Nonlinear Convex Anal Research Articles

Alternating mann iterative algorithms for the split common fixed-point problem of quasi-nonexpansive mappings

Very recently, Moudafi (Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal. ) introduced an alternating CQ-algorithm with weak convergence for the following split common fixed-point problem. Let H 1 , H 2 , H 3 be real Hilbert spaces, let A: H 1 → H 3 , B: H 2 → H 3 be two bounded linear operators. Find x∈F(U),y∈F(T) such that Ax=By, (1) where U: H 1 → H 1 and T: H 2 → H 2 are two firmly quasi-nonexpansive operators with nonempty fixed-point sets F(U)={x∈ H 1 :Ux=x} and F(T)={x∈ H 2 :Tx=x}. Note that by taking H 2 = H 3 and B=I, we recover the split common fixed-point problem originally introduced in Censor and Segal (J. Convex Anal. 16:587-600, 2009) and used to model many significant real-world inverse problems in sensor net-works and radiation therapy treatment planning. In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of quasi-nonexpansive operators. We introduce two alternating Mann iterative algorithms and prove the weak convergence of algorithms. At last, we provide some applications. Our results improve and extend the corresponding results announced by many others.MSC:47H09, 47H10, 47J05, 54H25.

Strong and Δ-convergence for mixed type total asymptotically nonexpansive mappings in CAT(0) spaces

It is our purpose in this paper first to introduce the class of total asymptotically nonexpansive nonself mappings and to prove the demiclosed principle for such mappings in CAT(0) spaces. Then, a new mixed Agarwal-O’Regan-Sahu type iterative scheme for approximating a common fixed point of two total asymptotically nonexpansive mappings and two total asymptotically nonexpansive nonself mappings is constructed. Under suitable conditions, some strong convergence theorems and Δ-convergence theorems are proved in a CAT(0) space. Our results improve and extend the corresponding results of Agarwal, O’Regan and Sahu (J. Nonlinear Convex Anal. 8(1):61-79, 2007), Guo et al. (Fixed Point Theory Appl. 2012:224, 2012. doi:10.1186/1687-1812-2012-224), Sahin et al. (Fixed Point Theory Appl. 2013:12, 2013. doi:10.1186/1687-1812-2013-12), Chang et al. (Appl. Math. Comput. 219:2611-2617, 2012), Khan and Abbas (Comput. Math. Appl. 61:109-116, 2011), Khan et al. (Nonlinear Anal. 74:783-791, 2011), Xu (Nonlinear Anal., Theory Methods Appl. 16(12):1139-1146, 1991), Chidume et al. (J. Math. Anal. Appl. 280:364-374, 2003) and others.MSC:47J05, 47H09, 49J25.

Split feasibility problem for quasi-nonexpansive multi-valued mappings and total asymptotically strict pseudo-contractive mapping

The purpose of this article is to study the split feasibility problem for a family of quasi-nonexpansive multi-valued mappings and a total asymptotically strict pseudo-contractive mapping in infinitely dimensional Hilbert spaces. The main results presented in the paper improve and extend some recent results of Censor et al. [Numer. Algorithms 8 (1994) 221–239; Inverse Problem 21 (2005) 2071–2084; J. Math. Anal. Appl. 327 (2007) 1244–1256], Byrne [Inverse Problem 18 (2002) 441–453], Yang [Inverse Problem 20 (2004) 1261–1266], Moudafi [Inverse Problem 26 (2010) 055007], Xu [Inverse Problem 26 (2010) 105018], Censor and Segal [J. Convex Anal. 16 (2009) 587–600], Masad and Reich [J. Nonlinear Convex Anal. 8 (2007) 367–371] and others.

Fixed point theorems for nonlinear non-self mappings in Hilbert spaces and applications

Abstract Recently, Kawasaki and Takahashi (J. Nonlinear Convex Anal. 14:71-87, 2013) defined a broad class of nonlinear mappings, called widely more generalized hybrid, in a Hilbert space which contains generalized hybrid mappings (Kocourek et al. in Taiwan. J. Math. 14:2497-2511, 2010) and strict pseudo-contractive mappings (Browder and Petryshyn in J. Math. Anal. Appl. 20:197-228, 1967). They proved fixed point theorems for such mappings. In this paper, we prove fixed point theorems for widely more generalized hybrid non-self mappings in a Hilbert space by using the idea of Hojo et al. (Fixed Point Theory 12:113-126, 2011) and Kawasaki and Takahashi fixed point theorems (J. Nonlinear Convex Anal. 14:71-87, 2013). Using these fixed point theorems for non-self mappings, we proved the Browder and Petryshyn fixed point theorem (J. Math. Anal. Appl. 20:197-228, 1967) for strict pseudo-contractive non-self mappings and the Kocourek et al. fixed point theorem (Taiwan. J. Math. 14:2497-2511, 2010) for super hybrid non-self mappings. In particular, we solve a fixed point problem. MSC:Primary 47H10; secondary 47H05.

