Pseudononspreading Mapping Research Articles

Weak and strong convergence theorems for a new class of enriched strictly pseudononspreading mappings in Hilbert spaces

Let Ω be a nonempty closed convex subset of a real Hilbert space H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak{H}$\\end{document}. Let ℑ be a nonspreading mapping from Ω into itself. Define two sequences {ψn}n=1∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\{\\psi _{{n}}\\}_{n=1}^{\\infty}$\\end{document} and {ϕn}n=1∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\{\\phi _{{n}}\\}_{n=1}^{\\infty}$\\end{document} as follows: {ψn+1=πnψn+(1−πn)ℑψn,ϕn=1n∑nt=1ψt,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} \ extstyle\\begin{cases} \\psi _{n+1}=\\pi _{n}\\psi _{{n}}+(1-\\pi _{n})\\Im \\psi _{{n}}, \\\\ \\phi _{{n}}=\\dfrac{1}{n}\\underset{t=1}{\\overset{n}{\\sum}}\\psi _{t}, \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} for n∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n\\in \\mathit{N}$\\end{document}, where 0≤πn≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$0\\leq \\pi _{n}\\leq 1$\\end{document}, and πn→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\pi _{n} \\rightarrow 0$\\end{document}. In 2010, Kurokawa and Takahashi established weak and strong convergence theorems of the sequences developed from the above Baillion-type iteration method (Nonlinear Anal. 73:1562–1568, 2010). In this paper, we prove weak and strong convergence theorems for a new class of (η,β)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\eta ,\\beta )$\\end{document}-enriched strictly pseudononspreading ((η,β)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\eta ,\\beta )$\\end{document}-ESPN) maps, more general than that studied by Kurokawa and W. Takahashi in the setup of real Hilbert spaces. Further, by means of a robust auxiliary map incorporated in our theorems, the strong convergence of the sequence generated by Halpern-type iterative algorithm is proved thereby resolving in the affirmative the open problem raised by Kurokawa and Takahashi in their concluding remark for the case in which the map ℑ is averaged. Some nontrivial examples are given, and the results obtained extend, improve, and generalize several well-known results in the current literature.

Fixed point results involving a finite family of enriched strictly pseudocontractive and pseudononspreading mappings

In this study, we introduce a method for finding common fixed points of a finite family of (ηi,ki)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\eta _{i}, k_{i})$\\end{document}-enriched strictly pseudocontractive (ESPC) maps and (ηi,βi)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\eta _{i}, \\beta _{i})$\\end{document}-enriched strictly pseudononspreading (ESPN) maps in the setting of real Hilbert spaces. Further, we prove the strong convergence theorem of the proposed method under mild conditions on the control parameters. Our main results are also applied in proving strong convergence theorems for ηi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\eta _{i}$\\end{document}-enriched nonexpansive, strongly inverse monotone, and strictly pseudononspreading maps. Some nontrivial examples are given, and the results obtained extend, improve, and generalize several well-known results in the current literature.

Multiple-set split feasibility problems for asymptotically pseudo-nonspreading mappings in Hilbert spaces

Multiple-set split feasibility problems for asymptotically pseudo-nonspreading mappings in Hilbert spaces

An iterative algorithm for minimization and fixed point problems of two families of pseudononspreading mappings in Hadamard spaces

An iterative algorithm for minimization and fixed point problems of two families of pseudononspreading mappings in Hadamard spaces

Split Feasibility Problem and Fixed Point Problem for Asymptotically Strictly Pseudo Nonspreading Mapping

The purpose of this paper is to introduce an iterative algorithm for finding a common element of the solution set of split feasibility problem and the fixed point set of a asymptotically strictly pseudo nonspreading mapping in the Hilbert Space.

