Nonself-mappings In Banach Spaces Research Articles

New Iterative Methods for Finding Common Fixed Points of Two Non-self Mappings in a Real Hilbert Space

Abstract In this paper, we introduce new iterative methods for approximating common fixed points of two non-self mappings in the framework of real Hilbert spaces. We establish weak and strong convergence results for approximating common fixed points of two nonexpansive non-self mappings. In addition, we establish strong convergence results for approximating common fixed points of two quasi-nonexpansive non-self mappings under appropriate conditions. Our results improve and generalize many of the results in the literature. Moreover, our findings will open the way forward for the study of iterative methods for finding common fixed points of two non-self mappings in Banach spaces more general than Hilbert spaces.

Convergence Theorems for an Iteration of Non-Lipschitzian Nonself Mappings in Banach Spaces

Convergence Theorems for an Iteration of Non-Lipschitzian Nonself Mappings in Banach Spaces

The Krasnosel'skiĭ-Mann algorithm for a countable family of non-self Lipschitzian mappings

In this paper, we address an open question regarding common fixed points of a family of non-self mappings. More precisely, we employ the Krasnosel'skiĭ-Mann algorithm to approximate common fixed points of a countable family of Lipschitzian non-self mappings in Banach spaces under different conditions.

Fixed point theorems for self and non-self F-contractions in metric spaces endowed with a graph

The main results obtained in this paper are fixed point theorems for self and non-self GF-contractions on metric spaces endowed with a graph. Our new results are generalization of recently fixed point theorems for self mappings on metric spaces and also fixed point theorems for non-self mappings in Banach spaces by using the concept of new type of contractive mappings namely F-contractions.

A novel iteration algorithm for a hybrid pair of total asymptotically non-expansive single-valued and total asymptotically quasi-non-expansive multi-valued non-self mappings in Banach spaces

A novel iteration algorithm for a hybrid pair of total asymptotically non-expansive single-valued and total asymptotically quasi-non-expansive multi-valued non-self mappings in Banach spaces

A new approximation method for common fixed points of a finite family of nonexpansive non-self mappings in Banach spaces

In this paper, we introduce a new iterative scheme to approximate a common fixed point for a finite family of nonexpansive non-self mappings. Strong convergence theorems of the proposed iteration in Banach spaces.

Iterative algorithm for common minimum-norm fixed point of a finite family of asymptotically nonextensive nonself mappings in Banach spaces

Iterative algorithm for common minimum-norm fixed point of a finite family of asymptotically nonextensive nonself mappings in Banach spaces

Weak and strong convergence theorems for a finite family of non-Lipschitzian nonself mappings in Banach spaces

In this paper, several weak and strong convergence theorems are established for a new modified iteration with errors for a finite family of nonself mappings which are asymptotically nonexpansive in the intermediate sense in Banach spaces. Mann-type, Ishikawa-type and Noor-type iterations are covered by this new iteration scheme. Our convergence theorems improve, unify and generalize many important results in the current literature.MSC:47H10, 47H09, 46B20.

Strong convergence theorems for two total asymptotically nonexpansive nonself mappings in Banach spaces

In this paper, we define and study the convergence theorems of a new two-steps iterative scheme for two total asymptotically nonexpansive nonself-mappings in Banach spaces. The results of this paper can be viewed as an improvement and extension of the corresponding results of (Shahzad in Nonlinear Anal. 61:1031-1039, 2005; Thianwan in Thai J. Math. 6:27-38, 2008; Ozdemir et al. in Discrete Dyn. Nat. Soc. 2010:307245, 2010) and all the others. MSC: 47H09; 47H10; 46B20

Convergence of a Viscosity Iterative Method for Multivalued Nonself-Mappings in Banach Spaces

Let be a reflexive Banach space having a weakly sequentially continuous duality mapping with gauge function , a nonempty closed convex subset of , and a multivalued nonself-mapping such that is nonexpansive, where . Let be a contraction with constant . Suppose that, for each and , the contraction defined by has a fixed point . Let , and be three sequences in satisfying approximate conditions. Then, for arbitrary , the sequence generated by for all converges strongly to a fixed point of .

Strong convergence theorems for a countable family of totally quasi-ϕ-asymptotically nonexpansive nonself mappings in Banach spaces with applications

In this paper, we introduce a class of totally quasi-ϕ-asymptotically nonexpansive nonself mappings and study the strong convergence under a limit condition only in the framework of Banach spaces. Meanwhile, our results are applied to study the approximation problem of a solution to a system of equilibrium problems. The results presented in the paper improve and extend the corresponding results of Chang et al. (Appl. Math. Comput. 218:7864-7870, 2012). MSC:47J05, 47H09, 49J25.

On convergence theorems for total asymptotically nonexpansive nonself-mappings in Banach spaces

In this paper, we define and study new strong convergence theorems of the modified Mann and the modified Ishikawa iterative scheme with errors for nonself-mappings which are total asymptotically nonexpansive in a uniformly convex Banach space.

Weak and strong convergence theorems of a multistep iteration to a common fixed point of a family of nonself asymptotically nonexpansive mappings in banach spaces

In this paper we have defined a multistep iterative scheme with errors involving a family of asymptotically nonexpansive nonself mappings in Banach spaces. A retraction has been used in the construction of the iteration. We prove here weak and strong convergences of the iteration to common fixed points of the family of asymptotically nonexpansive nonself mappings. We have used several concepts of Banach space geometry. Our results improve and extend some recent results.

Weak and strong convergence theorems for a finite family of generalized asymptotically quasinonexpansive nonself-mappings

In this paper, we introduce and study a new iterative scheme toapproximate a common xed point for a nite family of generalized asymptoticallyquasi-nonexpansive nonself-mappings in Banach spaces. Several strong and weakconvergence theorems of the proposed iteration are established. The main resultsobtained in this paper generalize and re ne some known results in the currentliterature.

