A viscosity approximation scheme for finite mixed equilibrium problems and variational inequality problems and fixed point problems

In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of finite mixed equilibrium problems, the set of solutions of variational inequalities for two cocoercive mappings, the set of common fixed points of an infinite family of nonexpansive mappings and the set of common fixed points of a nonexpansive semigroup in Hilbert space. Then we prove a strong convergence theorem under some suitable conditions. The results obtained in this paper extend and improve many recent ones announced by many others.     

Approximating common fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces

We introduce three-step iterative schemes with errors for two and three nonexpansive maps and establish weak and strong convergence theorems for these schemes. Mann-type and Ishikawa-type convergence results are included in the analysis of these new iteration schemes. The results presented in this paper substantially improve and extend the results due to [S.H. Khan, H. Fukhar-ud-din, Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear Anal. 8 (2005) 1295–1301], [N. Shahzad, Approximating fixed points of non-self nonexpansive mappings in Banach spaces, Nonlinear Anal. 61 (2005) 1031–1039], [W. Takahashi, T. Tamura, Convergence theorems for a pair of nonexpansive mappings, J. Convex Anal. 5 (1995) 45–58], [K.K. Tan, H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993) 301–308] and [H.F. Senter, W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974) 375–380].     

Strong convergence of the modified Ishikawa iterative method for infinitely many nonexpansive mappings in Banach spaces

In this paper, we introduce a new modified Ishikawa iterative process for computing fixed points of an infinite family nonexpansive mapping in the framework of Banach spaces. Then, we establish the strong convergence theorem of the proposed iterative scheme under some mild conditions which solves a variational inequality. The results obtained in this paper extend and improve on the recent results of Qin et al. [Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces, Journal of Computational and Applied Mathematics 230 (1) (2009) 121–127], Cho et al. [Approximation of common fixed points of an infinite family of nonexpansive mappings in Banach spaces, Computers and Mathematics with Applications 56 (2008) 2058–2064] and many others.     

Strong convergence for a minimization problem on solutions of systems of equilibrium problems and common fixed points of an infinite family and semigroup of nonexpansive mappings

In this paper, we introduce a suitable Mann type algorithm for finding a common element of the set of solutions of systems of equilibrium problems and the set of common fixed points of an infinite family and left amenable semigroup of nonexpansive mappings in Hilbert spaces. Then we prove the strong convergence of the proposed iterative scheme to the unique solution of the minimization problem on the solution of systems of equilibrium problems and the common fixed points of an infinite family and left amenable semigroup of nonexpansive mappings. Our results extend and improve the recent result of Colao and Marino [V. Colao and G. Marino, Strong convergence for a minimization problem on points of equilibrium and common fixed points of an infinite family of nonexpansive mappings, Nonlinear Anal. 73 (2010) 3513–3524] and many others.     

On the approximation problem of common fixed points for a finite family of non-self asymptotically quasi-nonexpansive-type mappings in Banach spaces

The purpose of this paper is to introduce the concept of non-self asymptotically quasi-nonexpansive-type mappings and to construct a iterative sequence to converge to a common fixed point for a finite family of non-self asymptotically quasi-nonexpansive-type mappings in Banach spaces. The results presented in this paper improve and extend the corresponding results in Alber, Chidume and Zegeye [Ya.I. Alber, C.E. Chidume, H. Zegeye, Approximating of total asymptotically nonexpansive mappings, Fixed Point Theory and Applications (2006) 1–20. Article ID10673], Ghosh and Debnath [M.K. Ghosh, L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings, Journal of Mathematical Analysis and Applications 207 (1997) 96–103], Liu [Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive type mappings, Journal of Mathematical Analysis and Applications 259 (2001) 1–37; Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, Journal of Mathematical Analysis and Applications 259 (2001) 18–24; Q.H. Liu, Iteration sequences for asymptotically quasi-nonexpansive mapping with an error member of uniform convex Banach space, Journal of Mathematical Analysis and Applications 266 (2002) 468–471], Petryshyn [W.V. Petryshyn, T.E. Williamson Jr., Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, Journal of Mathematical Analysis and Applications 43 (1973) 459–497], Quan and Chang [J. Quan, S.S. Chang, X.J. Long, Approximation common fixed point of asymptotically quasi-nonexpansive type mappings by the finite steps iterative sequences, Fixed Point Theory and Applications V (2006) 1–38. Article ID 70830], Shahzad and Udomene [N. Shahzad, A. Udomene, Approximating common fixed point of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications (2006) 1–10. Article ID 18909] Xu [B.L. Xu, M.A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications 267 (2002) 444–453], Zhang [S.S. Zhang, Iterative approximation problem of fixed points for asymptotically nonexpansive mappings in Banach spaces, Acta Mathematicae Applicatae Sinica 24 (2001) 236–241] and Zhou and Chang [Y.Y. Zhou, S.S. Chang, Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numerical Functional Analysis and Optimization 23 (2002) 911–921].     

Two extragradient methods for generalized mixed equilibrium problems, nonexpansive mappings and monotone mappings

In this paper, we introduce iterative schemes based on the extragradient method for finding a common element of the set of solutions of a generalized mixed equilibrium problem and the set of fixed points of an infinite (a finite) family of nonexpansive mappings and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. We obtain some weak convergence theorems for the sequences generated by these processes in Hilbert spaces. The results in this paper generalize, extend and unify some well-known weak convergence theorems in the literature.     

Halpern-type iterations for strongly relatively nonexpansive mappings in Banach spaces

In this paper, we note that the main convergence theorem in Zhang et al. (2011) [21] is incorrect and we prove a correction. We also modify Halpern’s iteration for finding a fixed point of a strongly relatively nonexpansive mapping in a Banach space. Consequently, two strong convergence theorems for a relatively nonexpansive mapping and for a mapping of firmly nonexpansive type are deduced. Using the concept of duality theorems, we obtain analogue results for strongly generalized nonexpansive mappings and for mappings of firmly generalized nonexpansive type. In addition, we study two strong convergence theorems concerning two types of resolvents of a maximal monotone operator in a Banach space.     

