Iterative Algorithm For Nonexpansive Mappings Research Articles

Fixed points of continuous pseudocontractive mappings on convex, closed unbounded domains

The purpose in this article is to discuss under what conditions a continuous pseudocontractive mapping, in particular a nonexpansive map, has a fixed point when its domain is a closed, convex and unbounded subset of a Banach space. In fact, we will give a characterization of this property. The Halpern iterative algorithm for nonexpansive mappings is also investigated.

The Generalized Implicit Iterative Process for Approximating Fixed Points of Nonexpansive Mappings in CAT(0) Spaces

The aim of this paper is to investigate implicit viscosity iterative algorithm for nonexpansive mappings in the framework of CAT(0) spaces. We prove strong convergence theorems of such algorithm under some suitable assumptions on the sequences of parameters involved therein. We also proved that the presented algorithm is faster than a number of existing iteration processes in literature. Our results extend and improve some recent results of Ahmad et al.[J. Appl. Math. Informatics 35 (5-6)(2017), 423-438] and others.

A modified iterative algorithm for nonexpansive mappings

A modified iterative algorithm for nonexpansive mappings

Approximating fixed points of a countable family of strict pseudocontractions in Banach spaces

We prove the strong convergence of the modified Mann-type iterative scheme for a countable family of strict pseudocontractions in q-uniformly smooth Banach spaces. Our results mainly improve and extend the results announced in (Y. Yao, H. Zhou, Y.-C. Liou, Strong convergence of a modified Krasnoselski-Mann iterative algorithm for non-expansive mappings, J. Appl. Math. Comput. 29 (2009), 383-389).

Modified Krasnoselskii–Mann iterative algorithm for nonexpansive mappings in Banach spaces

In this paper, we prove that the sequence {xn} generated by modified Krasnoselskii–Mann iterative algorithm introduced by Yao et al. [J Appl Math Comput 29:383–389, 2009] converges strongly to a fixed point of a nonexpansive mapping T in a real uniformly convex Banach space with uniformly Gâteaux differentiable norm. Furthermore, we present an example that illustrates our result in the setting of a real uniformly convex Banach space with uniformly Gâteaux differentiable norm. The results of this paper extend and improve several results presented in the literature in the recent past. Open image in new window

IMPLICIT AND EXPLICIT ALGORITHMS FOR MINIMUM-NORM FIXED POINTS OF PSEUDOCONTRACTIONS IN HILBERT SPACES

We introduce implicit and explicit iterative algorithms for the construction of fixed points of pseudocontractions $T$ in Hilbert spaces. We prove that the proposed iterative algorithms converge strongly to the minimum-norm fixed point of $T$. Moreover we show that some of the existing iterative algorithms for nonexpansive mappings fail to converge when applied to pseudocontractions.

MODIFIED HALPERN ITERATIVE ALGORITHM FOR NONEXPANSIVE MAPPINGS II

Halpern iterative algorithm is one of the most cited in the literature of approximation of fixed points of nonexpansive mappings. Different authors modified this iterative algorithm in Banach spaces to approximate fixed points of nonexpansive mappings. One of which is Yao et al. [16] modification of Halpern iterative algorithm for nonexpansive mappings in uniformly smooth Banach spaces. Unfortunately, some deficiencies are found in the Yao et al. [16] control conditions imposed on the modified iteration to obtain strong convergence. In this paper, counterexamples are constructed to prove that the strong convergence conditions of Yao et al. [16] are not sufficient and it is also proved that with some additional control conditions on the parameters strong convergence of the iteration is obtained.

Strong convergence for a countable family of strict pseudocontractions in [formula omitted]-uniformly smooth Banach spaces

We introduce a new iterative algorithm for finding a common fixed point of a countable family of strict pseudocontractions in q -uniformly smooth and uniformly convex Banach spaces. We then prove that the sequence generated by the proposed algorithm converges strongly to a common fixed point of an infinite family of strict pseudocontractions. Our results mainly improve and extend the results announced by Yao et al. [Y. Yao, Y.-C. Liou, G. Marino, Strong convergence of two iterative algorithms for nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl. 2009 (2009) 7 pages. doi:10.1155/2009/279058. Art. ID 279058].

