We prove that certain Lipschitz properties of the inverse F-1 of a set-valued map F are inherited by the map (f+F)~x when / has vanishing strict derivative. In this paper, we present an inverse mapping theorem for set-valued maps F acting from a complete metric space I toa linear space Y with a (translation) invariant metric. We prove that, for any function f: X -> Y with "vanishing strict deri...

Let (X, ‖ · ‖) be a real Banach space. Let C be a closed convex set in X. By a drop D(x, C) determined by a point x ∈ X, x / ∈ C, we shall mean the convex hull of the set {x} ∪ C. We say that C has the drop property if C 6= X and if for every nonvoid closed set A disjoint with C, there exists a point a ∈ A such that D(a, C) ∩ A = {a}. For a given C a sequence {xn} in X will be called a stream i...

where tn → 1 n → ∞ . We may assume that tn ≥ 1 for all n 1, 2, 3, . . . . Denote by F T the set of fixed points of T . Throughout this paper T and S : C → C are two commutative asymptotically nonexpansive mappings with asymptotical coefficients {tn} and {sn}, respectively. Suppose that F : F T ∩F S / ∅ 1, Goebel and Kirk’s theorem makes it possible . It is well known that F T and F S are convex...

We give a complete characterization of closed sets F ⊂ ℝ2 whose distance function dF:= dist(·, F) is DC (i.e., the difference two convex functions on ℝ2). Using this characterization, number properties such proved.

Motivated by Schur-concavity, we introduce the notion of G-concavity where G is a closed subgroup of the orthogonal group O(V ) on a finite dimensional real inner product space V . The triple (V,G, F ) is an Eaton triple if F ⊂ V is a nonempty closed convex cone such that (A1) Gx ∩ F is nonempty for each x ∈ V . (A2) maxg∈G(x, gy) = (x, gy) for all x, y ∈ F . If W := spanF and H := {g|W : g ∈ G...

We prove the following completion theorem for closed maps between metrizable spaces: Let f : X → Y be a closed surjection between metrizable spaces with e-dimf ≤ K, e-dimX ≤ LX and e-dimY ≤ LY for some countable CW -complexes K, LX and LY . Then there exist completions X̃ and Ỹ of X and Y , respectively, and a closed surjection f̃ : X̃ → Ỹ extending f such that e-dimf̃ ≤ K, e-dimX̃ ≤ LX and e-dimỸ ≤...

We consider the problem F = f(ν) for strictly convex, closed hypersurfaces in H and solve it for curvature functions F the inverses of which are of class (K).

We prove that if f is an orientation-preserving homeomorphism of a closed orientable surface M whose singular set Σ(f) is totally disconnected, then f is topologically conjugate to a conformal transformation.

LET f (z,w) ≡ f0(z)w + f1(z)w + · · · + fn(z) ∈ C[z,w]. Classically, the equation f (z,w) = 0 was said to define w as an (n-valued) algebraic function of z, provided that f0(z) was not identically 0 and that f (z,w) was squarefree and without factors of the form z− c. Then, indeed, the singular set B = {z: there are not n distinct solutions w to f (z,w) = 0} is finite; and as z varies in any si...

We construct some non-essential H-closed epireflections of W, the category of archimedean l-groups with weak order unit, that are not comparable with any known H-closed epireflections of W other than the divisible hull and the epicompletion. In preparation, we show that the free objects in any Hclosed epireflective subcategory of W must be closed under composition (see section 2 for a precise d...