Strong convergence to fixed points of non-Lipschitzian mappings

In this paper, we first show the strong convergence of the modified Moudafi iteration process when E is a real uniformly convex Banach space, S is AQT self-mapping and T is ANI self-mapping satisfying Condition (B). Next, we show the strong convergence of the modified Mann iteration process when T is ANI self-mapping satisfying Condition (A), which generalizes the result due to Kim (J. Nonlinear Convex Anal. 13(3):449–457, 2012). Finally, we show the strong convergence of the Schu iteration process when T is ANI self-mapping satisfying Condition (A), which generalizes the result due to Rhoades (J. Math. Anal. Appl. 183:118–120, 1994). Open image in new window

Some results on a tripled fixed point for nonlinear contractions in partially ordered G-metric spaces

Berinde and Borcut (Nonlinear Anal. 74(15):4889-4897, 2011) have quite recently defined the notion of a triple fixed point and proved some interesting results related to this concept in a partially ordered metric space. In this work we prove some triple fixed point theorem for a mixed monotone mapping satisfying nonlinear contractions in the framework of a generalized metric space endowed with partial order while the idea of a generalized metric space introduced by Mustafa and Sims (J. Nonlinear Convex Anal. 7:289-297, 2006). Further we prove the uniqueness of a coupled fixed point for such a mapping in this setting.

Split feasibility problems for total quasi-asymptotically nonexpansive mappings

Abstract The purpose of this paper is to propose an algorithm for solving the split feasibility problems for total quasi-asymptotically nonexpansive mappings in infinite-dimensional Hilbert spaces. The results presented in the paper not only improve and extend some recent results of Moudafi [Nonlinear Anal. 74:4083-4087, 2011; Inverse Problem 26:055007, 2010], but also improve and extend some recent results of Xu [Inverse Problems 26:105018, 2010; 22:2021-2034, 2006], Censor and Segal [J. Convex Anal. 16:587-600, 2009], Censor et al. [Inverse Problems 21:2071-2084, 2005], Masad and Reich [J. Nonlinear Convex Anal. 8:367-371, 2007], Censor et al. [J. Math. Anal. Appl. 327:1244-1256, 2007], Yang [Inverse Problem 20:1261-1266, 2004] and others. MSC:47J05, 47H09, 49J25.

General Class of Implicit Variational Inclusions and Graph Convergence on A-Maximal Relaxed Monotonicity

Based on the generalized graph convergence, first a general framework for an implicit algorithm involving a sequence of generalized resolvents (or generalized resolvent operators) of set-valued A-maximal monotone (also referred to as A-maximal (m)-relaxed monotone, and A-monotone) mappings, and H-maximal monotone mappings is developed, and then the convergence analysis to the context of solving a general class of nonlinear implicit variational inclusion problems in a Hilbert space setting is examined. The obtained results generalize the work of Huang, Fang and Cho (in J. Nonlinear Convex Anal. 4:301–308, 2003) involving the classical resolvents to the case of the generalized resolvents based on A-maximal monotone (and H-maximal monotone) mappings, while the work of Huang, Fang and Cho (in J. Nonlinear Convex Anal. 4:301–308, 2003) added a new dimension to the classical resolvent technique based on the graph convergence introduced by Attouch (in Variational Convergence for Functions and Operators, Applied Mathematics Series, Pitman, London 1984). In general, the notion of the graph convergence has potential applications to several other fields, including models of phenomena with rapidly oscillating states as well as to probability theory, especially to the convergence of distribution functions on ℜ. The obtained results not only generalize the existing results in literature, but also provide a certain new approach to proofs in the sense that our approach starts in a standard manner and then differs significantly to achieving a linear convergence in a smooth manner.

