Pseudo-contractions In Hilbert Spaces Research Articles

An asynchronous inertial algorithm for solving convex feasibility problems with strict pseudo-contractions in Hilbert spaces

AbstractFinding a common point in a finite intersection of sets, say, $C=\cap _{i=1}^{n} F(T_i)$, where each $T_i$ is a non-expansive-type mapping, is a central task in mathematics as it cuts across different areas of application, such as road design and medical image reconstruction. There are many algorithms for approximating solutions of such problems. Of particular interest in the implementation of these algorithms are cost and speed. This is due to the large computations to be performed at each step of the iterative process. One of the most efficient methods that optimizes the time of computation and cost of implementation is the asynchronous-parallel algorithm method. In this paper, we prove a weak convergence theorem for the asynchronous sequential inertial (ASI) algorithm (introduced by Heaton and Censor in [H. Heaton and Y. Censor, Asynchronous sequential inertial iterations for common fixed points problems with an application to linear systems, J. Glob. Optim. 74 (2019), 95–119.] ) for strictly pseudo-contractive mappings in Hilbert spaces. Under additional mild conditions, we also obtain a strong convergence theorem. Finally, we apply the ASI algorithm to solving convex minimization problems and Hammerstein integral equations.

Weak and strong convergence theorems for the Krasnoselskij iterative algorithm in the class of enriched strictly pseudocontractive operators

Abstract In this paper, we introduce and study the class of enriched strictly pseudocontractive mappings in Hilbert spaces and extend some convergence theorems, i.e., Theorem 12 in [Brow-der, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197–228] and Theorem 3.1 in [Marino, G., Xu, H.-K., Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), no. 1, 336–346], from the class of strictly pseudocontractive mappings to that of enriched strictly pseudocontractive mappings and thus include many other important related results from literature as particular cases.

Strong convergence theorems for a countable family of pseudocontractions in Hilbert spaces

Strong convergence theorems for a countable family of pseudocontractions in Hilbert spaces

Strong convergence theorem for strict pseudo-contractions in Hilbert spaces

In this paper, inspired by Hussain et al. (Fixed Point Theory Appl. 2015:17, 2015), we study a modified Mann method to approximate strongly fixed points of strict pseudo-contractive mappings. In (Hussain et al. in Fixed Point Theory Appl. 2015:17, 2015) it is shown that the same algorithm converges strongly to a fixed point of a nonexpansive mapping under suitable hypotheses on the coefficients. Here the assumptions on the coefficients are different, as well as the techniques of the proof.

Convergence Theorems for Pointwise Asymptotically Strict Pseudo-Contractions in Hilbert Spaces

ABSTRACTA new class of contractive mappings called pointwise asymptotically ϕ-strict pseudo-contractions in Hilbert spaces is introduced and weak convergence of the sequence generated by Mann's iterative scheme to a fixed point of a uniformly Lipschitzian and pointwise asymptotically ϕ-strict pseudo-contractive mapping T in a Hilbert space is established. Also, a new kind of monotone hybrid method which is a modification of Mann's iterative scheme for finding a common fixed point of an infinitely countable family of uniformly Lipschitzian and pointwise asymptotically ϕ-strict pseudo-contractive mappings is proposed. Strong convergence of the sequence generated by the proposedmonotone hybrid method for an infinitely countable family of uniformly Lipschitzian and pointwise asymptotically ϕ-strict pseudo-contractive mappings in a Hilbert space is also shown. The results presented in this article extend and improve some known results in the literature.

An intermixed algorithm for strict pseudo-contractions in Hilbert spaces

AbstractAn intermixed algorithm for two strict pseudo-contractions in Hilbert spaces have been presented. It is shown that the suggested algorithms converge strongly to the fixed points of two strict pseudo-contractions, independently. As a special case, we can find the common fixed points of two strict pseudo-contractions in Hilbert spaces.

Construction of minimum-norm fixed points of pseudocontractions in Hilbert spaces

Abstract An iterative algorithm is introduced for the construction of the minimum-norm fixed point of a pseudocontraction on a Hilbert space. The algorithm is proved to be strongly convergent. MSC:47H05, 47H10, 47H17.

