An Optimal Krasnosel'skii-Type Theorem for the Dimension of the Kernel of a Starshaped Set

1995 ◽  
Vol 27 (3) ◽  
pp. 249-256 ◽  
Author(s):  
J. Cel
Keyword(s):  
Type Theorem ◽  
Starshaped Set

(D-2)-Extreme Points and a Helly-Type Theorem for Starshaped Sets

1980 ◽  
Vol 32 (3) ◽  
pp. 703-713 ◽  
Author(s):  
Marilyn Breen
Keyword(s):  
Extreme Point ◽  
Type Theorem ◽  
Extreme Points ◽  
Compact Set ◽  
Dimensional Simplex ◽  
Starshaped Set ◽  
Starshaped Sets

We begin with some preliminary definitions. Let S be a subset of Rd. For points x and y in S, we say x sees y via S if and only if the corresponding segment [x, y] lies in S. The set Sis said to be starshaped if and only if there is some point p in S such that, for every x in S, p sees x via S. The collection of all such points p is called the kernel of S, denoted ker S. Furthermore, if we define the star of x in S by Sx = {y: [x, y] ⊆ S}, it is clear that ker S = ⋂{Sx: x in S}.Several interesting results indicate a relationship between ker S and the set E of (d – 2)-extreme points of S. Recall that for d ≧ 2, a point x in S is a (d – 2)-extreme point of S if and only if x is not relatively interior to a (d – 1)-dimensional simplex which lies in S. Kenelly, Hare et al. [4] have proved that if S is a compact starshaped set in Rd, d ≧ 2, then ker S = ⋂{Se: eE}. This was strengthened in papers by Stavrakas [6] and Goodey [2], and their results show that the conclusion follows whenever S is a compact set whose complement ~S is connected.

A Perron-Frobenius-type Theorem for Positive Matrix Semigroups

Positivity ◽  
2016 ◽  
Vol 21 (1) ◽  
pp. 61-72
Author(s):  
L. Livshits ◽  
G. MacDonald ◽  
H. Radjavi
Keyword(s):  
Type Theorem ◽  
Positive Matrix ◽  
Matrix Semigroups

scholarly journals A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian

2021 ◽  
Author(s):  
Yunru Bai ◽  
Nikolaos S. Papageorgiou ◽  
Shengda Zeng
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Solution Map ◽  
Continuity Properties ◽  
Variational Tools ◽  
Superlinear Reaction

AbstractWe consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

scholarly journals B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces

2020 ◽  
Vol 18 (1) ◽  
pp. 715-730
Author(s):  
Javanshir J. Hasanov ◽  
Rabil Ayazoglu ◽  
Simten Bayrakci
Keyword(s):  
Differential Operator ◽  
Singular Integral ◽  
Riesz Potential ◽  
Type Theorem ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Singular Integral Operators ◽  
Sobolev Type ◽  
Riesz Potentials ◽  
Bessel Differential Operator

Abstract In this article, we consider the Laplace-Bessel differential operator {\Delta }_{{B}_{k,n}}=\mathop{\sum }\limits_{i=1}^{k}\left(\frac{{\partial }^{2}}{\partial {x}_{i}^{2}}+\frac{{\gamma }_{i}}{{x}_{i}}\frac{\partial }{\partial {x}_{i}}\right)+\mathop{\sum }\limits_{i=k+1}^{n}\frac{{\partial }^{2}}{\partial {x}_{i}^{2}},{\gamma }_{1}\gt 0,\ldots ,{\gamma }_{k}\gt 0. Furthermore, we define B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials associated with the Laplace-Bessel differential operator. Moreover, we also obtain the boundedness of the B-maximal commutator {M}_{b,\gamma } and the commutator {[}b,{A}_{\gamma }] of the B-singular integral operator and Hardy-Littlewood-Sobolev-type theorem for the commutator {[}b,{I}_{\alpha ,\gamma }] of the B-Riesz potential on B-Morrey spaces {L}_{p,\lambda ,\gamma } , when b\in {\text{BMO}}_{\gamma } .

scholarly journals A Class of Integral Operators that Fix Exponential Functions

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Carlo Bardaro ◽  
Ilaria Mantellini ◽  
Gumrah Uysal ◽  
Basar Yilmaz
Keyword(s):  
General Class ◽  
Pointwise Convergence ◽  
Type Theorem ◽  
Integral Operators ◽  
Exponential Functions ◽  
Convergence Theorems ◽  
Korovkin Type Theorem ◽  
Type Formula

AbstractIn this paper we introduce a general class of integral operators that fix exponential functions, containing several recent modified operators of Gauss–Weierstrass, or Picard or moment type operators. Pointwise convergence theorems are studied, using a Korovkin-type theorem and a Voronovskaja-type formula is obtained.

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