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1
Content available remote On the functional equation associated with unbounded complete paraconcave entropies
100%
Dudek Z.
Demonstratio Mathematica
|
2001
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tom Vol. 34, nr 3
641-650
EN
A paraconcave entropy function, as defined in [2], is represented by a pair of two real functions of a real variable satisfying certain natural conditions. The subject of this paper is the functional equation, L [ sigma j f(pj)]= sigma j g(pj), that describes equivalence between two representations of a paraconcave entropy function with / and g satisfying the condition for unbounded entropy. We show that the above equation has a unique solution up to a multiplicative constant in the class of unbounded and complete paraconcave entropy functions. This is a generalization of the result [4] proven already for incomplete paraconcave entropy functions. This is done by invoking a special version of theCauchy classical functional equation [1]. Additionally, we prove some theorems about ranges of the sums sigma j f(pj) that occur in the investigated equation.
2
Content available remote On Popoviciu-Ionescu Functional Equation
80%
Almira J. M.
Annales Mathematicae Silesianae
|
2016
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tom 30
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nr 1
5-15
EN
We study a functional equation first proposed by T. Popoviciu [15] in 1955. It was solved for the easiest case by Ionescu [9] in 1956 and, for the general case, by Ghiorcoiasiu and Roscau [7] and Radó [17] in 1962. Our solution is based on a generalization of Radó’s theorem to distributions in a higher dimensional setting and, as far as we know, is different than existing solutions. Finally, we propose several related open problems.
3
Content available Iteration groups, commuting functions and simultaneous systems of linear functional equations
80%
Matkowski J.
Opuscula Mathematica
|
2008
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tom Vol. 28, no. 4
529-539
EN
Let (ƒt)t∈R be a measurable iteration group on an open interval I. Under some conditions, we prove that the inequalies g o ƒa ≤ ƒa o g and g o ƒb ≤ ƒb o g for some a, b ∈ R imply that g must belong to the iteration group. Some weak conditions under which two iteration groups have to consist of the same elements are given. An extension theorem of a local solution of a simultaneous system of iterative linear functional equations is presented and applied to prove that, under some conditions, if a function g commutes in a neighbourhood of ƒ with two suitably chosen elements ƒa and ƒb of an iteration group of ƒ then, in this neighbourhood, g coincides with an element of the iteration group. Some weak conditions ensuring equality of iteration groups are considered.
4
Content available remote Semicontinuous solutions of systems of functional equations
80%
Herzog G.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
|
2003
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tom [Z] 43(2)
207-219
EN
We use the method of upper and lower solutions to prove the existence of upper and lower semicontinuous solutions of functional equations of the form F(w,u(w),u(g_1(w)),...,u(g_m(w)) = ) in R^n under monotonicity and quasimonotonicity assumptions on F, and for w from a metrizable topological spaces.
5
Content available remote Singular linear differential equations and Laurent series
80%
Łysik G.
Demonstratio Mathematica
|
1999
|
tom Vol. 32, nr 3
503-520
EN
Linear ordinary differential operators with meromorphic coefficients at zero are studied. It is well known that in the case when zero is a regular or regular singular point then fundamental system of solutions consists of convergent series of the Taylor type. On the other hand in the case of irregular singular point power series solution, in general, does not converge; however it can be asymptotically sum up in sectors to an exact solution. The aim of the paper is to show that for a class of operators with irregular singular point the fundamental system of solutions can be found in a form of convergent Laurent type series of a Gevrey order. Under suitable conditions the convergence of the approximation scheme for a functional equation Wj ( z -j) G ( z - j) = H ( z 't j is also j=-k derived and properties of its solution G are described.
6
Content available remote General solution of some functional equations related to the determinant of some symmetric matrices
80%
Chung J. K. , Sahoo P. K.
Demonstratio Mathematica
|
2002
|
tom Vol. 35, nr 3
539-544
EN
In this paper, we determine the general solution of the functional equation f(ux+vy, uy+vx) = g(x, y) h(u, v) where f, g, h: R2 -R are unknown functions. We also treat the equation f(ux + vy, uy + vx, zw) = g(x, y, z) h(u, v, w) where f,g, h : R3 -R are unknown functions. Our method is elementary and we do not use any regularity conditions.
7
Content available remote Entire solutions of a functional equation of pxider type
80%
Smajdor W.