Strong convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces

The purpose of the paper is to introduce modified Halpern and Ishikawa iteration for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in Banach spaces. We also consider two strong convergence theorems for relatively nonexpansive mappings with some proper restriction.     

On solving the minimization problem and the fixed-point problem for nonexpansive mappings in CAT(0) spaces

We propose a new modified proximal point algorithm combined with Halpern's iteration process for nonexpansive mappings in the framework of CAT(0) spaces. We establish a strong convergence theorem under some mild conditions. Our results extend some known results which appeared in the literature.     

Existence and iteration for a mixed equilibrium problem and a countable family of nonexpansive mappings in Banach spaces

We prove the existence of a solution of the mixed equilibrium problem (MEP) by using the KKM mapping in a Banach space setting. Then, by virtue of this result, we construct a hybrid algorithm for finding a common element in the solutions set of a mixed equilibrium problem and the fixed points set of a countable family of nonexpansive mappings in the frameworks of Banach spaces. By using a projection technique, we also prove that the sequences generated by the hybrid algorithm converge strongly to a common element in the solutions set of MEP and common fixed points set of nonexpansive mappings. Moreover, some applications concerning the equilibrium and the convex minimization problems are obtained.     

The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces

In this paper, we discuss the strong convergence of the viscosity approximation method, in Hilbert spaces, relatively to the computation of fixed points of operators in the wide class of quasi-nonexpansive mappings. Our convergence results improve previously known ones obtained for the class of nonexpansive mappings.     

Convergence of an Ishikawa type iterative scheme for asymptotically quasi-nonexpansive mappings

In convex metric spaces, the Ishikawa iteration process and the Ishikawa iteration process with errors is defined for asymptotically quasi-nonexpansive mappings. It is proved some sufficient and necessary conditions for the iterative scheme converges to the fixed point of the asymptotically quasi-nonexpansive mappings. These results generalize and unify many important known results in recent literature.     

Erratum to “Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces” [Comput. Math. Appl. 54 (2007) 872–877]

We show strong convergence for Mann and Ishikawa iterates of multivalued nonexpansive mapping T under some appropriate conditions, which revises a gap in Panyanak [B. Panyanak, Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces, Comput. Math. Appl. 54 (2007) 872–877]. Furthermore, we also give an affirmative answer to Panyanak’s open question.     

A strong convergence theorem for a general split equality problem with applications to optimization and equilibrium problem

The purpose of this paper is to introduce and study an iterative algorithm for solving a general split equality problem. The problem consists of finding a common element of the set of common zero points for a finite family of maximal monotone operators, the set of common fixed points for a finite family of demimetric mappings and the set of common solutions of variational inequality problems for a finite family of inverse strongly monotone mappings in the setting of infinite-dimensional Hilbert spaces. Using our iterative algorithm, we state and prove a strong convergence theorem for approximating a solution of the split equality problem. As special cases, we shall utilize our results to study the split equality equilibrium problems and the split equality optimization problems. Our result complements and extends some related results in literature.     

Stability and convergence of F iterative scheme with an application to the fractional differential equation

In this paper, we prove that F iterative scheme is almost stable for weak contractions. Further, we prove convergence results for weak contractions as well as for generalized non-expansive mappings due to Hardy and Rogers via F iterative scheme. We also prove that F iterative scheme converges faster than the some known iterative schemes for weak contractions. An illuminative numerical example is formulated to support our assertion. Finally, utilizing our main result the solution of nonlinear fractional differential equation is approximated.

     

Common fixed points of a countable family of multivalued quasi-nonexpansive mappings in uniformly convex Banach spaces

In this paper, we introduce a one-step iterative scheme for finding a common fixed point of a countable family of multivalued quasi-nonexpansive mappings in a real uniformly convex Banach space. We establish weak and strong convergence theorems of the proposed iterative scheme under some control conditions.     

Mann and Ishikawa iterations with errors for asymptotically nonexpansive mappings

In a recent paper, Rhoades [1] presented some generalizations of Schu [2] on the convergence of the Mann and Ishikawa iterations of asymptotically nonexpansive mappings in uniformly convex Banach spaces. We continue the study on the Ishikawa (and Mann) iteration process with errors and prove that if X is a uniformly convex Banach space, ø ≠ E ⊂ X closed bounded and convex, and T : E → E is an asymptotically nonexpansive mapping, then the Ishikawa (and Mann) iteration process with errors converges strongly to some fixed point of T.     

Convergence theorems by a new iteration process for a finite family of nonself asymptotically nonexpansive mappings with errors in Banach spaces

In this paper, we study a new iteration process for a finite family of nonself asymptotically nonexpansive mappings with errors in Banach spaces. We prove some weak and strong convergence theorems for this new iteration process. The results of this paper improve and extend the corresponding results of Chidume et al. (2003) [10], Osilike and Aniagbosor (2000) [3], Schu (1991) [4], Takahashi and Kim (1998) [9], Tian et al. (2007) [18], Wang (2006) [11], Yang (2007) [17] and others.     

A modified hybrid projection method for solving generalized mixed equilibrium problems and fixed point problems in Banach spaces

In this paper, we introduce a modified new hybrid projection method for finding the set of solutions of the generalized mixed equilibrium problems and the convex feasibility problems for an infinite family of closed and uniformly quasi-?-asymptotically nonexpansive mappings. Strong convergence theorems are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. Our results improve and extend the corresponding results announced by Qin et al. (2010) and many authors.