Theorems of Bernstein-Doetsch, Piccard and Mehdi and semilinear topology

Kominek, Zygfryd; Kuczma, Marek

doi:10.1007/bf01237573

Cited by 26 publications

(30 citation statements)

References 6 publications

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“…Let x 0 ∈ X \ ker L and let Π 1 , Π 2 : X → X be given by (24) and (25), respectively. Furthermore, let a : X → S be of the form (27) and φ : [0, ∞) → S be given by…”

Section: Semigroup-valued Solutions Of Some Composite Equations 189mentioning

confidence: 99%

Semigroup-valued solutions of some composite equations

Chudziak

2013

Aequat. Math.

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“…Let x 0 ∈ X \ ker L and let Π 1 , Π 2 : X → X be given by (24) and (25), respectively. Furthermore, let a : X → S be of the form (27) and φ : [0, ∞) → S be given by…”

Section: Semigroup-valued Solutions Of Some Composite Equations 189mentioning

confidence: 99%

Semigroup-valued solutions of some composite equations

Chudziak

2013

Aequat. Math.

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“…In former literature the topology S~ is called semilinear (cf. [4,5,7]). These assumptions are weaker than those for X to be a topological vector space.…”

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confidence: 99%

On approximately convex functions

Ng¹,

Nikodem²

1993

Proc. Amer. Math. Soc.

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Abstract.The Bernstein-Doetsch theorem on midconvex functions is extended to approximately midconvex functions and to approximately Wright convex functions.Let X be a real vector space, D be a convex subset of X, and £ be a nonnegative constant. A function f:D ->R is said to be e-convex if f(tx + (1 -t)y) < tf{x) + (1 -t)f{y) + e for all x,y &D ande-Wright-convex if f{tx + (1 -t)y) + /((1 -t)x + ty) < f(x) + f{y) + 2e for all x,y €D and t e[0, 1]; e-midconvex if f{^)< \{f{x) + f{y)) + e for all x,y eD. Notice that e-convexity implies e-Wright-convexity, which in turn implies e-midconvexity, but not the converse. The usual notions of convexity, Wrightconvexity, and midconvexity correspond to the case e = 0. A comprehensive review on this subject can be found in [1,6,[8][9][10]. The Bernstein-Doetsch theorem relates local boundedness, midconvexity, and convexity (cf. [6,10]). In order to extend this result to approximately midconvex functions, we first specify the assumptions on the topology 3~ to be imposed on X: the map (t, x, y) -+ tx + y from lxlxl-»l is continuous in each of its three variables. Here the scalar field R is under the usual topology. In former literature the topology S~ is called semilinear (cf. [4, 5, 7]). These assumptions are weaker than those for X to be a topological vector space. The finest JonI is formed by taking all subsets A c X with the property that if xo € A , x e X, then there exists a 3 > 0 such that tx + (1 -t)xo e A for all t e ] -S, S[. (1) f(k2-"x + (1 -k2-")y) < k2~nf{x) + (1 -kl~n)f{y) + (2 -2~n+x)e for all x,yeD, «eN = {l

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“…Namely, by the triangle inequality, we have −N (x − y) ≤ N x − N y ≤ N (x − y), so by (2), N (N x − N y) ≤ N (x − y).…”

Section: Lemma 2 Let a Be A Lattice Ordered 2-divisible Group Then mentioning

confidence: 99%

A topology on lattice ordered groups

Gusić

1998

Proc. Amer. Math. Soc.

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Abstract. We show that a lattice ordered group can be topologized in a natural way. The topology depends on the choice of a set C of admissible elements (C-topology). If a lattice ordered group is 2-divisible and satisfies a version of Archimedes' axiom (C-group), then we show that the C-topology is Hausdorff. Moreover, we show that a C-group with the C-topology is a topological group.In section 1 we recall the definition as well as the elementary properties of lattice ordered groups, especially the properties of the norm N on such groups.In section 2 we introduce the notion of a set of admissible elements C (definition 2) in a lattice ordered 2-divisible group A. It is proved (lemma 2) that the sets U x0,r = {x ∈ A : r − N (x − x 0 ) ∈ C} constitute a base of a topology on A (the C-topology).In section 3 we describe the properties of the C-topology on lattice ordered 2-divisible C-Archimedean groups (such groups are called C-groups). It is proved that C-groups are Hausdorff topological groups (theorem 1 and theorem 2).

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Theorems of Bernstein-Doetsch, Piccard and Mehdi and semilinear topology

Cited by 26 publications

References 6 publications

Semigroup-valued solutions of some composite equations

Semigroup-valued solutions of some composite equations

On approximately convex functions

A topology on lattice ordered groups

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