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.
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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 $Γ: X → 2^Φ$ 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.
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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 $Γ: X → 2^{Φ}$ be a cyclic $Φ^{γ(·,·)}$-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 $Φ^{γ(·,·)}$-subdifferential of f, $Γ(x) ⊂ ∂_{Φ}^{γ(·,·)} f|_{x}$.
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Let (X,||·||) be a separable real Banach space. Let f be a real-valued strongly α(·)-paraconvex function defined on an open convex subset Ω ⊂ X, i.e. such that $f(tx + (1-t)y) ≤ tf(x) + (1-t)f(y) + min[t,(1-t)]α(||x-y||)$. Then there is a dense $G_{δ}$-set $A_{G} ⊂ Ω$ such that f is Gateaux differentiable at every point of $A_{G}$.
The notions of smooth points of the boundary of an open set and α(·) intrinsically paraconvex sets are introduced. It is shown that for an α(·) intrinsically paraconvex open set the set of smooth points is a dense $G_{δ}$-set of the boundary.
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We introduce the notion of uniform Fréchet differentiability of mappings between Banach spaces, and we give some sufficient conditions for this property to hold.
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Let X be an arbitrary set. Let 𝚽 be a family of real-valued functions defined on X. Let γ:X × X → ℝ. Set [𝚽 + γ] = {Φ(·) + γ(·,x) | Φ ∈ 𝚽, x ∈ X}$. We give conditions guaranteeing the equivalence of $𝚽^{γ(·,·)}$-subdifferentiability and [𝚽 + γ]-subdifferentiability.
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Let (X,d) be a metric space. Let Φ be a linear family of real-valued functions defined on X. Let $Γ: X → 2^{Φ}$ 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(·), $Γ(x)=∂^{-α}_{Φ}f|_{x}$.
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The notion of local completeness is extended to locally pseudoconvex spaces. Then a general version of the Borwein-Preiss variational principle in locally complete locally pseudoconvex spaces is given, where the perturbation is an infinite sum involving differentiable real-valued functions and subadditive functionals. From this, some particular versions of the Borwein-Preiss variational principle are derived. In particular, a version with respect to the Minkowski gauge of a bounded closed convex set in a locally convex space is presented. In locally convex spaces it can be shown that the relevant perturbation only consists of a single summand if and only if the bounded closed convex set has the quasi-weak drop property if and only if it is weakly compact. From this, a new description of reflexive locally convex spaces is obtained.
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In the framework of locally p-convex spaces, two versions of Ekeland's variational principle and two versions of Caristi's fixed point theorem are given. It is shown that the four results are mutually equivalent. Moreover, by using the local completeness theory, a p-drop theorem in locally p-convex spaces is proven.
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Let X, Y be two separable F-spaces. Let (Ω,Σ,μ) be a measure space with μ complete, non-atomic and σ-finite. Let $N_F$ be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function $F:Ω × X → 2^{Y}$. It is shown that if $N_F$ maps a modular space $(N(L(Ω,Σ,μ;X)), ϱ_{N,μ})$ into subsets of a modular space $(M(L(Ω,Σ,μ;Y)),ϱ_{M,μ})$, then $N_F$ is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that $r_K = sup{ϱ_{N,μ}(x) : x ∈ K} < ∞$ we have $sup{ϱ_{M,μ}(y): y ∈ N_F(K)} < ∞$.