Differentiable Norm Research Articles

Strong convergence theorems for a sequence of nonexpansive mappings with gauge functions

Abstract In this paper, we first prove a path convergence theorem for a nonexpansive mapping in a reflexive and strictly convex Banach space which has a uniformly Gˆateaux differentiable norm and admits the duality mapping jφ, where φ is a gauge function on [0,∞). Using this result, strong convergence theorems for common fixed points of a countable family of nonexpansive mappings are established.

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

Regularization methods for nonlinear Ill-posed equations involving m-accretive mappings in Banach spaces

In this paper we prove strong convergence of the Browder-Tikhonov regularization method and the regularization inertial proximal point algorithm to a solution of nonlinear ill-posed equations involving m-accretive mappings in real, reflexive, and strictly convex Banach spaces with a uniformly Gâteaux differentiable norm without weak sequential continuous duality mapping.

Mann-Type Viscosity Approximation Methods for Multivalued Variational Inclusions with Finitely Many Variational Inequality Constraints in Banach Spaces

We introduce Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of strictly pseudocontractive mappings and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

Composite Iterative Algorithms for Variational Inequality and Fixed Point Problems in Real Smooth and Uniformly Convex Banach Spaces

We introduce composite implicit and explicit iterative algorithms for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a real smooth and uniformly convex Banach space. These composite iterative algorithms are based on Korpelevich's extragradient method and viscosity approximation method. We first consider and analyze a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another composite explicit iterative algorithm in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures.

Viscosity Approximation Methods for Two Accretive Operators in Banach Spaces

We introduced a viscosity iterative scheme for approximating the common zero of two accretive operators in a strictly convex Banach space which has a uniformly Gâteaux differentiable norm. Some strong convergence theorems are proved, which improve and extend the results of Ceng et al. (2009) and some others.

Iterative Methods for Pseudocontractive Mappings in Banach Spaces

LetEa reflexive Banach space having a uniformly Gâteaux differentiable norm. LetCbe a nonempty closed convex subset ofE,T:C→Ca continuous pseudocontractive mapping withF(T)≠∅, andA:C→Ca continuous bounded strongly pseudocontractive mapping with a pseudocontractive constantk∈(0,1). Let{αn}and{βn}be sequences in(0,1)satisfying suitable conditions and for arbitrary initial valuex0∈C, let the sequence{xn}be generated byxn=αnAxn+βnxn-1+(1-αn-βn)Txn, n≥1.If either every weakly compact convex subset ofEhas the fixed point property for nonexpansive mappings orEis strictly convex, then{xn}converges strongly to a fixed point ofT, which solves a certain variational inequality related toA.

Application of Gabor Dictionary and SL0 Norm in Disguise Face Recognition

A fast sparse representation face recognition algorithm based on Gabor dictionary and SL0 norm is proposed in this paper. The Gabor filters, which could effectively extract local directional features of the image at multiple scales, are less sensitive to variations of illumination, expression and camouflage. SL0 algorithm, with the advantages of calculation speed，require fewer measurement values by continuously differentiable function approximation L0 norm and reconstructed sparse signal by minimizing the approximate L0 norm. The algorithm obtain the local feature face by extracting the Gabor face feature, reduce the dimensions by principal component analysis, fast sparse classify by the SL0 norm. Under camouflage condition, The algorithm block the Gabor facial feature and improve the speed of formation of the Gabor dictionary. The experimental results on AR face database show that the proposed algorithm can improve recognition speed and recognition rate to some extent and can generalize well to the face recognition, even with a few training image per class.