Viscosity methods for common solutions for equilibrium and hierarchical fixed point problems

Implicit and explicit viscosity methods for finding common solutions of equilibrium and hierarchical fixed points are presented. These methods are used to solve systems of equilibrium problems and variational inequalities where the involving operators are complements of nonexpansive mappings. The results here are situated on the lines of the research of the corresponding results of Moudafi [Krasnoselski-Mann iteration for hierarchical fixed-point problems, Inverse Probl. 23 (2007), pp. 1635–1640; Weak convergence theorems for nonexpansive mappings and equilibrium problems, to appear in JNCA], Moudafi and Maingé [Towards viscosity approximations of hierarchical fixed-points problems, Fixed Point Theory Appl. Art ID 95453 (2006), 10 pp.; Strong convergence of an iterative method for hierarchical fixed point problems, Pac. J. Optim. 3 (2007), pp. 529–538; Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems, to appear in JNCA], Yao and Liou [Weak and strong convergence of Krasnosel'skiĭ–Mann iteration for hierarchical fixed point problems, Inverse Probl. 24 (2008), 015015 8 pp.], S. Takahashi and W. Takahashi [Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006), pp. 506–515], Xu [Viscosity method for hierarchical fixed point approach to variational inequalities, preprint.], Combettes and Hirstoaga [Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), pp. 117–136] and Plubtieng and Pumbaeang [A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (2007), pp. 455–469.].

Convergence to common fixed points by a modified iteration process

We modify an iteration process of Agarwal et al. (J. Nonlinear Convex Anal. 8(1):61–79, 2007) to the case of two mappings and prove some weak and strong convergence theorems for two asymptotically nonexpansive mappings. We also point out that this process cannot be used for three mappings in its existing form even for nonexpansive mappings. We have to impose an extra condition to get convergence. We give an example to show that there do exist two nonexpansive mappings satisfying that condition.

A new iterative algorithm of solution for equilibrium problems, variational inequalities and fixed point problems in a Hilbert space

In this paper, we introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem, the set of solutions of the variational inequality for β-inverse-strongly monotone mappings and the set of fixed points of nonexpansive mappings in a Hilbert space. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. As applications, at the end of paper we utilize our results to study some convergence problem for finding the zeros of maximal monotone operators. Our results are generalizations and extensions of the results of Yao and Liou (Fixed Point Theory Appl. Article ID 384629, 10 p., 2008), Yao et al. (J. Nonlinear Convex Anal. 9(2):239–248, 2008) and Su and Li (Appl. Math. Comput. 181(1):332–341, 2006) and some recent results.

Convergence of composite iterative methods for finding zeros of accretive operators

Strong convergence theorems on composite iterative schemes by the viscosity approximation methods for finding a zero of an accretive operator are established in Banach spaces. The main results generalize the recent corresponding results of Aoyama et al. [K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in Banach spaces, Nonlinear Anal. 67 (2007) 2350–2360], Ceng et al. [L.C. Ceng, A.R. Khan, Q.H. Ansari, J.C. Yao, Strong convergence of composite iterative schemes for zeros of m-accretive operators in Banach spaces, Nonlinear Anal. 70 (2009) 1830–1840], Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60], and Xu [H.K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006) 631–643] to viscosity methods in a strictly convex and reflexive Banach space having a uniformly Gâteaux differentiable norm. Our results also improve the corresponding results of [T.D. Benavides, G.L. Acedo, H.K. Xu, Iterative solutions for zeros of accretive operators, Math. Nachr. 248–249 (2003) 62–71; R. Chen, Z. Zhu, Viscosity approximation fixed points for nonexpansive and m -accretive operators, Fixed Point Theory Appl. 2006 (2006) 1–10; S. Kamimura, W. Takahashi, Approximation solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory 106 (2000) 226–240; P.E. Maingé, Viscosity methods for zeroes of accretive operators, J. Approx. Theory 140 (2) (2006) 127–140; K. Nakajo, Strong convergence to zeros of accretive operators in Banach spaces, J. Nonlinear Convex Anal. 7 (2006) 71–81].

Weak convergence theorems for a countable family of Lipschitzian mappings

This paper is concerned with convergence of an approximating common fixed point sequence of countable Lipschitzian mappings in a uniformly convex Banach space. We also establish weak convergence theorems for finding a common element of the set of fixed points, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain and improve the corresponding results recently proved by Moudafi [A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9 (2008) 37–43], Tada–Takahashi [A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133 (2007) 359–370], and Plubtieng–Kumam [S. Plubtieng and P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings. J. Comput. Appl. Math. (2008) doi:10.1016/j.cam.2008.05.045]. Some of our results are established with weaker assumptions.

A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces

The purpose of this paper is to study the strong convergence of a general iterative scheme to find a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of variational inequality for a relaxed cocoercive mapping and the set of solutions of an equilibrium problem. Our results extend the recent results of Takahashi and Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515], Marino and Xu [G. Marino, H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43–52], Combettes and Hirstoaga [P.L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) 486–491], Iiduka and Takahashi, [H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005) 341–350] and many others.

Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces

In this paper, we introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Then, we prove a strong convergence theorem which is connected with Combettes and Hirstoaga's result [P.L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) 117–136] and Wittmann's result [R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486–491]. Using this result, we obtain two corollaries which improve and extend their results.