Infinite dimensional Banach spaces of functions with nonlinear properties

The aim of this paper is to show that there exist infinite dimensional Banach spaces of functions that, except for 0, satisfy properties that apparently should be destroyed by the linear combination of two of them. Three of these spaces are: a Banach space of differentiable functions on ?ⁿ failing the Denjoy‐Clarkson property; a Banach space of non Riemann integrable bounded functions, but with antiderivative at each point of an interval; a Banach space of infinitely differentiable functions that vanish at infinity and are not the Fourier transform of any Lebesgue integrable function (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dual variational methods in critical point theory and applications

This paper contains some general existence theorems for critical points of a continuously differentiable functional I on a real Banach space. The strongest results are for the case in which I is even. Applications are given to partial differential and integral equations.     

Morse theory on Banach manifolds

LetM be aC ²-Finsler manifold modeled on a Banach space, and letf be aC ²-real-valued function defined onM. Using theA-gradient vector field which was introduced in [31] we give a suitable definition for nondegenegacy of critical points off, then generalize the Morse handle-body decomposition theorem and the Morse inequalities to a kind of Banach manifolds. A generalization in the reflexive case has been done in [31].     

Origin and evolution of the Palais–Smale condition in critical point theory

In 1963–64, Palais and Smale have introduced a compactness condition, namely condition (C), on real functions of class C ¹ defined on a Riemannian manifold modeled upon a Hilbert space, in order to extend Morse theory to this frame and study nonlinear partial differential equations. This condition and some of its variants have been essential in the development of critical point theory on Banach spaces or Banach manifolds, and are referred as Palais–Smale-type conditions. The paper describes their evolution.     

Discrete chaos in Banach spaces

This paper is concerned with chaos in discrete dynamical systems governed by continuously Frech@t differentiable maps in Banach spaces. A criterion of chaos induced by a regular nondegenerate homoclinic orbit is established. Chaos of discrete dynamical systems in the n-dimensional real space is also discussed, with two criteria derived for chaos induced by nondegenerate snap-back repellers, one of which is a modified version of Marotto's theorem. In particular, a necessary and sufficient condition is obtained for an expanding fixed point of a differentiate map in a general Banach space and in an n-dimensional real space, respectively. It completely solves a long-standing puzzle about the relationship between the expansion of a continuously differentiable map near a fixed point in an n-dimensional real space and the eigenvalues of the Jacobi matrix of the map at the fixed point.     

一类Banach空间中的分析性质及反例

本文对具有Schauder基的无穷维Banach空间上的映射定义偏导数,并讨论f的可徽性与偏导数存在并连续的关系;同时,本文还时论象所在空间为具有Schauder基的空间时,映射f与坐标映射f_i在可微性、连续性方面的关系.     

APPROXIMATION OF COMMON FIXED POINT OF FAMILIES OF NONLINEAR MAPPINGS WITH APPLICATIONS

It is our purpose in this paper to show that some results obtained in uniformly convex real Banach space with uniformly G?ateaux differentiable norm are extendable to more general reflexive and strictly convex real Banach space with uniformly G?ateaux differentiable norm. Demicompactness condition imposed in such results is dispensed with. Furthermore,Applications of our theorems to approximation of common fixed point of countable infinite family of continuous pseudocontractive mappings and approximation of common solution of countable infinite family of generalized mixed equilibrium problems are also discussed. Our theorems improve, generalize, unify and extend several recently announced results.     

Regularized learning in Banach spaces as an optimization problem: representer theorems

We view regularized learning of a function in a Banach space from its finite samples as an optimization problem. Within the framework of reproducing kernel Banach spaces, we prove the representer theorem for the minimizer of regularized learning schemes with a general loss function and a nondecreasing regularizer. When the loss function and the regularizer are differentiable, a characterization equation for the minimizer is also established.     

Conditional extrema of functionals in topological linear spaces

A theorem of Lyusternik on conditional extrema of a differentiable functional defined on a Banach space is generalized to the case of a topological linear space.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 375–382, Septembers 1973.     

Product integrals in strongly differentiable power associative groupoids

The existence of product integrals in differentiable semigroups with neighborhood of 1 homeomorphic to a Banach space and with multiplication strongly differentiable at (1,1) was shown by Garrett Birkhoff in 1938. This result was then used to transform the multiplication into canonical parameters. We show here a sufficient condition to guarantee the existence of product integrals in differentiable power associative groupoids with multiplication strongly differentiable at (1,1).     