Iterative algorithm for a family of generalized strictly pseudononspreading mappings in CAT(0) spaces

Iterative algorithm for a family of generalized strictly pseudononspreading mappings in CAT(0) spaces

An Inertial Generalized Viscosity Approximation Method for Solving Multiple-Sets Split Feasibility Problems and Common Fixed Point of Strictly Pseudo-Nonspreading Mappings

In this paper, we propose a generalized viscosity iterative algorithm which includes a sequence of contractions and a self adaptive step size for approximating a common solution of a multiple-set split feasibility problem and fixed point problem for countable families of k-strictly pseudononspeading mappings in the framework of real Hilbert spaces. The advantage of the step size introduced in our algorithm is that it does not require the computation of the Lipschitz constant of the gradient operator which is very difficult in practice. We also introduce an inertial process version of the generalize viscosity approximation method with self adaptive step size. We prove strong convergence results for the sequences generated by the algorithms for solving the aforementioned problems and present some numerical examples to show the efficiency and accuracy of our algorithm. The results presented in this paper extends and complements many recent results in the literature.

A viscosity-type algorithm for an infinitely countable family of (f,g)-generalized k-strictly pseudononspreading mappings in CAT(0) spaces

Abstract In this paper, we propose a viscosity-type algorithm to approximate a common solution of a monotone inclusion problem, a minimization problem and a fixed point problem for an infinitely countable family of ( f , g ) {(f,g)} -generalized k-strictly pseudononspreading mappings in a CAT ⁢ ( 0 ) {\mathrm{CAT}(0)} space. We obtain a strong convergence of the proposed algorithm to the aforementioned problems in a complete CAT ⁢ ( 0 ) {\mathrm{CAT}(0)} space. Furthermore, we give an application of our result to a nonlinear Volterra integral equation and a numerical example to support our main result. Our results complement and extend many recent results in literature.

Fixed points of a new class of pseudononspreading mappings

Abstract We extend the notion of k-strictly pseudononspreading mappings introduced in Nonlinear Analysis 74 (2011) 1814-1822 to the notion of the more general pseudononspreading mappings. It is shown with example that the class of pseudononspreading mappings is more general than the class of k-strictly pseudonon-spreading mappings. Furthermore, it is shown with explicit examples that the class of pseudononspreading mappings and the important class of pseudocontractive mappings are independent. Some fundamental properties of the class of pseudononspreading mappings are proved. In particular, it is proved that the fixed point set of certain class of pseudononspsreading selfmappings of a nonempty closed and convex subset of a real Hilbert space is closed and convex. Demiclosedness property of such class of pseudonon-spreading mappings is proved. Certain weak and strong convergence theorems are then proved for the iterative approximation of fixed points of the class of pseudononspreading mappings.

Strong convergence theorem for monotone operators and strict pseudo-nonspreading mapping

In this paper, based on the recent results of Osilike et al. [9] and motivated by the results of Liu et al. [10] and Takahashi et al. [13], we introduce an iterative sequence and prove that the sequence converges strongly to a common element of the set of fixed points of strict pseudo-non spreading mapping, T and the set of zeros of sum of an α−inverse strongly monotone mapping A and a maximal monotone operator B in a real Hilbert space. Our results improve and generalize many recent important results.

Split equality problem for κ-asymptotically strictly pseudo-nonspreading mapping in Hilbert space

Split equality problem for κ-asymptotically strictly pseudo-nonspreading mapping in Hilbert space

Alternating Iterative Algorithms for Split Equality Problem of Strictly Pseudononspreading Mapping

The purpose of this article is to prove the weak convergence theorems for solving split equality problem of strictly pseudononspreading mapping by introducing two alternating iterative algorithms. Furthermore, we apply our iterative algorithms to some convex and nonlinear problems. The results shown in this paper improve and extend the recent ones announced by many others.

Convergence Theorems for Nonspreading Type Mappings in Hilbert Spaces

In this paper first we give a weak convergence ergodic theorem for K-strictly pseudo-nonspreading and nonspreading mapping in Ishikawa iteration scheme, motivated by Kurokawa and Takahashi in9. Then we deal with a strong convergence theorem for nonspreading mappings in a Hilbert space. Our results improve and extend Kurokawa and Takahashi result.

Strong convergence theorem for a common fixed point of a finite family of strictly pseudo-contractive mappings and a strictly pseudononspreading mapping

AbstractIn this paper, we introduce a new mapping in a real Hilbert space to prove a strong convergence theorem for finding a common fixed point of a finite family of strictly pseudo-contractive mappings and a strictly pseudononspreading mapping. Moreover, we also obtain a strong convergence theorem for a finite family of inverse-strongly monotone mappings and a strictly pseudononspreading mapping.

