APPROXIMATION OF FIXED POINTS AND VARIATIONAL SOLUTIONS FOR PSEUDO-CONTRACTIVE MAPPINGS IN BANACH SPACES

Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. Assume that every nonempty closed con- vex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive map- pings which is also a unique solution to variational inequality problem involving φ-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, con- vex optimization problems, and split feasibility problems. Our result extends many recent important results.

THE BALL-COVERING PROPERTY ON DUAL SPACES AND BANACH SEQUENCE SPACES

In this paper,we prove that(X,p)is separable if and only if there exists a w^(*)-lower semicontinuous norm sequence{p_(n)}_(n=1)^(∞)of(X^(*),p)such that(1)there exists a dense subset G_(n)of X^(*)such that p_(n)is Gateaux differentiable on G_(n)and dp_(n)(Gn_(n))■X for all n∈N;(2)p_(n)≤p and p_(n)→p uniformly on each bounded subset of X^(*);(3)for anyα∈(0,1),there exists a ball-covering{B(x^(*)i,n,Ti,n)}∞i=1 of(X^(*),p_(n))such that it isα-off the origin and x_(i,n)^(*)∈Gn_(n).Moreover,we also prove that if Xi is a Gateaux differentiability space,then there exist a real numberα>0 and a ball-covering(B)i of Xi such that(B)i isα-off the origin if and only if there exist a real numberα>0 and a ball-covering B of l^(∞)(X_(i))such that(B)isα-off the origin.

ON THE PRODUCT OF GTEAUX DIFFERENTIABILITY LOCALLY CONVEX SPACES

A locally convex space is said to be a Gateaux differentiability space (GDS) provided every continuous convex function defined on a nonempty convex open subset D of the space is densely Gateaux differentiable in .D.This paper shows that the product of a GDS and a family of separable Prechet spaces is a GDS,and that the product of a GDS and an arbitrary locally convex space endowed with the weak topology is a GDS.

BRUCK FORMULA FOR A PERTURBED LIPSCHITZIAN ITERATION OF LIPSCHITZ PSEUDOCONTRACTIVE MAPS

The solution to evolution equations has developed an independent theory within nonlinear analysis dealing with the existence and approximation of such solution （ fixed point） of pseudocontractive operators and its variants. The object is to introduce a perturbed iteration method for proving the convergence of sequence of Lipschitzian pseudocontractive mapping using approximate fixed point technique. This iteration can be ued for nonlinear operators which are more general than Lipschitzian pseudocontractive operator and Bruck iteration fails for proving their convergence. Our results generalize the results of Chidume and Zegeye.

UGD PROPERTY OF MUSIELAK-ORLICZ SEQUENCE SPACES

By the properties of the Musielak_Orlicz funciton's sequence, the necessary and sufficient condition for uniform Gateaux differential (UGD) property of Musielak_Orlicz sequence spaces equipped with the Luxemburg norm and a criterion for weakly uniform rotundity of Musielak_Orlicz sequence space with Orlicz norm are given.

Smulyan Lemma and Differentiability of the Support Function

The purpose of this paper is to verify the Smulyan lemma for the support function, and also the Gateaux differentiability of the support function is studied on its domain. Moreover, we provide a characterization of Frechet differentiability of the support function on the extremal points.

A Note on Approximating Fixed Points of Pseudocontractive Mappings

Let K be a nonempty bounded closed convex subset of a real reflexive Banach space E with a uniformly Gateaux differentiable norm. Let T ： K →K be a uniformly continuous pseudocontractive mapping. Suppose every closed convex and bounded subset of K has the fixed point property for nonexpansive mappings. Let {λn} C （0,1/2] be a sequence satisfying the conditions： （i） limn→∞λn=0; （ii） ∑n=0^∞ λn=∞. Let the sequence {xn} be generated from arbitrary x1∈K by xn＋1 = （1 -λn）xn ＋ λnTxn -λn（xn - x1）, n ≥ 1. Suppose limn→∞‖xn - Txn‖ = 0. Then {xn} converges strongly to a fixed point of T.

Ball-covering property of Banach spaces that is not preserved under linear isomorphisms

By a ball-covering B of a Banach space X, we mean that it is a collection of open balls off the origin whose union contains the sphere of the unit ball of X. The space X is said to have a ball-covering property, if it admits a ball-covering consisting of countably many balls. This paper, by constructing the equivalent norms on l~∞, shows that ball-covering property is not invariant under isomorphic mappings, though it is preserved under such mappings if X is a Gateaux differentiability space; presents that this property of X is not heritable by its closed subspaces; and the property is also not preserved under quotient mappings.

Differentiability of Functions on Spheres and Criterions of Convexity

Some basic concepts for functions defined on subsets of the unit sphere,such as the s-directional derivative,s-gradient and s-Gateaux and s-Frechet differentiability etc,are introduced and investigated.These concepts are different from the usual ones for functions defined on subsets of Euclidean spaces,however,the results obtained here are very similar.Then,as applications,we provide some criterions of s-convexity for functions defined on unit spheres which are improvements or refinements of some known results.

ON THE SETS OF GTEAUX NON-DIFFERENTIABILITY OF LIPSCHITZ ISOMORPHISM BETWEEN BANACH SPACES

We prove that for every Lipschitz isomorphism f from a separable Hilbert space H to a Banach space Y with Radon-Nikodym property, there is a bounded surjective linear operator T： H → Y so that （f ＋ T）-1 （NG（f-1）） is a r-null set of H, where NG（f-1） is the set of all the points of Gateaux non-diiTerentiability of f -1.