Search

We prove some existence theorems for solutions of a certain system of multivariate nonexpansive operator equations and calculate the solutions by using the generalized Mann and Halpern iterative algorithms in uniformly convex and uniformly smooth Banach spaces. The results of this paper improve and extend the previously known ones in the literature.

Let K be a nonempty closed convex and bounded subset of a reflexive Banach space X. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A_{1}, A_{2},\ldots,A_{N}$\end{document}A1,A2,…,AN be N-variables monotone demi-continuous mappings from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K^{N}$\end{document}KN into X. Then: (1) the system of multivariate variational inequalities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textstyle\begin{cases} \langle A_{1}(x_{1},x_{2},\ldots,x_{N}), y_{1}-x_{1} \rangle\geq0, &\forall y_{1} \in K,\\ \langle A_{2}(x_{1},x_{2},\ldots,x_{N}), y_{2}-x_{2} \rangle\geq0, &\forall y_{2} \in K,\\ \cdots\\ \langle A_{N}(x_{1},x_{2},\ldots,x_{N}), y_{N}-x_{N} \rangle\geq0, &\forall y_{N} \in K,\\ \end{cases} $$\end{document}{〈A1(x1,x2,…,xN),y1−x1〉≥0,∀y1∈K,〈A2(x1,x2,…,xN),y2−x2〉≥0,∀y2∈K,⋯〈AN(x1,x2,…,xN),yN−xN〉≥0,∀yN∈K, has a solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(x_{1}^{*},x_{2}^{*},\ldots,x_{N}^{*}) \in K^{N}$\end{document}(x1∗,x2∗,…,xN∗)∈KN; (2) the set of solutions of this system of multivariate variational inequalities is closed convex in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K^{N}$\end{document}KN; (3) if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A_{1}, A_{2},\ldots,A_{N}$\end{document}A1,A2,…,AN are also strictly monotone, this system of multivariate variational inequalities has a unique solution.

The purpose of this paper is to present the notion of weak relatively nonexpansive multi-valued mapping and to prove the strong convergence theorems of fixed point for weak relatively nonexpansive multivalued mappings in Banach spaces. The weak relatively nonexpansive multivalued mappings are more generalized than relatively nonexpansive multivalued mappings. In this paper, an example will be given which is a weak relatively nonexpansive multivalued mapping but not a relatively nonexpansive multivalued mapping. In order to get the strong convergence theorems for weak relatively nonexpansive multivalued mappings, a new monotone hybrid iteration algorithm with generalized (metric) projection is presented and is used to approximate the fixed point of weak relatively nonexpansive multivalued mappings. In this paper, the notion of multivalued resolvent of maximal monotone operator has been also presented which is a weak relatively nonexpansive multivalued mapping and can be used to find the zero point of maximal monotone operator.

In this paper, under the framework of real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E ∗ , we study the strong convergence of an implicit and an explicit composite viscosity approximation algorithm (I) and (II), respectively for a pseudocontractive mapping T by using the weakly contractive mapping f as follows: (I) x t , s = t f ( x t , s ) + ( 1 − t ) y t , s y t , s = s x t , s + ( 1 − s ) T x t , s and (II) x n + 1 = α n f ( x n ) + ( 1 − α n ) y n , y n = β n x n + ( 1 − β n ) T x n , n ≥ 0 . Our results unify, improve and complement several recent ones existing in the current literature.

The purpose of this article is to prove strong convergence theorems for common fixed points of two countable families of weak relatively nonexpansive mappings in Banach spaces. In order to get the strong convergence theorems, the monotone hybrid algorithms are presented and are used to approximate the common fixed points. Using this result, we also discuss the problem of strong convergence concerning the maximal monotone operators in a Banach space. The results of this article modify and improve the results of Matsushita and Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005) 257–266] and the results of Plubtieng and Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space, J. Approx. Theory 149 (2007) 103–115] and the results of Su et al. [Y. Su, Z. Wang and H. Xu, Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings, Nonlinear Anal. 71 (2009) 5616–5628], and many others.

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a weak relatively nonexpansive mapping, the set of solutions of the variational inequality for the monotone mapping and the set of solutions of an equilibrium problem in a 2-uniformly convex and uniformly smooth Banach space. Then we show that the iterative sequence converges strongly to a common element of the three sets. In this paper, we also give an example which is a weak relatively nonexpansive mapping but not a relatively nonexpansive mapping in Banach space l 2 .

The purpose of this article is to prove strong convergence theorems for common fixed points of two closed hemi-relatively nonexpansive mappings in Banach spaces. In order to get the strong convergence theorems, the monotone hybrid algorithms are presented and are used to approximate the common fixed points. Finally, a new simplified hybrid algorithm has been proposed and relative convergence theorem has been proved by using the new method for proofs. The results of this article modify and improve the results of Matsushita, Takahashi [S. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005) 257–266] and the results of Plubtieng, Ungchittrakool [S. Plubtieng, K. Ungchittrakool, Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space, J. Approx. Theory 149 (2007) 103–115], and many others.