Strong convergence of three-step iteration methods for a countable family of generalized strict pseudocontractions in Hilbert spaces

Abstract In this paper, we introduce a new class of generalized strict pseudocontractions in a real Hilbert space, and we consider a three-step Ishikawa-type iteration method { z n = ( 1 − γ n ) x n + γ n T n x n , y n = ( 1 − β n ) x n + β n T n z n , x n + 1 = ( 1 − α n ) x n + α n T n y n , for finding a common fixed point of a countable family { T n } of uniformly Lipschitz generalized λ n -strict pseudocontractions. Under mild conditions imposed on the parameter sequences { α n } , { β n } and { γ n } , we prove the strong convergence of { x n } to a common fixed point of a countable family { T n } of uniformly Lipschitz generalized strict pseudocontractions. On the other hand, we also introduce three-step hybrid viscosity approximation method for finding a common fixed point of a countable family { T n } of uniformly Lipschitz generalized λ n -strict pseudocontractions with λ n = 0 , i.e., a countable family { T n } of uniformly Lipschitz pseudocontractions. Under appropriate conditions we derive the strong convergence results for this method. The results presented in this paper improve and extend the corresponding results in the earlier and recent literature. MSC:47H06, 47H09, 47J20, 47J30.

CONVERGENCE THEOREMS FOR GENERALIZED EQUILIBRIUM PROBLEMS AND ASYMPTOTICALLY κ-STRICT PSEUDO-CONTRACTIONS IN HILBERT SPACES

Abstract. In this paper, we introduce an iterative scheme for ndinga common element of the set of solutions of a generalized equilibriumproblem and the set of common xed points of a nite family of asymp-totically k-strict pseudo-contractions in Hilbert spaces. Weak and strongconvergence theorems are established for the iterative scheme. 1. IntroductionLet Hbe a real Hilbert space with inner product h;iand induced norm kk.Let Cbe a nonempty closed convex subset of H. Assume that a bifunctionF: C C!Rsatis es the following conditions:(A1) F(x;x) = 0;8x2C;(A2)Fis monotone, i.e., F(x;y) + F(y;x) 0;8x;y2C;(A3)lim t#0 F(tz+ (1 t)x;y) F(x;y);8x;y;z2C;(A4) for each x2C;y7!F(x;y) is convex and lower semicontinuous.Let A: C !H be a nonlinear mapping. Then, we consider the followinggeneralized equilibrium problem(GEP) which is to nd z2Csuch thatGEP:F(z;y) + hAz;y zi0;8y2C: (1:1)In the case of A0, this problem (1.1) reduces to the equilibrium problem(EP),which is to nd z2Csuch thatEP: F(z;y) 0;8y2C: (1:2)In the case of F 0, this problem (1.1) reduces to the variational inequalityproblem(VIP), which is to nd z2Csuch thatVIP: hAz;y zi0;8y2C: (1:3)

New Hybrid Steepest Descent Algorithms for Equilibrium Problem and Infinitely Many Strict Pseudo-Contractions in Hilbert Spaces

We propose an explicit iterative scheme for finding a common element of the set of fixed points of infinitely many strict pseudo-contractive mappings and the set of solutions of an equilibrium problem by the general iterative method, which solves the variational inequality. In the setting of real Hilbert spaces, strong convergence theorems are proved. The results presented in this paper improve and extend the corresponding results reported by some authors recently. Furthermore, two numerical examples are given to demonstrate the effectiveness of our iterative scheme.

A HYBRID PROJECTION METHOD FOR RELAXED COCOERCIVE MAPPINGS AND STRICTLY PSEUDO-CONTRACTIVE MAPPINGS

The purpose of this paper is to introduce a hybrid projection method for finding a common element of the set of solutions of a generalized equilibrium problem, the set of solutions of a variational inclusion problem and the set of common fixed points of a finite family of strict pseudo-contractions in Hilbert spaces.

Strong convergence of a new general iterative method for variational inequality problems in Hilbert spaces

Abstract In this article, we introduce a new iterative scheme with Meir-Keeler contractions for strict pseudo-contractions in Hilbert spaces. We also discuss the strong convergence theorems of the new iterative scheme for variational inequality problems in Hilbert spaces. The methods in this article are interesting and are different from those given in many other articles. Our results improve and extend the corresponding results announced by many others. MR(2000) Subject Classification: 47H09; 47H10.