A general composite iterative algorithm for nonexpansive mappings in Hilbert spaces

Let H be a real Hilbert space. Suppose that T is a nonexpansive mapping on H with a fixed point, f is a contraction on H with coefficient α ∈ ( 0 , 1 ) , F : H → H is a k -Lipschitzian and η -strongly monotone operator with k > 0 , η > 0 , and A : H → H is a strongly positive bounded linear operator with coefficient γ ̄ ∈ ( 1 , 2 ) . Let 0 < μ < 2 η / k 2 , 0 < γ < μ ( η − μ k 2 2 ) / α = τ / α . It is shown that the sequence { x n } generated by the following general composite iterative method: { y n = ( I − α n μ F ) T x n + α n γ f ( x n ) , x n + 1 = ( I − β n A ) T x n + β n y n , ∀ n ≥ 0 , where { α n } ⊂ [ 0 , 1 ] and { β n } ⊂ ( 0 , 1 ] , converges strongly to a fixed point x ̄ ∈ Fix ( T ) , which solves the variational inequality 〈 ( I − A ) x ̄ , x − x ̄ 〉 ≤ 0 , ∀ x ∈ Fix ( T ) .

Halpern iteration in CAT(κ) spaces

In this paper we show that an iterative sequence generated by the Halpern algorithm converges to a fixed point in the case of complete CAT(κ) spaces. Similar results for Hadamard manifolds were obtained in [Li, C., Lopez, G., Martin-Marquez, V.: Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwanese J. Math., 14, 541–559 (2010)], but we study a much more general case. Moreover, we discuss the Halpern iteration procedure for set-valued mappings.

A general iterative algorithm for nonexpansive mappings in Banach spaces

Let $E$ be a real $q$-uniformly smooth Banach space whose duality map is weakly sequentially continuous. Let $T:E\to E$ be a nonexpansive mapping with $F(T)\neq\emptyset.$ Let $A:E\to E$ be an $\eta$-strongly accretive map which is also $\kappa$-Lipschitzian. Let $f:E\to E$ be a contraction map with coefficient $0\lt \alpha\lt1.$ Let a sequence $\{y_{n}\}$ be defined iteratively by $y_{0}\in E,~~ y_{n+1}=\alpha_n\gamma f(y_n)+(I-\alpha_n\mu A)Ty_n,n\geq0,$ where $\{\alpha_n\},~~\gamma$ and $\mu$ satisfy some appropriate conditions. Then, we prove that $\{y_{n}\}$ converges strongly to the unique solution $x^{*} \in F(T)$ of the variational inequality $\langle(\gamma f-\mu A)x^{*},j(y-x^{*})\rangle\leq0,~\forall~y\in F(T).$ Convergence of the correspondent implicit scheme is also proved without the assumption that $E$ has weakly sequentially continuous duality map. Our results are applicable in $l_{p}$ spaces, $1\lt p \lt \infty$.

MODIFIED HALPERN ITERATIVE ALGORITHMS FOR NONEXPANSIVE MAPPINGS

Halpern iterative algorithm is one of the most cited in the literature of approximation of fixed points of nonexpansive mappings. Different authors modified this iterative algorithm in Banach spaces to approximate fixed points of nonexpansive mappings. One of which is Hu [8] and Yao et al [21] modification of Halpern iterative algorithm for nonexpansive mappings in Banach spaces. It is the purpose of this paper to thoroughly analyze this modification and its convergence conditions. Unfortunately, Hu [8] and Yao et al [21] control conditions imposed on the modified Halpern iterative algorithm to have strong convergence are found to be not sufficient. In this paper, counterexamples are constructed to prove that the strong convergence conditions of Hu [8] and Yao et al [21] are not sufficient. It is also proved that with some additional conditions on the control parameters, strong convergence of the defined iterative algorithm is obtained in different Banach space settings.

A general iterative algorithm for nonexpansive mappings in Hilbert spaces

Let H be a real Hilbert space. Suppose that T is a nonexpansive mapping on H with a fixed point, f is a contraction on H with coefficient 0 < α < 1 , and F : H → H is a k -Lipschitzian and η -strongly monotone operator with k > 0 , η > 0 . Let 0 < μ < 2 η / k 2 , 0 < γ < μ ( η − μ k 2 2 ) / α = τ / α . We proved that the sequence { x n } generated by the iterative method x n + 1 = α n γ f ( x n ) + ( I − μ α n F ) T x n converges strongly to a fixed point x ̃ ∈ F i x ( T ) , which solves the variational inequality 〈 ( γ f − μ F ) x ̃ , x − x ̃ 〉 ≤ 0 , for x ∈ F i x ( T ) .

ITERATIVE ALGORITHMS FOR NONEXPANSIVE MAPPINGS ON HADAMARD MANIFOLDS

Two iterative algorithms for nonexpansive mappings on Hadamard manifolds, which are extensions of the well-known Halpern's and Mann's algorithms in Euclidean spaces, are proposed and proved to be convergent to a fixed point of the mapping. Some numerical examples are provided.