New iteration algorithms for an asymptotically nonexpansive mapping

Under the framework of a real Banach space with an uniformly Gâteaux differentiable norm, two new iteration algorithms are introduced to obtain strong convergence to a fixed point of an asymptotically nonexpansive mapping. Furthermore, the proof technique is independent of the implicit iteration-path.     

A variant of Newton's method and its application

This paper details an existence and uniqueness theorem for solving an operator equation of the form F(x)=0, where F is a Gateaux differentiable operator defined on an open convex subset of a Banach space proved. From the main theorem, an earlier theorem of Argyros follows as a consequence. Other corollaries constitute the semilocal versions of the theorems due to Ozban and Weerakoon and Fernando in a general Banach space. Our main theorem leads to the existence of solutions for a class of nonlinear Urysohn-type integral equations in the n-dimensional Euclidean space.     

Banach空间的p— Asplund 伴随空间

我们称一个定义在Banach空间E上的连续凸函数f具有Frechet可微性质（FDP）,如果E上的每个实值凸函数g≤f均在E一个稠密的Gδ－子集上Frechet可微。本文主要证明了：对任何Banach空间E,均存在一个局部凸相容拓扑p使得1）（E,p)是Hausdorff局部凸空间;2） E上的每个范数连续具有FDP的凸函数均是p－连续的;3）每个p-连续的凸函数均具有FDP ;4）p等价某个范数拓扑当且仅不E是Asplund空间。     

A Newton-like method and its application

In this paper we prove an existence and uniqueness theorem for solving the operator equation F(x)+G(x)=0, where F is a Gateaux differentiable continuous operator while the operator G satisfies a Lipschitz-condition on an open convex subset of a Banach space. As corollaries, a theorem of Tapia on a weak Newton's method and the classical convergence theorem for modified Newton-iterates are deduced. An existence theorem for a generalized Euler-Lagrange equation in the setting of Sobolev space is obtained as a consequence of the main theorem. We also obtain a class of Gateaux differentiable operators which are nowhere Frechet differentiable. Illustrative examples are also provided.     

Note on symmetry and bifurcation near families of solutions

We give conditions for an orbit under continuous, but not everywhere differentiable action of a compact Lie group on a Banach space to have an invariant tubular neighborhood of a prescribed form. The result is important for the study of bifurcation problems near families of solutions. This note completes and corrects part of the earlier paper [6].     

Strong convergence of iterative sequences for nonexpansive mappings

An iterative algorithm is proposed for finding a fixed point of a nonexpansive self-mapping of a closed convex subset of a Banach space with a uniformly Gâteaux differentiable norm, and an estimate of the convergence speed also is given.     

Evolution Systems for Differential Equations with Variable Time Lags

We consider nonautonomous retarded functional differential equations under hypotheses which are designed for the application to equations with variable time lags, which may be unbounded, and construct an evolution system of solution operators which are continuously differentiable. These operators are defined on manifolds of continuously differentiable functions. The results apply to pantograph equations and to nonlinear Volterra integro-differential equations, for example. For linear equations we also provide a simpler evolution system with solution operators on a Banach space of continuous functions.     

Perturbations of eventually differentiable and eventually norm-continuous semigroups on Banach spaces

In this paper we discuss perturbations of eventually differentiable and eventually norm-continuous semigroups on a Banach space. Two kinds of new perturbation theorems are obtained.     

Continuous bijective mappings in topological and Banach manifolds

Continuity and monotonicity of the inverse operator is investigated for the case when the original operator is a bijective mapping acting in topological manifolds or in Banach spaces. For finite-dimensional topological manifolds, the inverse operator is always continuous; for Banach spaces, continuity of the inverse operator is guaranteed only if the space is finite-dimensional. For an arbitrary differentiable bijective mapping acting in the Euclidean space, the Jacobian preserves its sign.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 117–124, 1985;     

无穷可数族非自射非扩张映象之公共不动点的带误差逼近问题

设E具Gateaux可微的严格凸的自反Banach空间,C是E的一非空闭凸子集.受姚永红等2007年文献[1]的启发.本文在此Banach空间框架下引进了一涉及无穷可数族非自射非扩张映象{T:C→E)<'∞><,t=1>的含误差的显式迭代算法,并且在非常少的限制条件下证明了该迭代序列的强收敛于无穷可数族非自射非扩张映象...