Hybrid and Relaxed Mann Iterations for General Systems of Variational Inequalities and Nonexpansive Mappings

We introduce hybrid and relaxed Mann iteration methods for a general system of variational inequalities with solutions being also common solutions of a countable family of variational inequalities and common fixed points of a countable family of nonexpansive mappings in real smooth and uniformly convex Banach spaces. Here, the hybrid and relaxed Mann iteration methods are based on Korpelevich’s extragradient method, viscosity approximation method, and Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for hybrid and relaxed Mann iteration algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gateaux differentiable norm. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

Hybrid Extragradient Methods for Finding Zeros of Accretive Operators and Solving Variational Inequality and Fixed Point Problems in Banach Spaces

We introduce and analyze hybrid implicit and explicit extragradient methods for finding a zero of an accretive operator and solving a general system of variational inequalities and a fixed point problem of an infinite family of nonexpansive self-mappings in a uniformly convex Banach spaceXwhich has a uniformly Gateaux differentiable norm. We establish some strong convergence theorems for hybrid implicit and explicit extra-gradient algorithms under suitable assumptions. Furthermore, we derive the strong convergence of hybrid implicit and explicit extragradient algorithms for finding a common element of the set of zeros of an accretive operator and the common fixed point set of an infinite family of nonexpansive self-mappings and a self-mapping whose complement is strictly pseudocontractive and strongly accretive inX. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

Viscosity Approximation Methods and Strong Convergence Theorems for the Fixed Point of Pseudocontractive and Monotone Mappings in Banach Spaces

Suppose thatCis a nonempty closed convex subset of a real reflexive Banach spaceEwhich has a uniformly Gateaux differentiable norm. A viscosity iterative process is constructed in this paper. A strong convergence theorem is proved for a common element of the set of fixed points of a finite family of pseudocontractive mappings and the set of solutions of a finite family of monotone mappings. And the common element is the unique solution of certain variational inequality. The results presented in this paper extend most of the results that have been proposed for this class of nonlinear mappings.

On the computational content of convergence proofs via Banach limits

This paper addresses new developments in the ongoing proof mining programme, i.e. the use of tools from proof theory to extract effective quantitative information from prima facie ineffective proofs in analysis. Very recently, the current authors developed a method of extracting rates of metastability (as defined by Tao) from convergence proofs in nonlinear analysis that are based on Banach limits and so (for all that is known) rely on the axiom of choice. In this paper, we apply this method to a proof due to Shioji and Takahashi on the convergence of Halpern iterations in spaces X with a uniformly Gâteaux differentiable norm. We design a logical metatheorem guaranteeing the extractability of highly uniform rates of metastability under the stronger condition of the uniform smoothness of X. Combined with our method of eliminating Banach limits, this yields a full quantitative analysis of the proof by Shioji and Takahashi. We also give a sufficient condition for the computability of the rate of convergence of Halpern iterations.

Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type with Lipschitz and Bounded Nonlinear Operators

Let E be a reflexive real Banach space with uniformly Gâteaux differentiable norm and F, K : be Lipschitz accretive maps with Suppose that the Hammerstein equation has a solution. An explicit iteration method is shown to converge strongly to a solution of the equation. No invertibility assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our theorems are significant improvements on important recent results (e.g., (Chiume and Djitte, 2012)).

Viscosity approximation methods for pseudo-contractive semigroups in Banach spaces

We study the strong convergence of two viscosity iteration processes for pseudo-contractive semigroup and for ϕ-strongly pseudo-contractive mapping in uniformly convex Banach spaces with uniformly Gâteaux differentiable norm. As special cases, we get strong convergence of two viscosity iteration processes for approximating common fixed points of nonexpansive semigroups in certain Banach spaces. The results presented in this paper extend and generalize previous results.

Convergence of iteration schemes for continuous pseudocontractive multi-valued mappings

In this paper, an iteration scheme for continuous pseudocontractive multi-valued mappings is introduced. The strong convergence of this iteration scheme is proved in reflexive real Banach spaces with a uniformly Gâteaux differentiable norm. This iteration scheme can be viewed as a multi-valued version of the corresponding one of Song and Chen [Y. Song, R. Chen, Convergence theorems of iterative algorithms for continuous pseudocontractive mappings, Nonlinear Anal. 67 (2007) 486–497].