Modified Iterative Algorithms for Nonexpansive Mappings

Let H be a real Hilbert space, let S, T be two nonexpansive mappings such that F(S)∩F(T)≠∅, let f be a contractive mapping, and let A be a strongly positive linear bounded operator on H. In this paper, we suggest and consider the strong converegence analysis of a new two-step iterative algorithms for finding the approximate solution of two nonexpansive mappings as xn+1=βnxn+(1−βn)Syn, yn=αnγf(xn)+(I−αnA)Txn, n≥0 is a real number and {αn}, {βn} are two sequences in (0,1) satisfying the following control conditions: (C1) lim⁡n→∞ αn=0, (C3) 0&lt;lim⁡inf⁡n→∞ βn≤lim⁡sup⁡n→∞ βn&lt;1, then ‖xn+1−xn‖→0. We also discuss several special cases of this iterative algorithm.

Strong Convergence of Two Iterative Algorithms for Nonexpansive Mappings in Hilbert Spaces

We introduce two iterative algorithms for nonexpansive mappings in Hilbert spaces. We prove that the proposed algorithms strongly converge to a fixed point of a nonexpansive mapping .

On Picard iterations for strongly accretive and strongly pseudo-contractive Lipschitz mappings

In this note, we speed up the convergence of the Picard sequence of iterations for strongly accretive and strongly pseudo-contractive mappings. Our results improve the results of Chidume [C.E. Chidume, Picard iteration for strongly accretive and strongly pseudo-contractive Lipschitz maps, ICTP Preprint no. IC2000098; C.E. Chidume, Iterative Algorithms for Non-expansive Mappings and Some of Their Generalizations, in: Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80th Birthday, vol. 1, 2, Kluwer Acad. Publ, Dordrecht, 2003, pp. 383–429], and Liu [L. Liu, Approximation of fixed points of a strictly pseudo-contractive mapping, Proc. Amer. Math. Soc. 125 (2) (1997) 1363–1366], and some other known results. The technique of the proof, presented in this paper, is different from the technique used by Chidume.

Strong convergence of a modified Krasnoselski-Mann iterative algorithm for non-expansive mappings

In this paper, we introduce a modified Krasnoselski-Mann iterative algorithm for non-expansive mappings. Furthermore, we prove that the proposed iterative algorithm converges strongly to a fixed point of a non-expansive mapping in Hilbert spaces.

Iterative Algorithms for Nonexpansive Mappings

We suggest and analyze two new iterative algorithms for a nonexpansive mapping T in Banach spaces. We prove that the proposed iterative algorithms converge strongly to some fixed point of T.

On Mann implicit iterations for strongly accretive and strongly pseudo-contractive mappings

For a Lipschitz strongly accretive map considered by Chidume in [C.E. Chidume, Picard iteration for strongly accretive and strongly pseudo-contractive Lipschitz maps, ICTP Preprint No. IC2000098; C.E. Chidume, Iterative algorithms for nonexpansive mappings and some of their generalizations, Nonlinear analysis and applications: to V. Lakshmikantam on his 80th birthday, vols. 1 and 2, Kluwer Acad. Publ., Dordrecht, 2003, pp. 383–429], it is known that a classical Picard-type iteration process converges strongly to a zero of the operator. He also proved that the rate of convergence, in this case, is at least as fast as a geometric progression. In this paper we study the Mann implicit iteration sequence for strongly accretive and strongly pseudo-contractive mappings. We showed that this implicit scheme gives better convergence rate estimate. Presented results improve the corresponding results of [C.E. Chidume, Picard iteration for strongly accretive and strongly pseudo-contractive Lipschitz maps, ICTP Preprint No. IC2000098; C.E. Chidume, Iterative algorithms for nonexpansive mappings and some of their generalizations, Nonlinear analysis and applications: to V. Lakshmikantam on his 80th birthday, vols. 1 and 2, Kluwer Acad. Publ., Dordrecht, 2003, pp. 383–429; L. Liu, Approximation of fixed points of a strictly pseudo-contractive mapping, Proc. Am. Math. Soc. 125 (2) (1997) 1363–1366; W.R. Sastry, G.V.R. Babu, Approximation of fixed points of strictly pseudo-contractive mappings on arbitrary closed, convex sets in a Banach space, Proc. Amer. Math. Soc. 128 (2000) 2907–2909; Y. Song, R. Chen, Viscosity approximative methods to Cesàro means for non-expansive mappings, Appl. Math. Comput. 186 (2) (2007) 1120–1128].