In this paper, we discuss some properties of iterates generated by a strict pseudo-contraction or a finite family of strict pseudo-contractions in a real q -uniformly smooth Banach space. We prove the weak convergence and strong convergence of pseudo-contractive mappings. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–349], Acedo and Xu [G.L. Acedo, H.K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 67 (2007) 2258–2271] and Zhou [H. Zhou, Convergence theorems for λ -strict pseudo-contractions in 2-uniformly smooth Banach spaces, Nonlinear Anal. 69 (2008) 3160–3173].

In the present paper, we propose two kinds of new algorithms for a family of quasi-asymptotic pseudo-contractions in real Hilbert spaces. By using the proposed algorithms, we prove several strong convergence theorems for a family of quasi-asymptotic pseudo-contractions. The results of this paper are interesting extensions of those known results.

Let H be a Hilbert space and f a fixed contractive mapping with coefficient 0 α 1 , A a strongly positive linear bounded operator with coefficient γ > 0 . Consider two iterative methods that generate the sequences { x n } , { y n } by the algorithm, respectively. (I) x n = ( I − α n A ) 1 t n ∫ 0 t n T ( s ) x n d s + α n γ f ( x n ) (II) y n + 1 = ( I − α n A ) 1 t n ∫ 0 t n T ( s ) y n d s + α n γ f ( y n ) where { α n } and { t n } are two sequences satisfying certain conditions, and ℑ = { T ( s ) : s ≥ 0 } is a one-parameter nonexpansive semigroup on H . It is proved that the sequences { x n } , { y n } generated by the iterative method (I) and (II) , respectively, converge strongly to a common fixed point x ∗ ∈ F ( ℑ ) which solves the variational inequality 〈 ( A − γ f ) x ∗ , x ∗ − z 〉 ≤ 0 z ∈ F ( ℑ ) .

In this paper, we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of the variational inequality for an inverse-strongly monotone mapping and the set of solutions of an equilibrium problem in a Hilbert space. We show that the iterative sequence converges strongly to a common element of the three sets. The results of this paper extended and improved the results of H. Iiduka and W. Takahashi [Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350] and S. Takahashi and W. Takahashi [Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515]. Therefore, by using the above result, an iterative algorithm for the solution of a optimization problem was obtained.

Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379] proved strong convergence theorems for nonexpansive mappings, nonexpansive semigroups and the proximal point algorithm for zero-point of monotone operators in Hilbert spaces by the CQ iteration method. The purpose of this paper is to modify the CQ iteration method of K. Nakajo and W. Takahashi using the monotone CQ method, and to prove strong convergence theorems. In the proof process of this article, the Cauchy sequence method is used, so we proceed without use of the demiclosedness principle and Opial’s condition, and other weak topological techniques.

The purpose of this article is to prove strong convergence theorems for fixed points of closed hemi-relatively nonexpansive mappings. In order to get these convergence theorems, the monotone hybrid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S. Matsushita and W. Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping. The results of this paper modify and improve the results of S. Matsushita and W. Takahashi (2005), and some others.

Let be nonempty closed convex subset of real Hilbert space . Consider a nonexpansive semigroup with a common fixed point, a contraction with coefficient , and a strongly positive linear bounded operator with coefficient . Let . It is proved that the sequence generated iteratively by converges strongly to a common fixed point which solves the variational inequality for all .

X. Qin and Y. Su proved a strong convergence theorems of modified Ishikawa iteration by CQ method for relatively nonexpansive mappings in a Banach space [Xiaolong Qin, Yongfu Su, Nonlinear Anal. 67 (2007), no. 6, 1958-1965]. The result of this paper extends and improves the result of X. Qin and Y. Su in the two respects: (1). By using the monotone CQ method to modify the CQ method, so that the new method of proof is used. (2). Relax the restriction on $T$ from uniformly continuous to continuous. The result of this paper also extends and improves the recent ones announced by Nakajo, Takahashi, Kim, Martinez-Yanes, Xu and some others.

In this paper, we prove two strong convergence theorems of modified Ishikawa iteration and modified Halpern iteration for relatively nonexpansive mappings in a Banach space. Our results extend and improve the recent ones announced by Nakajo, Takahashi, Kim, Martinez-Yanes, Xu and some others.

