On the points of non-differentiability of convex functions
We characterize sets of non-differentiability points of convex functions on . This completes (in ) the result by Zajíček [2] which gives a characterization of the magnitude of these sets.
On the range of the derivative of a real-valued function with bounded support
We study the set f’(X) = f’(x): x ∈ X when f:X → ℝ is a differentiable bump. We first prove that for any C²-smooth bump f: ℝ² → ℝ the range of the derivative of f must be the closure of its interior. Next we show that if X is an infinite-dimensional separable Banach space with a -smooth bump b:X → ℝ such that is finite, then any connected open subset of X* containing 0 is the range of the derivative of a -smooth bump. We also study the finite-dimensional case which is quite different. Finally,...
On the reproducing kernel for harmonic functions and the space of Bloch harmonic functions on the unit ball in
On the singular support of the distributional determinant
On the singularities of convex functions.
On the size of approximately convex sets in normed spaces
Let X be a normed space. A set A ⊆ X is approximately convexif d(ta+(1-t)b,A)≤1 for all a,b ∈ A and t ∈ [0,1]. We prove that every n-dimensional normed space contains approximately convex sets A with and , where ℋ denotes the Hausdorff distance. These estimates are reasonably sharp. For every D>0, we construct worst possible approximately convex sets in C[0,1] such that ℋ(A,Co(A))=(A)=D. Several results pertaining to the Hyers-Ulam stability theorem are also proved.
On the symmetry of Dini derivates of arbitrary functions
On the theorems of Y. Mibu and G. Debs on separate continuity.
On the uniqueness result for the Dirichlet problem and invexity.
On totalization of the Kurzweil-Henstock integral in the multidimensional space
In this paper a full totalization is presented of the Kurzweil-Henstock integral in the multidimensional space. A residual function of the total Kurzweil-Henstock primitive is defined.
On weak Hessian determinants
We consider and study several weak formulations of the Hessian determinant, arising by formal integration by parts. Our main concern are their continuity properties. We also compare them with the Hessian measure.
On weakly Gibson -measurable mappings
A function f: X → Y between topological spaces is said to be a weakly Gibson function if for any open connected set U ⊆ X. We prove that if X is a locally connected hereditarily Baire space and Y is a T₁-space then an -measurable mapping f: X → Y is weakly Gibson if and only if for any connected set C ⊆ X with dense connected interior the image f(C) is connected. Moreover, we show that each weakly Gibson -measurable mapping f: ℝⁿ → Y, where Y is a T₁-space, has a connected graph.
Open mapping theorem and inversion theorem for γ-paraconvex multivalued mappings and applications
We extend the open mapping theorem and inversion theorem of Robinson for convex multivalued mappings to γ-paraconvex multivalued mappings. Some questions posed by Rolewicz are also investigated. Our results are applied to obtain a generalization of the Farkas lemma for γ-paraconvex multivalued mappings.
Optimal embeddings of critical Sobolev-Lorentz-Zygmund spaces
We establish the embedding of the critical Sobolev-Lorentz-Zygmund space into the generalized Morrey space with an optimal Young function Φ. As an application, we obtain the almost Lipschitz continuity for functions in . O’Neil’s inequality and its reverse play an essential role in the proofs of the main theorems.
Oscillations, quasi-oscillations and joint continuity.
Outer -convex functions on a normed space.
Parametric and Non-Parametric Minima.
Parametric extension of the Poincaré theorem
Parametric monotone generalized equations in reflexive Banach spaces.
Partial Differentiation, Differentiation and Continuity on n -Dimensional Real Normed Linear Spaces
In this article, we aim to prove the characterization of differentiation by means of partial differentiation for vector-valued functions on n-dimensional real normed linear spaces (refer to [15] and [16]).