Let (X,τ) be a topological space. Let Φ be a class of real-valued functions defined on X. A function ϕ ∈ Φ is called a local Φ-subgradient of a function f:X → ℝ at a point $x_0$ if there is a neighbourhood U of $x_0$ such that f(x) - f($x_0$) ≥ ϕ(x) - ϕ($x_0$) for all x ∈ U. A function ϕ ∈ Φ is called a global Φ-subgradient of f at $x_0$ if the inequality holds for all x ∈ X. The following properties of the class Φ are investigated: (a) when the existence of a local Φ-subgradient of a function f at each point implies...

Let (X,d) be a metric space. Let Φ be a linear family of real-valued functions defined on X. Let $\Gamma :X\to 2^{\Phi }$ be a maximal cyclic α(·)-monotone multifunction with non-empty values. We give a sufficient condition on α(·) and Φ for the following generalization of the Rockafellar theorem to hold. There is a function f on X, weakly Φ-convex with modulus α(·), such that Γ is the weak Φ-subdifferential of f with modulus α(·), $\Gamma \left(x\right)={\partial }_{\Phi }^{-\alpha }{f|}_x$.

In Rolewicz (2002) it was proved that every strongly α(·)-paraconvex function defined on an open convex set in a separable Asplund space is Fréchet differentiable on a residual set. In this paper it is shown that the assumption of separability is not essential.

The aim of the paper is to present three-variable generalizations of fuzzy metric spaces in sense of George and Veeramani from functional and topological points of view, respectively. From the viewpoint of functional generalization, we introduce a notion of generalized fuzzy 2-metric spaces, study their topological properties, and point out that it is also a common generalization of both tripled fuzzy metric spaces proposed by Tian et al. and $ℳ$-fuzzy metric spaces proposed by Sedghi and Shobe. Since...

The concept of separation by hyperplanes and halfspaces is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question which semispaces are hyperplanes and when it is possible to “classically” separate by hyperplanes in max-min convex geometry.

Let X be an arbitrary set, and γ: X × X → ℝ any function. Let Φ be a family of real-valued functions defined on X. Let $\Gamma :X\to 2^{\Phi }$ be a cyclic ${\Phi }^{\gamma (·,·)}$-monotone multifunction with non-empty values. It is shown that the following generalization of the Rockafellar theorem holds. There is a function f: X → ℝ such that Γ is contained in the ${\Phi }^{\gamma (·,·)}$-subdifferential of f, $\Gamma \left(x\right)\subset {\partial }_{\Phi }^{\gamma (·,·)}{f|}_x$.

Let (X,d) be a metric space. Let Φ be a family of real-valued functions defined on X. Sufficient conditions are given for an α(·)-monotone multifunction $\Gamma :X\to 2^{\Phi }$ to be single-valued and continuous on a weakly angle-small set. As an application it is shown that a γ-paraconvex function defined on an open convex subset of a Banach space having separable dual is Fréchet differentiable on a residual set.