Existence and approximation of fixed points of nonlinear mappings in spaces with weak uniform normal structure

It is shown that in a Banach space X with weak uniform normal structure, every demicontinuous asymptotically regular nearly Lipschitzian self-mapping T with lim supn→∞η(Tn)<WCS(X) defined on a weakly compact convex subset C of X satisfies the (ω)-fixed point property. We show that if X has a uniformly Gâteaux differentiable norm, then the set of fixed points of every asymptotically pseudocontractive nearly nonexpansive mapping T is nonempty and a sunny nonexpansive retract of C. Further, we study the approximation of fixed points of T by Halpern type iteration process: xn+1=αnu+(1−αn)Tnxnfor alln∈N, where u∈C and {αn} is a sequence in (0,1) satisfying appropriate conditions. Our results improve several known existence and convergence fixed point theorems in general Banach spaces for a wider class of nonlinear mappings which are not necessarily Lipschitzian.

The general iterative methods for nonexpansive semigroups in Banach spaces

Let E be a real reflexive strictly convex Banach space which has uniformly Gâteaux differentiable norm. Let $${\mathcal{S} = \{T(s): 0 \leq s 1 and $${0 < \gamma < \min\left\{\frac{\delta}{\alpha}, \frac{1-\sqrt{ \frac{1-\delta}{\lambda} }}{\alpha} \right\} }$$ . When the sequences of real numbers {? n } and {t n } satisfy some appropriate conditions, the three iterative processes given as follows : $${\left.\begin{array}{ll}{x_{n+1} = \alpha_n \gamma f(x_n) + (I - \alpha_n F)T(t_n)x_n,\quad n\geq 0,}\\ {y_{n+1} = \alpha_n \gamma f(T(t_n)y_n) + (I - \alpha_n F)T(t_n)y_n,\quad n\geq 0,}\end{array}\right.}$$ and $$ z_{n+1} = T(t_n)( \alpha_n \gamma f(z_n) + (I - \alpha_n F)z_n),\quad n\geq 0 $$ converge strongly to $${\tilde{x}}$$ , where $${\tilde{x}}$$ is the unique solution in $${Fix(\mathcal{S})}$$ of the variational inequality $${ \langle (F - \gamma f)\tilde {x}, j(x - \tilde{x}) \rangle \geq 0,\quad x\in Fix(\mathcal{S}).}$$ Our results extend and improve corresponding ones of Li et al. (Nonlinear Anal 70:3065---3071, 2009) and Chen and He (Appl Math Lett 20:751---757, 2007) and many others.

A Unified Hybrid Iterative Method for Solving Variational Inequalities Involving Generalized Pseudocontractive Mappings

We study in this paper the existence and the approximation of solutions of variational inequalities involving generalized pseudocontractive mappings in Banach spaces. The convergence analysis of a proposed hybrid iterative method for approximating common zeros or fixed points of a possibly infinitely countable or uncountable family of such operators will be conducted within the conceptual framework of the “viscosity approximation technique” in reflexive Banach spaces with uniform Gâteaux differentiable norms. This technique should make existing or new results in solving variational inequalities more applicable.

Convergence Theorem for a Family of Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces

Let 𝐸 be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let 𝔍={𝑇(𝑡)∶𝑡≥0} be a family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup of 𝐸, with functions 𝑢,𝑣∶[0,∞)→[0,∞). Let 𝐹∶=𝐹(𝔍)=∩𝑡≥0𝐹(𝑇(𝑡))≠∅ and 𝑓∶𝐾→𝐾 be a weakly contractive map. For some positive real numbers 𝜆 and 𝛿 satisfying 𝛿

Strong Convergence Theorems for Nonexpansive Semigroups and Variational Inequalities in Banach Spaces

Let X be a uniformly convex Banach space and 𝒮 = {T(s) : 0 ≤ s &lt; ∞} be a nonexpansive semigroup such that F(𝒮) = ⋂s&gt;0F(T(s)) ≠ ∅. Consider the iterative method that generates the sequence {xn} by the algorithm , where {αn}, {βn}, and {sn} are three sequences satisfying certain conditions, f : C → C is a contraction mapping. Strong convergence of the algorithm {xn} is proved assuming X either has a weakly continuous duality map or has a uniformly Gâteaux differentiable norm.