Demonstratio Mathematica
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2002
|
tom Vol. 35, nr 3
572-582
8
Content available remote On probability distribution solutions of a functional equation
80%
Morawiec J. , Reich L.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2005
|
tom Vol. 53, no 4
389-399
EN
Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation [WZÓR] and its solutions in two classes of functions, namely Z ={φ: R→ R∣ φ is increasing, φ|(−∞,0] = 0, φ|[1,∞) = 1}, C = {φ: R → R∣ φ is continuous, φ|(−∞,0] = 0, φ|[1,∞) = 1}. We prove that the above equation has at most one solution in C and that for some parameters α, β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in Z and we show the exact connection between solutions in both classes.
9
Content available remote On the stability of a mean value type functional equation
80%
Jung S. , Sahoo P.
Demonstratio Mathematica
|
2000
|
tom Vol. 33, nr 4
793-796
EN
In this paper, we prove the stability results of a mean value type functional equation, namely f (x) - g{y) = (x - y)h(x 4- y) which arises from the mean value theorem.
10
Content available remote Euler's Beta function diagonalized and a related functional equation
80%
Choczewski B. , Wach-Michalik A.
Opuscula Mathematica
|
2004
|
tom Vol. 24, no. 1
35-41
EN
Euler's Gamma function is the unique logarithmically convex solution of the functional equation (1), cf. the Proposition. In this paper we deal with the function beta: R+ → R+, beta(x) := B(x, x), where B(x, y) is the Euler Beta function. We prove that, whenever a function h is asymptotically comparable at the origin with the function a log +b, a > 0, if varphi: R+ → R+ satisfies equation (5) and the function h o varphi is continuous and ultimately convex, then varphi = beta.
11
Content available remote A localization principle for classes of means
80%
Berrone L.
Demonstratio Mathematica
|
2000
|
tom Vol. 33, nr 3
557-566
EN
Several families of continuous means defined on a square I x I have the remarkable property of being entirely determined when their values in an arbitrary small neigborhood of the diagonal {(x,x) : x G 1} of the square are known. Some examples are given of application of this property in solving functional equations.
12
Content available remote Extension theorems for the Matkowski-Suto problem
80%
Daroczy Z. , Maksa G. , Pales Z.
Demonstratio Mathematica
|
2000
|
tom Vol. 33, nr 3
547-556
13
Content available remote Solutions of some functional equations related to the determinant of certain matrices
80%
Akkouchi M. , Rhali M.
Demonstratio Mathematica
|
2010
|
tom Vol. 43, nr 1
97-106
EN
We solve the functional equation : [....] where K is a field which is not of characteristic 2 and f, g, h : K4 ->o K are unknown functions. We study also a class of functional equations of 2n variables generalizing the two equations above. This work is motivated by a paper of J. K. Chung and P. K. Sahoo published in 2002 and a recent paper of the authors published in 2005.
14
Content available The equality case in some recent convexity inequalities
80%
Maksa G. , Pales Z.
Opuscula Mathematica
|
2011
|
tom Vol. 31, no. 2
269-277
EN
In this paper, we investigate a functional equation related to some recently introduced and investigated convexity type inequalities.
15
Content available 18th International Conference on Functional Equations and Inequalities, Będlewo, Poland, July 9-15, 2019
80%
Meeting R. o.
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
|
2019
|
tom 18
167-192
EN
Report from the conference.
16
Content available remote Fischer-Muszély additivity on Abelian groups
80%
Ger R.
Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae
|
2004
|
tom Vol. 44, nr spec.
83-96
EN
A description of a general solution f : X -> Y mapping a commutative group (X, +) into a real normed linear space (Y, || o ||) of the functional equation [formula] is given in terms of isometrics and additive mappings. Several results describing the solutions of this equation that were obtained earlier under some alternative assumptions regarding the domains, ranges and//or by imposing some regularity upon the map f become special cases of our main result. To gain a proper proof tool we have also established an improvement of E. Berz's [4] representation theorem for sublinear functionals on Abelian groups.
17
Content available remote On solutions of some system of functional equations
80%
Midura S.
Demonstratio Mathematica
|
1998
|
tom Vol. 31, nr 2
299-304
18
Content available remote On a problem concerning the indicator plurality function
70%
Bahyrycz A.
Opuscula Mathematica
|
2001
|
tom Vol. 21
11-30
19
Content available remote Linear approximation and asymptotic expansion associated with the system of functional equations
60%
Long N.
Demonstratio Mathematica
|
2004
|
tom Vol. 37, nr 2
349--362
20
Content available remote Positivity of Schilling functions
60%
Girgensohn R. , Morawiec J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2000
|
tom Vol. 48, no 4
407-412
EN
We prove that every non-trivial [L^1]-solution of the Schilling problem is either positive or negative almost everywhere on its support.
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