…
Monotone operators, especially in the form of subdifferential operators, are of basic importance in optimization. It is well known since Minty, Rockafellar, and Bertsekas-Eckstein that in Hilbert space, monotone operators can be understood and analyzed from the alternative viewpoint of firmly nonexpansive mappings, which were found to be precisely the resolvents of monotone operators. For example, the proximal mappings in the sense of Moreau are precisely the resolvents of subdifferential operators. More general notions of "resolvent", "proximal mapping" and "firmly nonexpansive" have been stu…
Finding the solutions of nonlinear operator equations has been a subject of research for decades but has recently attracted much attention. This paper studies the convergence of a newly introduced viscosity implicit iterative algorithm to a fixed point of a nonexpansive mapping in Banach spaces. Our technique is indispensable in terms of explicitly clarifying the associated concepts and analysis. The scheme is effective for obtaining the solutions of various nonlinear operator equations as it involves the generalized contraction. The results are applied to obtain a fixed point of λ-strictly ps…
We introduce a new extragradient iterative process, motivated and inspired by [S. H. Khan, A Picard-Mann Hybrid Iterative Process, Fixed Point Theory and Applications, doi:10.1186/1687-1812-2013-69], for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality for an inverse strongly monotone mapping in a Hilbert space. Using this process, we prove a weak convergence theorem for the class of nonexpansive mappings in Hilbert spaces. Finally, as an application, we give some theorems by using resolvent operator and strictly…
Using the technique of enrichment of contractive type mappings by Krasnoselskij averaging, introduced in [Berinde, V., {\it Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces}, Carpathian J. Math. {\bf 35} (2019), no. 3, 277-288.], we introduce the class of enriched Chatterjea contractions and prove general fixed point theorems for such contractions in the setting of a Banach space. Examples to illustrate the richness of the new class of contractions and the relationship between enriched Banach contractions, enriched Kannan contractions an…
By iterative techniques,we present two fixed point theorems, whose modular formulations are relatively close to the Banach's fixed point theorem in the normed spaces.The first result concerns the fixed point of the strongly contraction mappings.The second result deals with the fixed point of the strict contraction mappings where the modular satisfies the delta-condition. For the nonexpansive mappings,where the modular satisfies the regular growth condition,we present a fixed point theorem of the Shauder's type,without boundedness condition on the domain of theses mappings.
We provide in a unified way quantitative forms of strong convergence results for numerous iterative procedures which satisfy a general type of Fejer monotonicity where the convergence uses the compactness of the underlying set. These quantitative versions are in the form of explicit rates of so-called metastability in the sense of T. Tao. Our approach covers examples ranging from the proximal point algorithm for maximal monotone operators to various fixed point iterations (x_n) for firmly nonexpansive, asymptotically nonexpansive, strictly pseudo-contractive and other types of mappings. Many o…
The purpose of this paper is to propose and analyze a multi-step iterative algorithm to solve a convex optimization problem and a fixed point problem posed on a Hadamard space. The convergence properties of the proposed algorithm are analyzed by employing suitable conditions on the control sequences of parameters and the structural properties of the under lying space. We aim to establish strong and del-convergence results of the proposed iterative algorithm and compute an optimal solution for a minimizer of proper convex lower semicontinuous function and a common fixed point of a finite family…
In this paper, we introduce an iterative process which converges strongly to a common element of sets of solutions of finite family of generalized equilibrium problems, sets of fixed points of finite family of continuous relatively nonexpansive mappings and sets of zeros of finite family of $γ$-inverse strongly monotone mappings in Banach spaces. Our theorems improve and generalize several results which are announced recently. Our iteration process, method of proof and corollaries are of independent interest
We introduce a large class of mappings, called enriched contractions, which includes, amongst many other contractive type mappings, the Picard-Banach contractions and some nonexpansive mappings. We show that any enriched contraction has a unique fixed point and that this fixed point can be approximated by means of an appropriate Kransnoselskij iterative scheme. Several important results in fixed point theory are shown to be corollaries or consequences of the main results in this paper. We also study the fixed points of local enriched contractions, asymptotic enriched contractions and Maia type…
A simpler and more efficient fixed point iterative scheme
本文提出了一种更简单高效的固定点迭代方案，通过理论分析和数值实验验证了其在收敛速度和计算效率上的优势。
This paper deals with a modifed iterative projection method for approximating a solution of hierarchical fixed point problems for nearly nonexpansive mappings. Some strong convergence theorems for the proposed method are presented under certain approximate assumptions of mappings and parameters. As a special case, this projection method solves some quadratic minimization problem. It should be noted that the proposed method can be regarded as a generalized version of Wang et.al. [15], Ceng et. al. [14], Sahu [4] and many other authors.
We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed poin…
The aim of this paper is to prove that, in an appropriate setting, every iterative sequence generated by the hybrid steepest descent method is convergent whenever so is every iterative sequence generated by the Halpern type iterative method.
For a bounded closed convex set $K$, in this note, we study the FPP for $α$-Hölder nonexpansive maps, i.e. mappings $T\colon K\to K$ for which $\|T x -Ty\| \leq\| x - y\|^α$ for all $x, y\in K$, $α\in (0,1)$. First, we note that only finite-dimensional spaces have the Hölder-FPP. Moreover, the unit ball $B_X$ of any infinite-dimensional space fails the FPP for Hölder maps with $\mathrm{d}(T, B_X)>0$, where $\mathrm{d}(T, K)$ denotes the minimal displacement of $T$. We further show that reflexivity and weak sequential continuity are sufficient conditions to capture fixed points of Hölder-Lipsch…
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