Papers by Kazimierz Nikodem
New characterizations of the set-valued solutions for a class of functional equations with involutions
Acta Mathematica Hungarica, Nov 21, 2022
Generalized convex functions and separation theorems
A sandwich with convexity
Page 1. Mathematica Pannonica 5/1 (1994), 139 - 144 A SANDWICH WITH CONVEXITY Karol Baron Instytu... more Page 1. Mathematica Pannonica 5/1 (1994), 139 - 144 A SANDWICH WITH CONVEXITY Karol Baron Instytut Mdtematyki, Uniwcrsyict glgski, ul. ... Ber. 316 (1992), 103-138. ROBERTS, AW and VARBERG, DE: Convex Functions, Academic Press, New York-London, 1973. ...
Ann. Funct. Anal. Volume 2, Number 2, 2011
arXiv (Cornell University), Jan 19, 2012
Recently Nikodem, Rajba and Wąsowicz compared the classes of n-Wrightconvex functions and n-Jense... more Recently Nikodem, Rajba and Wąsowicz compared the classes of n-Wrightconvex functions and n-Jensen-convex functions by showing that the first one is a proper subclass of the latter one, whenever n is an odd natural number. Till now the case of even n was an open problem. In this paper the complete solution is given: it is shown that the inclusion is proper for any natural n. The classes of strongly n-Wright-convex and strongly n-Jensen-convex functions are also compared (with the same assertion).
On quadratic stochastic processes (Short Communication)
Jensen and Hermite-Hadamard inequalities for strongly convex set-valued maps
Counterparts of the classical integral and discrete Jensen inequalities and the Hermite-Hadamard ... more Counterparts of the classical integral and discrete Jensen inequalities and the Hermite-Hadamard inequalities for strongly convex set-valued maps are presented.
Separation by monotonic functions
: It is shown that real functions f and g defined on an arbitraryinterval I can be separated by a... more : It is shown that real functions f and g defined on an arbitraryinterval I can be separated by a monotonic function ifff\Gammatx + (1 \Gamma t)y\Delta max\Phig(x); g(y)\Psiandg\Gammatx + (1 \Gamma t)y\Delta min\Phif (x); f (y)\Psifor all x; y 2 I and t 2 [0; 1]. Some results on the existence of monotonic selectionsof multifunctions and on the Hyers--Ulam
On approximately Jensen-convex and Wright-convex functions
Lagrangian multiplier rule for set-valued optimization
ABSTRACT We prove some separation and Farkas type alternatives theorems in the case where the rel... more ABSTRACT We prove some separation and Farkas type alternatives theorems in the case where the relative interior (instead of interior) of appropriate sets is involved. As an application we derive necessary conditions for the existence of relatively weak minimal solutions of a set-valued optimization problem. Our results contain, in particular, similar results obtained recently by [W. Song, Diss. Math. 375 (1988; Zbl 0934.90070) and J. B. G. Frenk and G. Kassay, J. Optim. Theory Appl. 102, 315-343 (1999; Zbl 0937.90127)].
On quadratic stochastic processes
Aequationes Mathematicae, Dec 1, 1980
arXiv (Cornell University), Jan 19, 2012
The classes of n-Wright-convex functions and n-Jensen-convex functions are compared with each oth... more The classes of n-Wright-convex functions and n-Jensen-convex functions are compared with each other. It is shown that for any odd natural number n the first one is the proper subclass of the second one. To reach this aim new tools connected with measure theory are developed.
Mathematical Inequalities & Applications, 2022
We introduce the class of (k,h)-convex set-valued maps defined on k-convex domains by h(t)G(x 1) ... more We introduce the class of (k,h)-convex set-valued maps defined on k-convex domains by h(t)G(x 1) + h(1 − t)G(x 2) ⊂ G(k(t)x 1 + k(1 − t)x 2), x 1 ,x 2 ∈ D, t ∈ [0,1], and prove a Hermite-Hadamard-type theorem for such maps. Many other properties of (k,h)convex set-valued maps are also presented.
Results in Mathematics, Jun 2, 2020
In this article, we obtain a characterizations and representations of set-valued solutions define... more In this article, we obtain a characterizations and representations of set-valued solutions defined on an Abelian group G with values in a Hausdorff topological vector space of the following generalized biquadratic functional equation: F (x + y, z + w) + F (x + y, z − w) + F (x − y, z + w) + F (x − y, z − w) = aF (x, z) + bF (x, w) + cF (y, z) + dF (y, w), for some nonnegative real numbers a, b, c, and d.
Proceedings of the American Mathematical Society, 1993
The Bernstein-Doetsch theorem on midconvex functions is extended to approximately midconvex funct... more The Bernstein-Doetsch theorem on midconvex functions is extended to approximately midconvex functions and to approximately Wright convex functions.
On convex stochastic processes
Aequationes Mathematicae, Dec 1, 1980
Converse Ohlin’s lemma for convex and strongly convex functions
Journal of Applied Analysis, Oct 26, 2022
Theorems which are converse to the Ohlin lemma for convex and strongly convex functions are prove... more Theorems which are converse to the Ohlin lemma for convex and strongly convex functions are proved. New proofs of probabilistic characterizations of convex and strongly convex functions are presented.
Journal of Mathematical Inequalities, 2021
The notion of fuzzy means of fuzzy numbers is introduced. Fuzzy counterparts of the arithmetic, g... more The notion of fuzzy means of fuzzy numbers is introduced. Fuzzy counterparts of the arithmetic, geometric and harmonic means are investigated and inequalities between them are presented.
Set-valued solutions of a two-variable functional equation with involutions
Aequationes mathematicae, 2021
In this work, we give some characterizations or representations of set-valued solutions defined o... more In this work, we give some characterizations or representations of set-valued solutions defined on a commutative monoid $$(M,+)$$ ( M , + ) with values in a Hausdorff topological vector space of the following two-variable functional equation with involutions: $$\begin{aligned} F(x+y,z+w)+F(x+\sigma (y),z+\tau (w)) =\alpha F(x,z)+\beta F(y,w), \end{aligned}$$ F ( x + y , z + w ) + F ( x + σ ( y ) , z + τ ( w ) ) = α F ( x , z ) + β F ( y , w ) , where $$\alpha ,\;\beta $$ α , β are fixed nonnegative real numbers and $$\sigma ,\tau : M\rightarrow M$$ σ , τ : M → M are involutions (i.e., $$\sigma (x+y)=\sigma (x)+\sigma (y)$$ σ ( x + y ) = σ ( x ) + σ ( y ) and $$\sigma \circ \sigma (x)=x$$ σ ∘ σ ( x ) = x for all $$x,y\in M$$ x , y ∈ M ).
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Papers by Kazimierz Nikodem