General Iterative Methods for Equilibrium Problems and Infinitely Many Strict Pseudocontractions in Hilbert Spaces

We propose an implicit iterative scheme and an explicit iterative scheme for finding a common element of the set of fixed point of infinitely many strict pseudocontractive mappings and the set of solutions of an equilibrium problem by the general iterative methods. In the setting of real Hilbert spaces, strong convergence theorems are proved. Our results improve and extend the corresponding results reported by many others.

Viscosity Approximation Methods for Equilibrium Problems, Variational Inequality Problems of Infinitely Strict Pseudocontractions in Hilbert Spaces

We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of the solutions of the equilibrium problem and the set of fixed points of infinitely strict pseudocontractive mappings. Strong convergence theorems are established in Hilbert spaces. Our results improve and extend the corresponding results announced by many others recently.

Strong convergence theorems for strictly asymptotically pseudocontractive mappings in Hilbert spaces

The purpose of this paper is by using CSQ method to study the strong convergence problem of iterative sequences for a pair of strictly asymptotically pseudocontractive mappings to approximate a common fixed point in a Hilbert space. Under suitable conditions some strong convergence theorems are proved. The results presented in the paper are new which extend and improve some recent results of Acedo and Xu [Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal., 67(7), 2258–2271 (2007)], Kim and Xu [Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal., 64, 1140–1152 (2006)], Martinez-Yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal., 64, 2400–2411 (2006)], Nakajo and Takahashi [Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl., 279, 372–379 (2003)], Marino and Xu [Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl., 329(1), 336–346 (2007)], Osilike et al. [Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps. J. Math. Anal. Appl., 326, 1334–1345 (2007)], Liu [Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal., 26(11), 1835–1842 (1996)], Osilike et al. [Fixed points of demi-contractive mappings in arbitrary Banach spaces. Panamer Math. J., 12 (2), 77–88 (2002)], Gu [The new composite implicit iteration process with errors for common fixed points of a finite family of strictly pseudocontractive mappings. J. Math. Anal. Appl., 329, 766–776 (2007)].

Convergence Analysis for a System of Generalized Equilibrium Problems and a Countable Family of Strict Pseudocontractions

We introduce a new iterative algorithm for a system of generalized equilibrium problems and a countable family of strict pseudocontractions in Hilbert spaces. We then prove that the sequence generated by the proposed algorithm converges strongly to a common element in the solutions set of a system of generalized equilibrium problems and the common fixed points set of an infinitely countable family of strict pseudocontractions.

An Application of Hybrid Steepest Descent Methods for Equilibrium Problems and Strict Pseudocontractions in Hilbert Spaces

We use the hybrid steepest descent methods for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudocontraction mapping in the setting of real Hilbert spaces. We proved strong convergence theorems of the sequence generated by our proposed schemes.

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].

Convergence theorems for λ-strict pseudo-contractions in q-uniformly smooth Banach spaces

In this paper, we continue to discuss the properties of iterates generated by a strict pseudocontraction or a finite family of strict pseudo-contractions in a real q-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [Marino, G., Xu, H. K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl., 324, 336–349 (2007)]. In order to get a strong convergence theorem, we modify the normal Mann’s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C. This result extends a recent result of Kim and Xu [Kim, T. H., Xu, H. K.: Strong convergence of modified Mann iterations. Nonl. Anal., 61, 51–60 (2005)] both from nonexpansive mappings to λ-strict pseudo-contractions and from Hilbert spaces to q-uniformly smooth Banach spaces.

Strong convergence theorems for solving equilibrium problems and fixed point problems of [formula omitted]-strict pseudo-contraction mappings by two hybrid projection methods

In this paper, we introduce an iterative scheme by using the hybrid projection methods for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a ξ -strict pseudo-contraction mapping in Hilbert spaces. We obtain two strong convergence theorems under mild assumptions on parameters for the sequences generated by these processes. The results presented in the paper extend and improve some recent results of Marino and Xu [G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–346], Tada and Takahashi [A. Tada, W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem, J. Optim. Theory Appl. 133 (2007) 359–370] and Ceng et al. [L.C. Ceng, S. Al-Homidan, Q.H. Ansari, J.C. Yao, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Comput. Appl. Math. 223 (2009) 967–974] and many others.