Differentiable Norm Research Articles

A novel algorithm with an inertial technique for fixed points of nonexpansive mappings and zeros of accretive operators in Banach spaces

<abstract><p>The purpose of this paper was to prove that a novel algorithm with an inertial approach, used to generate an iterative sequence, strongly converges to a fixed point of a nonexpansive mapping in a real uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Furthermore, zeros of accretive mappings were obtained. The proposed algorithm has been implemented and tested via numerical simulation in MATLAB. The simulation results showed that the algorithm converges to the optimal configurations and shows the effectiveness of the proposed algorithm.</p></abstract>

Gâteaux or Fréchet differentiable norms on duals of interpolation spaces

AbstractLet be an interpolation couple of Banach spaces, let , and let be the corresponding real interpolation space. We consider a norm on the space , equivalent to the usual interpolation norm, and we show that if the norms of the dual spaces and are Gâteaux (resp., Fréchet) differentiable, then the dual norm on has the same property. If we only assume that one of the two duals is Gâteaux (or Fréchet) differentiable, we are able to show that there is an equivalent Gâteaux (or Fréchet) differentiable dual norm on the space .

REGULARITY OF AML FUNCTIONS IN TWO-DIMENSIONAL NORMED SPACES

AbstractSavin [‘ $\mathcal {C}^{1}$ regularity for infinity harmonic functions in two dimensions’, Arch. Ration. Mech. Anal.3(176) (2005), 351–361] proved that every planar absolutely minimizing Lipschitz (AML) function is continuously differentiable whenever the ambient space is Euclidean. More recently, Peng et al. [‘Regularity of absolute minimizers for continuous convex Hamiltonians’, J. Differential Equations274 (2021), 1115–1164] proved that this property remains true for planar AML functions for certain convex Hamiltonians, using some Euclidean techniques. Their result can be applied to AML functions defined in two-dimensional normed spaces with differentiable norm. In this work we develop a purely non-Euclidean technique to obtain the regularity of planar AML functions in two-dimensional normed spaces with differentiable norm.

Gâteaux or Fréchet differentiable norms in duals of complex interpolation spaces

Gâteaux or Fréchet differentiable norms in duals of complex interpolation spaces

Modified inertial Ishikawa iterations for fixed points of nonexpansive mappings with an application

<abstract><p>This manuscript aims to prove that the sequence $ \{\nu _{n}\} $ created iteratively by a modified inertial Ishikawa algorithm converges strongly to a fixed point of a nonexpansive mapping $ Z $ in a real uniformly convex Banach space with uniformly Gâteaux differentiable norm. Moreover, zeros of accretive mappings are obtained as an application. Our results generalize and improve many previous results in this direction. Ultimately, two numerical experiments are given to illustrate the behavior of the purposed algorithm.</p></abstract>

On alpha -Firmly Nonexpansive Operators in r-Uniformly Convex Spaces

We introduce the class of alpha -firmly nonexpansive and quasi alpha -firmly nonexpansive operators on r-uniformly convex Banach spaces. This extends the existing notion from Hilbert spaces, where alpha -firmly nonexpansive operators coincide with so-called alpha -averaged operators. For our more general setting, we show that alpha -averaged operators form a subset of alpha -firmly nonexpansive operators. We develop some basic calculus rules for (quasi) alpha -firmly nonexpansive operators. In particular, we show that their compositions and convex combinations are again (quasi) alpha -firmly nonexpansive. Moreover, we will see that quasi alpha -firmly nonexpansive operators enjoy the asymptotic regularity property. Then, based on Browder’s demiclosedness principle, we prove for r-uniformly convex Banach spaces that the weak cluster points of the iterates x_{n+1}:=Tx_{n} belong to the fixed point set {{,mathrm{Fix},}}T whenever the operator T is nonexpansive and quasi alpha -firmly. If additionally the space has a Fréchet differentiable norm or satisfies Opial’s property, then these iterates converge weakly to some element in {{,mathrm{Fix},}}T. Further, the projections P_{{{,mathrm{Fix},}}T}x_n converge strongly to this weak limit point. Finally, we give three illustrative examples, where our theory can be applied, namely from infinite dimensional neural networks, semigroup theory, and contractive projections in L_p, p in (1,infty ) backslash {2} spaces on probability measure spaces.

Generalized variational inequalities for maximal monotone operators

In this paper, we present some new results on the existence of solutions of generalized variational inequalities for set-valued mappings in reflexive Banach spaces with Fréchet differentiable norms. Moreover, the structure of the solution sets is investigated. The result obtained in this paper improves and extends the ones announced by Fang and Peterson [S. C. Fang and E. L. Peterson, Generalized Variational Inequalities, J. Optim. Theory Appl., 38 (1982), 363-383] and others.

Strong Convergence of Mann’s Iteration Process in Banach Spaces

Mann’s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fréchet differentiable norm), but not strongly even in a Hilbert space. Strong convergence is therefore a nontrivial problem. In this paper we provide certain conditions either on the underlying space or on the mapping under investigation so as to guarantee the strong convergence of Mann’s iteration process and its variants.

Browder’s Convergence Theorem for Multivalued Mappings in Banach Spaces without the Endpoint Condition

We prove Browder’s convergence theorem for multivalued mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm by using the notion of diametrically regular mappings. Our results are significant improvement on results of Jung (2007) and Panyanak and Suantai (2020).

Steepest-descent Ishikawa iterative methods for a class of variational inequalities in Banach spaces

In this paper, for finding a fixed point of a nonexpansive mapping in either uniformly smooth or reflexive and strictly convex Banach spaces with a uniformly G?teaux differentiable norm, we present a new explicit iterative method, based on a combination of the steepest-descent method with the Ishikawa iterative one. We also show its several particular cases one of which is the composite Halpern iterative method in literature. The explicit iterative method is also extended to the case of infinite family of nonexpansive mappings. Numerical experiments are given for illustration.

