scholarly journals On locally Lipschitz vector-valued invex functions

1993 ◽  
Vol 47 (2) ◽  
pp. 259-272 ◽  
Author(s):  
N.D. Yen ◽  
P.H. Sach
Keyword(s):  
The Other ◽  
Joint Paper ◽  
Invex Functions ◽  
Locally Lipschitz ◽  
Invex Function ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Class Of Functions

The four types of invexity for locally Lipschitz vector-valued functions recently introduced by T. W. Reiland are studied in more detail. It is shown that the class of restricted K-invex in the limit functions is too large to obtain desired optimisation theorems and the other three classes are contained in the class of functions which are invex 0 in the sense of our previous joint paper with B. D. Craven and T. D. Phuong. We also prove that the extended image of a locally Lipschitz vector-valued invex function is pseudoconvex in the sense of J. Borwein at each of its points.

scholarly journals Entire multivariate vector-valued functions of bounded $\mathbf{L}$-index: analog of Fricke’s theorem

2020 ◽  
Vol 54 (1) ◽  
pp. 56-63
Author(s):  
A. I. Bandura ◽  
V. P. Baksa
Keyword(s):  
Complex Variable ◽  
Entire Functions ◽  
Satisfying Inequality ◽  
Vector Valued Function ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Entire Vector ◽  
Class Of Functions

We consider a class of vector-valued entire functions $F\colon \mathbb{C}^{n}\rightarrow \mathbb{C}^{p}$. For this class of functions there is introduced a concept of boundedness of $\mathbf{L}$-index in joint variables. Let $|\cdot|_p$ be a norm in $\mathbb{C}^p$. Let $\mathbf{L}(z)=(l_{1}(z),\ldots,l_{n}(z))$, where $l_{j}(z)\colon \mathbb{C}^{n}\to \mathbb{R}_+$ is a positive continuous function.An entire vector-valued function $F\colon \mathbb{C}^{n}\rightarrow \mathbb{C}^{p}$ is said to be ofbounded $\mathbf{L}$-index (in joint variables), if there exists $n_{0}\in \mathbb{Z}_{+}$ such that $\displaystyle \forall z\in G \ \ \forall J \in \mathbb{Z}^n_{+}\colon \quad\frac{|F^{(J)}(z)|_p}{J!\mathbf{L}^J(z)}\leq \max \left \{\frac{|F^{(K)}(z)|_p}{K!\mathbf{L}^K(z)} \colon K\in \mathbb{Z}^n_{+}, \|K\|\leq n_{0} \right \}.$ We assume the function $\mathbf{L}\colon \mathbb{C}^n\to\mathbb{R}^p_+$ such that $0< \lambda _{1,j}(R)\leq\lambda _{2,j}(R)<\infty$ for any $j\in \{1,2,\ldots, p\}$ and $\forall R\in \mathbb{R}_{+}^{p},$where $\lambda _{1,j}(R)=\inf\limits_{z_{0}\in \mathbb{C}^{p}} \inf \left \{{l_{j}(z)}/{l_{j}(z_{0})}\colon z\in \mathbb{D}^{n}[z_{0},R/\mathbf{L}(z_{0})]\right \},$ $\lambda _{2,j}(R)$ is defined analogously with replacement $\inf$ by $\sup$.It is proved the following theorem:Let $|A|_p=\max\{|a_j|\colon 1\leq j\leq p\}$ for $A=(a_1,\ldots,a_p)\in\mathbb{C}^p$. An entire vector-valued function $F$ has bounded $\mathbf{L}$-index in joint variables if and only if for every $R\in \mathbb{R}^{n}_+$ there exist $n_{0}\in \mathbb{Z}_{+}$, $p_0>0$ such that for all $z_{0}\in \mathbb{C}^{n}$ there exists $K_{0}\in \mathbb{Z}_{+}^{n}$, $\|K_0\|\leq n_{0}$, satisfying inequality $\displaystyle\!\max\!\left \{\frac{|F^{(K)}(z)|_p}{K!\mathbf{L}^{K}(z)} \colon \|K\|\leq n_{0},z\in \mathbb{D}^{n}[z_{0},R/\mathbf{L}(z_{0})]\right \}%\leq \nonumber\\\label{eq:5}\leq p_{0}\frac{|F^{(K_0)}(z_0)|_p}{K_0!\mathbf{L}^{K_0}(z_0)},$ where $\mathbb{D}^{n}[z_{0},R]=\{z=(z_1,\ldots,z_n)\in \mathbb{C}^{n}\colon |z_1-z_{0,1}|<r_{1},\ldots, |z_n-z_{0,n}|<r_{n}\}$ is the polydisc with $z_0=(z_{0,1},\ldots,z_{0,n}),$\ $R=(r_{1},\ldots,r_{n})$. This theorem is an analog of Fricke's Theorem obtained for entire functions of bounded index of one complex variable.

scholarly journals ALMOST TRANSITIVITY IN $\mathcal{C}_0$ SPACES OF VECTOR-VALUED FUNCTIONS

