Showing papers on "Gâteaux derivative published in 2006"

Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces

Abstract: Viscosity approximation methods for a family of finite nonexpansive mappings are established in Banach spaces. The main theorems extend the main results of Moudafi [Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46-55] and Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279-291] to the case of finite mappings. Our results also improve and unify the corresponding results of Bauschke [The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 202 (1996) 150-159], Browder [Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces, Archiv. Ration. Mech. Anal. 24 (1967) 82-90], Cho et al. [Some control conditions on iterative methods, Commun. Appl. Nonlinear Anal. 12 (2) (2005) 27-34], Ha and Jung [Strong convergence theorems for accretive operators in Banach spaces, J. Math. Anal. Appl. 147 (1990) 330-339], Halpern [Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957-961], Jung [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509-520], Jung et al. [Iterative schemes with some control conditions for a family of finite nonexpansive mappings in Banach space, Fixed Point Theory Appl. 2005 (2) (2005) 125-135], Jung and Kim [Convergence of approximate sequences for compositions of nonexpansive mappings in Banach spaces, Bull. Korean Math. Soc. 34 (1) (1997) 93-102], Lions [Approximation de points fixes de contractions, C.R. Acad. Sci. Ser. A-B, Paris 284 (1977) 1357-1359], O'Hara et al. [Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417-1426], Reich [Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 287-292], Shioji and Takahashi [Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (12) (1997) 3641-3645], Takahashi and Ueda [On Reich's strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984) 546-553], Wittmann [Approximation of fixed points of nonexpansive mappings, Arch. Math. 59 (1992) 486-491], Xu [Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2) (2002) 240-256], and Zhou et al. [Strong convergence theorems on an iterative method for a family nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput., in press] among others.

A Gâteaux differentiability space that is not weak Asplund

Abstract: In this paper we construct a Gâteaux differentiability space that is not a weak Asplund space. Thus we answer a question raised by David Larman and Robert Phelps from 1979.

The mathematics of functional differentiation under conservation constraint

Abstract: The mathematics of K-conserving functional differentiation, with K being the integral of some invertible function of the functional variable, is clarified. The most general form for constrained functional derivatives is derived from the requirement that two functionals that are equal over a restricted domain have equal derivatives over that domain. It is shown that the K-conserving derivative formula is the one that yields no effect of K-conservation on the differentiation of K-independent functionals, which gives the basis for its generalization for multiple constraints. Connections with the derivative with respect to the shape of the functional variable and with the shape-conserving derivative, together with their use in the density-functional theory of many-electron systems, are discussed. Yielding an intuitive interpretation of K-conserving functional derivatives, it is also shown that K-conserving derivatives emerge as directional derivatives along K-conserving paths, which is achieved via a generalization of the Gateaux derivative for that kind of paths. These results constitute the background for the practical application of K-conserving differentiation.

Scalar Derivatives in Hilbert Spaces

Abstract: Computational formulae for scalar derivatives of mappings and pairs of mappings in the direction of a set will be given. These formulae are very important for explicitly verifying conditions of existence for fixed point theorems, surjectivity theorems, integral equations, variational inequalities and complementarity problems under additional differentiability conditions. To emphasize this idea at the end of the paper we give an application to complementarity problems. Some theorems which extend the correspondence between monotone mappings and scalar derivatives from Euclidean spaces to Hilbert spaces will also be given.

On uniformly Gâteaux smooth norms and normal structure

Abstract: It is shown that every separable Banach space admits an equivalent norm that is uniformly Gâteaux smooth and yet lacks asymptotic normal structure.

Gâteaux derivative and orthogonality in -classes.

Abstract: The general problem in this paper is minimizing the Cp norm of suitable affine mappings from B(H) to Cp, using convex and differential analysis (Gateaux derivative) as well as input from operator theory. The mappings considered generalize the so-called elementary operators and in particular the generalized derivations, which are of great interest by themselves. The main results obtained characterize global minima in terms of (Banach space) orthogonality, and constitute an interesting combination of infinite-dimensional differential analysis, operator theory and duality. Note that the results obtained generalize all results in the literature concerning operator which are orthogonal to the range of a derivation and the techniques used have not been done by other authors.

Antiproximinal convex bounded sets in the space c0(Γ) equipped with the day norm

Abstract: We construct a convex smooth antiproximinal set in any infinite-dimensional space c 0(Γ) equipped with the Day norm; moreover, the distance function to the set is Gâteaux differentiable at each point of the complement.