A simultaneous iterative method for split equality problems of two finitefamilies of strictly pseudononspreading mappings without prior knowledge ofoperator norms

AbstractIn this article, we first introduce the concept of T-mapping of a finitefamily of strictly pseudononspreading mapping "Equation missing", and we show that the fixed point set of theT-mapping is the set of common fixed points of"Equation missing" and T is a quasi-nonexpansive mapping.Based on the concept of a T-mapping, we propose a simultaneousiterative algorithm to solve the split equality problem with a way of selectingthe stepsizes which does not need any prior information about the operatornorms. The sequences generated by the algorithm weakly converge to a solution ofthe split equality problem of two finite families of strictly pseudononspreadingmappings. Furthermore, we apply our iterative algorithms to some convex andnonlinear problems.MSC: 47H09, 47H05, 47H06, 47J25.

Algorithm of a new variational inclusion problem and strictly pseudononspreading mapping with application

The purpose of this research is to modify the variational inclusion problems and prove a strong convergence theorem for finding a common element of the set of fixed points of a κ-strictly pseudononspreading mapping and the set of solutions of a finite family of variational inclusion problems and the set of solutions of a finite family of equilibrium problems in Hilbert space. By using our main result, we prove a strong convergence theorem involving a κ-quasi-strictly pseudo-contractive mapping in Hilbert space. We give a numerical example to support some of our results.

Variational inequality problems over split fixed point sets of strict pseudo-nonspreading mappings and quasi-nonexpansive mappings with applications

In this paper, we first establish a strong convergence theorem for a variational inequality problem over split fixed point sets of a finite family of strict pseudo-nonspreading mappings and a countable family of quasi-nonexpansive mappings. As applications, we establish a strong convergence theorem of split fixed point sets of a finite family of strict pseudo-nonspreading mappings and a countable family of strict pseudo-nonspreading mappings without semicompact assumption on the strict pseudo-nonspreading mappings. We also study the variational inequality problems over split common solutions of fixed points for a finite family of strict pseudo-nonspreading mappings, fixed points of a countable family of pseudo-contractive mappings (or strict pseudo-nonspreading mappings) and solutions of a countable family of nonlinear operators. We study fixed points of a countable family of pseudo-contractive mappings with hemicontinuity assumption, neither Lipschitz continuity nor closedness assumption is needed.

Demiclosed principle and convergence theorems for asymptotically strictly pseudononspreading mappings and mixed equilibrium problems

In this paper, the demiclosed principle for a k-asymptotically strictly pseudononspreading mapping is shown. Meanwhile, an iterative scheme is introduced to approximate a common element of the set of common fixed points of k-asymptotically strictly pseudononspreading mappings and the set of solutions of mixed equilibrium problems in Hilbert spaces, and some weak and strong convergence theorems are proved. The results presented in this paper improve and extend some recent corresponding results.

Multiple-set split feasibility problems for κ-asymptotically strictly pseudo-nonspreading mappings in Hilbert spaces

Some weak and strong convergence theorems for solving multiple-set split feasibility problems for κ-asymptotically strictly pseudo-nonspreading mappings in infinite-dimensional Hilbert spaces are proved. The results presented in the paper extend and improve the corresponding results of Xu (Inverse Probl. 22(6):2021-2034, 2006), Osilike and Isiogugu (Nonlinear Anal. 74:1814-1822, 2011), Chang et al. (Abstr. Appl. Anal. 2012:491760, 2012), and others.

Strong Convergence of a Unified General Iteration fork-Strictly Pseudononspreading Mapping in Hilbert Spaces

We introduce a unified general iterative method to approximate a fixed point ofk-strictly pseudononspreading mapping. Under some suitable conditions, we prove that the iterative sequence generated by the proposed method converges strongly to a fixed point of ak-strictly pseudononspreading mapping with an idea of mean convergence, which also solves a class of variational inequalities as an optimality condition for a minimization problem. The results presented in this paper may be viewed as a refinement and as important generalizations of the previously known results announced by many other authors.