New Algorithms for a Class of Accretive Variational Inequalities in Banach Spaces

In this paper, for finding a common zero of a finite family of m-accretive mappings in uniformly convex Banach spaces with a uniformly Gâteaux differentiable norm, we propose an implicit iteration algorithm and an explicit one, based on a convex combination of the steepest-descent method and a composition of resolvents. We also show that our main algorithm contains some iterative ones in literature as special cases. Finally, we give numerical examples for illustration.

A hybrid method for solving variational inequalities over the common fixed point sets of infinite families of nonexpansive mappings in Banach spaces

ABSTRACT In this paper, we introduce a hybrid method, a combination of the steepest-descent method and the Krasnosel'skii-Mann one, for solving a variational inequality over the set of common fixed points of an infinite family of nonexpansive mappings in Banach spaces under two different conditions on the Banach space, either a uniformly smooth Banach space or a reflexive and strictly convex one with a uniformly Gâteaux differentiable norm, without imposing the sequential weak continuity of the normalized duality mapping. The method is an improvement and extension of some other published results. We also give a numerical example to illustrate the convergence analysis of the proposed method.

Convergence of a hybrid viscosity approximation method for finding zeros of m-accretive operators

For finding a zero of an m-accretive operator in a real Banach space, we consider an implicit hybrid viscosity approximation method and show that its explicit variant converges strongly to a zero under weaker assumptions than the ones used recently by Ceng et al. (Numer. Func. Anal. Opt. 35(2), 142–165, 2012). First, we examine the case of spaces which are uniformly smooth or reflexive and strictly convex with a uniformly Gâteaux differentiable norm. Afterwards, we improve our convergence results when the space is uniformly convex. Furthermore, we show that the strong convergence of some modified Halpern and Krasnosel’skii–Mann type methods can also be deduced from our results. Finally, a numerical example is also given to illustrate the convergence analysis of the considered method.

Application of the product net technique and Kadec\u2013Klee property to study nonlinear ergodic theorems and weak convergence theorems in uniformly convex multi-Banach spaces

Let Y be a uniformly convex multi-Banach space which has not a Frechet differentiable norm. We use the technique of product net to obtain the nonlinear ergodic theorems in Y. Finally, let the dual of uniformly convex multi-Banach space have the Kadec–Klee property, we instate the weak convergence theorem in the case of reversible semi-group.

Steepest‐descent proximal point algorithms for a class of variational inequalities in Banach spaces

AbstractIn this paper, we present a new approach to the problem of finding a common zero for a system of m‐accretive mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. We propose an implicit iteration method and two explicit ones, based on compositions of resolvents with the steepest‐descent method. We show that our results contain some iterative methods in literature as special cases. An extension of the Xu's regularization method for the proximal point algorithm from Hilbert spaces onto Banach ones under simple conditions of convergence and a new variant for the method of alternating resolvents are obtained. Numerical experiments are given to affirm efficiency of the methods.

A New Simple Parallel Iteration Method for a Class of Variational Inequalities

In this paper, we propose a new simple parallel iterative method to find a solution for variational inequalities over the set of common fixed points of an infinite family of nonexpansive mappings on real reflexive and strictly convex Banach spaces with a uniformly Gâteaux differentiable norm. Our parallel iterative method is simpler than the one proposed by Buong et al. (Numer. Algorithms 72, 467–481 2016). An iterative method of Halpern type for common zeros of an infinite family of m-accretive mappings is shown as a special case of our result. Two numerical examples are also given to illustrate the effectiveness and superiority of the proposed algorithm.

STRONG CONVERGENCE OF GENERAL ITERATIVE ALGORITHMS FOR NONEXPANSIVE MAPPINGS IN BANACH SPACES

In this paper, we introduce two general iterative algorithms (one implicit algorithm and other explicit algorithm) for nonexpansive mappings in a reflexive Banach space with a uniformly <TEX>$G{\hat{a}}teaux$</TEX> differentiable norm. Strong convergence theorems for the sequences generated by the proposed algorithms are established.

Correlation between the norm and the geometry of minimal networks

The paper is concerned with the inverse problem of the minimal Steiner network problem in a normed linear space. Namely, given a normed space in which all minimal networks are known for any finite point set, the problem is to describe all the norms on this space for which the minimal networks are the same as for the original norm. We survey the available results and prove that in the plane a rotund differentiable norm determines a distinctive set of minimal Steiner networks. In a two-dimensional space with rotund differentiable norm the coordinates of interior vertices of a nondegenerate minimal parametric network are shown to vary continuously under small deformations of the boundary set, and the turn direction of the network is determined. Bibliography: 15 titles.

Strong convergence of modified iterative algorithm for family of asymptotically nonexpansive mappings

In this paper we introduce new modified implicit and explicit algorithms and prove strong convergence of the two algorithms to a common fixed point of a family of uniformly asymptotically regular asymptotically nonexpansive mappings in a real reflexive Banach space with a uniformly G$hat{a}$teaux differentiable norm. Our result is applicable in $L_{p}(ell_{p})$ spaces, $1 < p <infty$ and consequently in sobolev spaces.

A steepest-descent Krasnosel’skii–Mann algorithm for a class of variational inequalities in Banach spaces

In this paper, in order to solve a variational inequality problem over the fixed point set of a nonexpansive mapping on uniformly smooth or reflexive and strictly convex Banach spaces with a uniformly Gâteaux differentiable norm, we investigate an explicit iteration method, based on the steepest-descent and Krasnosel’skii–Mann algorithms. We also show that some modifications of the last and Halpern-type algorithms are special cases of our result.