2005 ◽  
Vol 48 (3) ◽  
pp. 513-529 ◽  
Author(s):  
Antonio Aizpuru ◽  
Fernando Rambla
Keyword(s):  
The Other ◽  
Dimension Theory ◽  
Topological Properties ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Stone Property

AbstractBy means of $M$-structure and dimension theory, we generalize some known results and obtain some new ones on almost transitivity in $\mathcal{C}_0(L,X)$. For instance, if $X$ has the strong Banach–Stone property, then almost transitivity of $\mathcal{C}_0(L,X)$ is divided into two weaker properties, one of them depending only on topological properties of $L$ and the other being closely related to the covering dimensions of $L$ and $X$. This leads to some non-trivial examples of almost transitive $\mathcal{C}_0(L,X)$ spaces.

scholarly journals A class of nonconvex functions and mathematical programming

1988 ◽  
Vol 38 (2) ◽  
pp. 177-189 ◽  
Author(s):  
T. Weir ◽  
V. Jeyakumar
Keyword(s):  
Mathematical Programming ◽  
Optimality Conditions ◽  
Convex Functions ◽  
Duality Theorems ◽  
Nonconvex Functions ◽  
Invex Functions ◽  
Vector Valued ◽  
Class Of Functions

A class of functions, called pre-invex, is defined. These functions are more general than convex functions and when differentiable are invex. Optimality conditions and duality theorems are given for both scalar-valued and vector-valued programs involving pre-invex functions.

scholarly journals Invexity criteria for a class of vector-valued functions

1995 ◽  
Vol 51 (2) ◽  
pp. 249-262 ◽  
Author(s):  
Pham Huu Sach ◽  
Ta Duy Phuong
Keyword(s):  
Convex Cone ◽  
Jacobian Matrix ◽  
Gradient Mapping ◽  
Locally Lipschitz ◽  
Vector Valued Function ◽  
Vector Valued Functions ◽  
Vector Valued

This paper gives criteria, necessary or sufficient for a vector-valued function F = (f1, f2, …, fk) to be invex. Here each fi is of the -class (that is, each fi is a function whose gradient mapping is locally Lipschitz in a neighbourhood of x0) and the invexity of F means that F(x) − F(x0) ⊂ ˚F′(X) + Q for a fixed convex cone Q of Rk and every x near x0 (˚F′ being the Jacobian matrix of F at x0).

scholarly journals Farkas-Type Results for Vector-Valued Functions with Applications

2017 ◽  
Vol 173 (2) ◽  
pp. 357-390 ◽  
Author(s):  
N. Dinh ◽  
M. A. Goberna ◽  
M. A. López ◽  
T. H. Mo
Keyword(s):  
Vector Valued Functions ◽  
Vector Valued

scholarly journals Diameter-preserving linear bijections of function spaces

2001 ◽  
Vol 70 (3) ◽  
pp. 323-336 ◽  
Author(s):  
T. S. S. R. K. Rao ◽  
A. K. Roy
Keyword(s):  
Maximal Ideal ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Continuous Functions ◽  
Ideal Space ◽  
Function Algebras ◽  
Compact Convex Set ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractIn this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

scholarly journals THE LOCAL NON-HOMOGENEOUS Tb THEOREM FOR VECTOR-VALUED FUNCTIONS

2014 ◽  
Vol 57 (1) ◽  
pp. 17-82 ◽  
Author(s):  
TUOMAS P. HYTÖNEN ◽  
ANTTI V. VÄHÄKANGAS
Keyword(s):  
Singular Integrals ◽  
Local Version ◽  
Square Functions ◽  
Property A ◽  
Martingale Differences ◽  
Umd Spaces ◽  
Function Lattice ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Similar Extension

AbstractWe extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg to the setting of singular integrals with operator-valued kernel that act on vector-valued functions. Here, ‘vector-valued’ means ‘taking values in a function lattice with the UMD (unconditional martingale differences) property’. A similar extension (but for general UMD spaces rather than UMD lattices) of Nazarov-Treil-Volberg's global non-homogeneous Tb theorem was achieved earlier by the first author, and it has found applications in the work of Mayboroda and Volberg on square-functions and rectifiability. Our local version requires several elaborations of the previous techniques, and raises new questions about the limits of the vector-valued theory.

Strict Topologies for Vector-Valued Functions

1974 ◽  
Vol 26 (4) ◽  
pp. 841-853 ◽  
Author(s):  
Robert A. Fontenot
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Measure Theory ◽  
Strict Topology ◽  
Locally Compact ◽  
Topological Measure ◽  
Topological Measure Theory ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Compact Hausdorff Space

This paper is motivated by work in two fields, the theory of strict topologies and topological measure theory. In [1], R. C. Buck began the study of the strict topology for the algebra C*(S) of continuous, bounded real-valued functions on a locally compact Hausdorff space S and showed that the topological vector space C*(S) with the strict topology has many of the same topological vector space properties as C0(S), the sup norm algebra of continuous realvalued functions vanishing at infinity. Buck showed that as a class, the algebras C*(S) for S locally compact and C*(X), for X compact, were very much alike. Many papers on the strict topology for C*(S), where S is locally compact, followed Buck's; e.g., see [2; 3].

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