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.     

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.     

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.     

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.     

Hybrid algorithm for generalized mixed equilibrium problems and variational inequality problems and fixed point problems

In this paper, we introduce a new iterative algorithm by hybrid method for finding a common element of the set of solutions of finite general mixed equilibrium problems and the set of solutions of a general variational inequality problem for finite inverse strongly monotone mappings and the set of common fixed points of infinite family of strictly pseudocontractive mappings in a real Hilbert space. Then we prove strong convergence of the scheme to a common element of the three above described sets. Our result improves and extends the corresponding results announced by many others.     

Strong convergence theorems for an infinite family of quasi-nonexpansive mappings and generalized equilibrium problems and variational inequality problems

In this paper, we introduce a new iterative scheme for finding a common element of the set of common fixed points of an infinite family of uniformly continuous quasi-nonexpansive mappings and the set of solutions to a generalized equilibrium problem and the set of solutions to a variational inequality problem in a real Hilbert space. We then prove strong convergence of the scheme to a common element of the three mentioned sets. Our results extend important recent results.     

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.     

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.     

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.     

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].     

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].     

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.     

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.     

Strong convergence theorems based on a new modified extragradient method for variational inequality problems and fixed point problems in Banach spaces

In this paper, we introduce a new iterative algorithm by a modified extragradient method for finding a common element of the set of solutions of a general variational inequality and the set of common fixed points of an infinite family of k_i-strict pseudocontractions in a Banach space. We obtain some strong convergence theorems under suitable conditions. The results obtained in this paper improve and extend the recent ones announced by many others.     

A general iterative method with strongly positive operators for general variational inequalities

In this paper, we introduce and study a general iterative method with strongly positive operators for finding solutions of a general variational inequality problem with inverse-strongly monotone mapping in a real Hilbert space. The explicit and implicit iterative algorithms are proposed by virtue of the general iterative method with strongly positive operators. Under two sets of quite mild conditions, we prove the strong convergence of these explicit and implicit iterative algorithms to the unique common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality problem, respectively.     

Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method

In this paper, we introduce an iterative method based on the extragradient method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real Hilbert space. Furthermore, we prove that the studied iterative method strongly converges to a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping under some mild conditions imposed on algorithm parameters.     

Hybrid Ishikawa iterative methods for a nonexpansive semigroup in Hilbert space

In this paper, on the base of the Ishikawa iteration method and the hybrid method in mathematical programming, we give two new strong convergence methods for finding a point in the common fixed point set of a nonexpansive semigroup in Hilbert space.     

Nonzero solutions for a class of set-valued variational inequalities in reflexive Banach spaces

In this paper, we study the existence of nonzero solutions for a class of set-valued variational inequalities involving set-contractive mappings by using the fixed point index approach in reflexive Banach spaces. Some new existence theorems of nonzero solutions for this class of set-valued variational inequalities are established.