非一致连续的最近点映射

A uniformly continuous linear extension principle in topological vector spaces with an application to Lebesgue integration

We prove a uniformly continuous linear extension principle in topological vector spaces from which we derive a very short and canonical construction of the Lebesgue integral of Banach space valued maps on a finite measure space. The Vitali Convergence Theorem and the Riesz-Fischer Theorem are shown to follow as easy consequences from our construction.

A hybrid scheme for fixed points of a countable family of generalized nonexpansive-type maps and finite families of variational inequality and equilibrium problems, with applications

Let $C$ be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space $E$ with dual space $E^*$. We present a novel hybrid method for finding a common solution of a family of equilibrium problems, a common solution of a family of variational inequality problems and a common element of fixed points of a family of a general class of nonlinear nonexpansive maps. The sequence of this new method is proved to converge strongly to a common element of the families. Our theorem and its applications complement, generalize, and extend various results in literature.

The Bishop--Phelps--Bollobás property for Lipschitz maps

In this paper, we introduce and study a Lipschitz version of the Bishop-Phelps-Bollobás property (Lip-BPB property). This property deals with the possibility of making a uniformly simultaneous approximation of a Lipschitz map $F$ and a pair of points at which $F$ almost attains its norm by a Lipschitz map $G$ and a pair of points such that $G$ strongly attains its norm at the new pair of points. We first show that if $M$ is a finite pointed metric space and $Y$ is a finite-dimensional Banach space, then the pair $(M,Y)$ has the Lip-BPB property, and that both finiteness assumptions are needed.…

可数族广义非扩张型映射与广义混合平衡问题不动点的混合格式

A hybrid scheme for fixed points of a countable family of generalized non-expansive-type maps and generalized mixed equilibrium problems

提出一种混合算法，在一致光滑且一致凸的实Banach空间中寻找广义混合平衡问题解集与一族广义非扩张型映射公共不动点的公共元，并应用于优化问题。

Nonlinear biseparating maps

An additive map $T$ acting between spaces of vector-valued functions is said to be biseparating if $T$ is a bijection so that $f$ and $g$ are disjoint if and only if $Tf$ and $Tg$ are disjoint. Note that an additive bijection retains $\mathbb{Q}$-linearity. For a general nonlinear map $T$, the definition of biseparating given above turns out to be too weak to determine the structure of $T$. In this paper, we propose a revised definition of biseparating maps for general nonlinear operators acting between spaces of vector-valued functions, which coincides with the previous definition for additiv…

Error Estimates for Approximating Best Proximity Points for Cyclic Contractive Maps

We find a priori and a posteriori error estimates of the best proximity point for the Picard iteration associated to a cyclic contraction map, which is defined on a uniformly convex Banach space with modulus of convexity of power type. We find the rate of convergence for the Picard sequence.

Approximation of functions and their derivatives by analytic maps on certain Banach spaces

Let X be a separable Banach space which admits a separating polynomial; in particular X a separable Hilbert space. Let $f:X \rightarrow R$ be bounded, Lipschitz, and $C^1$ with uniformly continuous derivative. Then for each ε>0, there exists an analytic function $g:X \rightarrow R$ with $|g-f|<ε$ and $||g'-f'||<ε$.