This paper introduces a composite iteration scheme for approximating a zero point of accretive operator in the framework of uniformly smooth Banach spaces and the reflexive Banach space which has a weak continuous duality map, respectively. Strong convergence of the composite iteration scheme { x n } defined by { y n = β n x n + ( 1 − β n ) J r n x n , x n + 1 = α n u + ( 1 − α n ) y n , where J r n is the resolvent of m-accretive operator A and u ∈ C is an arbitrary (but fixed) element in C and sequences { α n } in ( 0 , 1 ) , { β n } in [ 0 , 1 ] is established. Under certain appropriate assumptions on the sequences { α n } , { β n } and { r n } , that { x n } defined by the above iteration scheme converges to a zero point of A is proved. The results improve and extend results of T.H. Kim, H.K. Xu and some others.

In this paper, we introduce a modified Ishikawa iterative process for approximating a fixed point of nonexpansive mappings in Hilbert spaces. we establish the strong convergence theorem of the general iteration scheme under some mild conditions. Our results extend and improve the results announced by many others.

Let E be a real Banach space and let K be a nonempty closed convex subset of E. Let { T i } i = 1 N be N strictly pseudocontractive self-maps of K such that F = ⋂ i = 1 N F ( T i ) ≠ ∅ , where F ( T i ) = { x ∈ K : T i x = x } and let { α n } n = 1 ∞ , { β n } n = 1 ∞ ⊂ [ 0 , 1 ] be two real sequence satisfying the conditions: ( i ) ∑ n = 1 ∞ ( 1 − α n ) = + ∞ ; ( ii ) ∑ n = 1 ∞ ( 1 − α n ) 2 + ∞ ; ( iii ) ∑ n = 1 ∞ ( 1 − β n ) + ∞ ; ( iv ) ( 1 − α n ) ( 1 − β n ) L 2 1 , ∀ n ⩾ 1 , where L ⩾ 1 is common Lipschitz constant of { T i } i = 1 N . For x 0 ∈ K , let { x n } n = 1 ∞ be new implicit process defined by x n = α n x n − 1 + ( 1 − α n ) T n y n , y n = β n x n − 1 + ( 1 − β n ) T n x n where T n = T n mod N , then ( i ) lim n → ∞ ‖ x n − p ‖ exists, for all p ∈ F ; ( ii ) lim inf n → ∞ ‖ x n − T n x n ‖ = 0 . The results of this paper generalize and improve the results of Osilike in 2004. In this paper, the proof methods of the main results are also different from that of Osilike.

In this paper, a hybrid projection algorithm for a total quasi-asymptotically pseudo-contractive mapping is introduced in a Hilbert space. A strong convergence theorem of the proposed algorithm to a fixed point of a total quasi-asymptotically pseudo-contractive mapping is proved. Our main result extends and improves many corresponding results.

In recent years, the definition of relatively nonexpansive multivalued mapping and the definition of weak relatively nonexpansive multivalued mapping have been presented and studied by many authors. In this paper, we give some results about weak relatively nonexpansive multivalued mappings and give two examples which are weak relatively nonexpansive multivalued mappings but not relatively nonexpansive multivalued mappings in Banach space ${l}^{2}$ and ${L}^{p}[0,1](1

Saewan and Kumam (2010) have proved the convergence theorems for finding the set of solutions of a general equilibrium problems and the common fixed point set of a family of closed and uniformly quasi--asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach spaceEwith Kadec-Klee property. In this paper, authors prove the convergence theorems and do not need the Kadec-Klee property of Banach space and some other conditions used in the paper of S. Saewan and P. Kumam. Therefore, the results presented in this paper improve and extend some recent results.

The following main results have been given. (1) LetEbe ap-uniformly convex Banach space and letT:E→E*be a(p-1)-L-Lipschitz mapping with condition0<(pL/c2)1/(p-1)<1. ThenThas a unique generalized duality fixed pointx*∈Eand (2) letEbe ap-uniformly convex Banach space and letT:E→E*be aq-α-inverse strongly monotone mapping with conditions1/p+1/q=1,0<(q/(q-1)c2)q-1<α. ThenThas a unique generalized duality fixed pointx*∈E. (3) LetEbe a2-uniformly smooth and uniformly convex Banach space with uniformly convex constantcand uniformly smooth constantband letT:E→E*be aL-lipschitz mapping with condition0<2b/c2<1. ThenThas a unique zero pointx*. These main results can be used for solving the relative variational inequalities and optimal problems and operator equations.

The purpose of this paper is to prove strong convergence theorems for common fixed points of two countable families of relatively quasi nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space using the properties of generalized $f$ -projection operator. In order to get the strong convergence theorems, a new iterative scheme by monotone hybrid method is presented and is used to approximate the common fixed points. Then, two examples of countable families of uniformly closed nonlinear mappings are given. The results of this paper modify and improve the results of Li et al. (2010), the results of Takahashi and Zembayashi (2008), and many others.