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1
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Stability of the Cauchy functional equation in quasi-Banach spaces

100%
Tabor J.
Annales Polonici Mathematici
|
2004
|
tom 83
|
nr 3
243-255
EN
Let X be a quasi-Banach space. We prove that there exists K > 0 such that for every function w:ℝ → X satisfying ||w(s+t)-w(s)-w(t)|| ≤ ε(|s|+|t|) for s,t ∈ ℝ, there exists a unique additive function a:ℝ → X such that a(1)=0 and ||w(s)-a(s)-sθ(log₂|s|)|| ≤ Kε|s| for s ∈ ℝ, where θ: ℝ → X is defined by $θ(k):= w(2^k)/2^k$ for k ∈ ℤ and extended in a piecewise linear way over the rest of ℝ.
2
Content available remote

Characterization of convex functions

64%
Tabor J., Tabor J.
Studia Mathematica
|
2009
|
tom 192
|
nr 1
29-37
EN
There are many inequalities which in the class of continuous functions are equivalent to convexity (for example the Jensen inequality and the Hermite-Hadamard inequalities). We show that this is not a coincidence: every nontrivial linear inequality which is valid for all convex functions is valid only for convex functions.
3
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Generalized midconvexity

51%
Tabor J., Tabor J., Misztal K.
Banach Center Publications
|
2013
|
tom 99
|
nr 1
207-215
EN
There are many types of midconvexities, for example Jensen convexity, t-convexity, (s,t)-convexity. We provide a uniform framework for all the above mentioned midconvexities by considering a generalized middle-point map on an abstract space X. We show that we can define and study the basic convexity properties in this setting.
4
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Generalized shadowing for discrete semidynamical systems

51%
Jabłoński W., Tabor J., Tabor J.
Annales Polonici Mathematici
|
2006
|
tom 88
|
nr 3
263-269
EN
We provide a unified approach to different types of shadowing. This enables us to generalize some known shadowing result.
5
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Generalized approximate midconvexity

23%
Tabor J., Tabor J.
Control and Cybernetics
|
2009
|
tom 38
|
nr 3
655-669
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