Weak convergence theorem for the three-step iterations of non-Lipschitzian nonself mappings in Banach spaces

Abstract In this article, we introduce a new three-step iterative scheme for the mappings which are asymptotically nonexpansive in the intermediate sense in Banach spaces. Weak convergence theorem is established for this three-step iterative scheme in a uniformly convex Banach space that satisfies Opial's condition or whose dual space has the Kadec-Klee property. Furthermore, we give an example of the nonself mapping which is asymptotically nonexpansive in the intermediate sense but not asymptotically nonexpansive. The results obtained in this article extend and improve many recent results in this area. AMS classification: 47H10; 47H09; 46B20.

Krasnoselski-Mann Iterations for Hierarchical Fixed Point Problems for a Finite Family of Nonself Mappings in Banach Spaces

This paper deals with a method for approximating a solution of the following problem: find hierarchically a common fixed point of a finite family of nonself nonexpansive mappings with respect to a nonexpansive self mapping on a closed convex subset of a smooth and reflexive Banach space X, which admits a weakly sequentially continuous duality mapping. First, we prove a weak convergence theorem which extends and improves one recent result proved by Yao and Liou (see Inverse Problems 24 (2008), doi:10.1088/0266-5611/24/1/015015). Secondly, when the self mapping is a contraction, we prove, under different restrictions on parameters, a strong convergence result which generalize some recent results in the literature.

New iterative schemes for asymptotically quasi-nonexpansive nonself-mappings in Banach spaces

In this paper, a new two-step iterative scheme with errors is introduced for two asymptotically quasi-nonexpansive nonself-mappings. Several convergence theorems are established in real Banach spaces and real uniformly convex Banach spaces. Our theorems improve and extend the results due to Thianwan [S. Thianwan, Common fixed point of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space, J. Comput. Appl. Math. 224 (2009) 685–695] and many other papers.

Viscosity approximative methods to Cesàro mean iterations for nonexpansive nonself-mappings in Banach spaces

Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to E∗E∗, C a closed convex subset of E which is also a sunny nonexpansive retract of E . Let T:C→ET:C→E be a nonexpansive nonself-mapping with F(T)≠∅F(T)≠∅, and f:C→Cf:C→C be a fixed contractive mapping. The implicit viscosity iterative sequence {zm}{zm} is defined by {tm}⊂(0,1){tm}⊂(0,1), zm=1m+1∑j=0mP(tmf(zm)+(1-tm)(TP)jzm), for m⩾0m⩾0 and two explicit viscosity iterative sequences {xn}{xn} and {yn}{yn} are given by x0∈C,xn+1=αnf(xn)+(1-αn)1n+1∑j=0n(PT)jxn and y0∈C,yn+1=1n+1∑j=0nP(αnf(yn)+(1-αn)(TP)jyn) for n⩾0n⩾0, where {αn}{αn} is a sequence in (0,1)(0,1) and P is a sunny nonexpansive retraction of E onto C . We prove that under appropriate conditions imposed on {tm}{tm} and {αn}{αn}, the sequences {zm}{zm}, {xn}{xn} and {yn}{yn} converge strongly to some fixed point of T which solves some variational inequalities. The results presented extend and improve the corresponding results of Matsushita and Kuroiwa [S. Matsushita, D. Kuroiwa, Strong convergence of averaging iterations of nonexpansive nonself-mappings, J. Math. Anal. Appl. 294 (2004) 206–214], Song and Chen [Y. Song, R. Chen, Viscosity approximation methods to Cesaro means for nonexpansive mappings, Appl. Math. Comput. 186 (2007) 1120–1128] and many authors.

Random fixed points of multivalued random operators with property (D)

The purpose of this paper is to prove some random fixed point theorem for multivalued nonexpansive non-self random operators. We will prove the existence of a random fixed point theorem for multivalued non-self random operator in the framework of a Banach space with property (D), and satisfying an inwardness condition. Our work also extends the stochastic version of the results of Kumam [P. Kumam, A note on some fixed point theorems for set-valued non-self mappings in Banach spaces. Int. Journal of Math. Analysis, to appear.], and improves the work of Kumam and Plubtieng [P. Kumam and S. Plubtieng, Random fixed point theorems for multivalued nonexpansive operators in uniformly nonsquare Banach spaces. Random Oper. and Stoch. Equ. 14(1) (2006), 35–44.].

Modified multistep iterative process for some common fixed point of a finite family of nonself asymptotically nonexpansive mappings

Let E be a real uniformly convex Banach space and K be a nonempty closed convex subset of E which is also a nonexpansive retract of E . Let T i : K → E ( i = 1 , 2 , … , r ) be nonself asymptotically nonexpansive mappings. It is proved that the modified multistep iterative sequence converges weakly and strongly to some common fixed point of { T i } i = 1 r under suitable conditions. The results of this paper improve and extend the corresponding results of [J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158 (1991) 407–413; M.O. Osilike, S.C. Aniagbosor, Weak and strong convergence theormes for fixed points of asymptotically nonexpansive mappings, Math. Comput. Modelling 32 (2000) 1181–1191; W. Takahashi, G.E. Kim, Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces, Nonlinear Anal. 3 (32) (1998) 447–454; C.E. Chidume, E.U. Ofoedu, H. Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 280 (2003) 364–374; L. Wang, Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl. 323 (2006) 550–557; B.L. Xu, M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002) 444–453]. On the other hand, it is shown that the necessary and sufficient conditions for the strong convergence of the modified multistep iterative sequence to some common fixed points of { T i } i = 1 r .