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].     

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.     

Fixed point theorems for generalized contractive multi-valued maps

In [N. Mizoguchi, W. Takahashi, Fixed point theorems for multi-valued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177–188] the authors gave a positive answer to the conjecture of S. Reich concerning the existence of fixed points of multi-valued mappings that satisfy certain contractive conditions. In this paper, we establish some results for multi-valued mappings that satisfy a generalized contractive condition in a way that it contains Mizoguchi’s result as one of its special cases. In addition, our results not only improve the results of Kiran and Kamran [Q. Kiran, T. Kamran, Nadler’s type principle with high order of convergence, Nonlinear Anal. TMA 69 (2008) 4106–4120] and some results of Agarwal et al. [R.P. Agarwal, Jewgeni Dshalalow, Donal O’Regan, Fixed point and homotopy results for generalized contractive maps of Reich type, Appl. Anal. 82 (4) (2003) 329–350] but also provide the high order of convergence of the iterative scheme and error bounds. As an application of our results, we obtain an existence result for a class of integral inclusions.     

Modified hybrid algorithms for Lipschitz quasi-pseudo-contractive mappings in Hilbert spaces

In this paper, strong convergence theorems are obtained by modified hybrid methods for Lipschitz quasi-pseudo-contractions in a Hilbert space. Besides, applications of these theorems are introduced. Finally, we use these methods to modify Ishikawa’s iteration process and get some strong convergence theorems which are different from Zhou [H.Y. Zhou, Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 343 (2008) 546–556].     

Existence and uniqueness of common fixed points for two multivalued operators in ordered metric spaces

In this paper, we discuss some new fixed point theorems for a pair of multivalued operators which satisfy weakly generalized contractive conditions. Our results are the extension and improvement of corresponding results of [J. Harjani and K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Analysis, 71 (2009) 3403–3410] and [X. Zhang, Common fixed point theorems for some new generalized contractive type mappings, J. Math. Anal. Appl. 333 (2007) 780–786]. Finally, some examples are given to illustrate the usability of our 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.     

Approximating solutions for the systems of strongly accretive operator equations

The purpose of this paper is to study a strong convergence of multi-step iterative scheme to a common fixed point for a finite family of strongly pseudo-contractive mappings. As a consequence, the strong convergence theorems for the modified multi-step iterative sequence to a solution of systems of strongly accretive operator equations are obtained in real uniformly smooth Banach spaces. The results presented in this paper improve and extend the corresponding Xu [Y. Xu, Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998) 91–101] and Noor, Rassias and Huang [M.A. Noor, T.M. Rassias, Z. Huang, Three-step iterations for nonlinear accretive operator equations, J. Math. Anal. Appl. 274 (2002) 59–68], and many others.     

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.     

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.     

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.     

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.     

Tauberian theorems for statistically convergent double sequences

In [F. Moricz, Tauberian theorems for Cesàro summable double sequences, Studia Math. 110 (1994) 83-96], Moricz has proved some Tauberian theorems for Cesàro summable double sequences and deduced the Tauberian theorems of Landau [E. Landau, Über die Bedeutung einiger neuerer Grenzwertsätze der Herren Hardy and Axer, Prac. Mat.-Fiz. 21 (1910) 97-177] and Hardy [G.H. Hardy, Divergent Series, Univ. Press, Oxford 1956] type. In [J.A. Fridy, M.K. Khan, Statistical extension of some classical Tauberian theorems, Proc. Amer. Math. Soc. 128 (2000) 2347-2355], Fridy and Khan have given statistical extensions of some classical Tauberian theorems. The concept of statistical convergence for double sequences has recently been introduced by Mursaleen and Edely [M. Mursaleen, Osama H. H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003) 223-231] and by Moricz [F. Moricz, Statistical convergence of multiple sequences, Arch. Math. 81 (2003) 82-89] independently. In this paper we give some Tauberian theorems for statistically convergent double sequences. We have also provided some examples including an example to Problem 1 of Moricz [F. Moricz, Tauberian theorems for Cesàro summable double sequences, Studia Math. 110 (1994) 83-96].     

The Bruck's ergodic iteration method for the Ky Fan inequality over the fixed point set

We introduce a new iteration algorithm for solving the Ky Fan inequality over the fixed point set of a nonexpansive mapping, where the cost bifunction is monotone without Lipschitz-type continuity. The algorithm is based on the idea of the ergodic iteration method for solving multi-valued variational inequality which is proposed by Bruck [On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space, J. Math. Anal. Appl. 61 (1977), pp. 159–164] and the auxiliary problem principle for equilibrium problems P.N. Anh, T.N. Hai, and P.M. Tuan. [On ergodic algorithms for equilibrium problems, J. Glob. Optim. 64 (2016), pp. 179–195]. By choosing suitable regularization parameters, we also present the convergence analysis in detail for the algorithm and give some illustrative examples.     

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.     

A double-sequence iteration process for fixed points of continuous pseudocontractions

Let C be a bounded closed convex nonempty subset of a (real) Hilbert space H. The idea of a double-sequence iteration is introduced, and it is proved that a Mann-type double-sequence iteration process converges strongly to a fixed point of a continuous pseudocontractive map T which maps C into C. Related results deal with the strong convergence of the iteration process to fixed points of nonexpansive maps.     

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.     

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 implicit algorithm for two families of nonexpansive mappings

In this paper, we study an implicit iterative algorithm for two nonexpansive mappings and two finite families of nonexpansive mappings in Banach spaces. We prove some weak and strong convergence theorems for these iterative algorithms. Our results extend some existing results.