Search Results

Journal: Annales Mathematicae Silesianae

An equation associated with the distance between probability distributions

PL. Kannappan , P.K. Sahoo , J.K. Chung

Language: EN | Published: 30-09-1994 | Abstract | pp. 39-58

In this paper, we solve the functional equation
f₁(pr,qs) + f₂(ps,qr) = g(p,q)h(r,s) (p,q,r,s∈]0,1])
where f₁,f₂,g,h are complex-valued functions defined on ]0,1]. This functional equation is a generalization of a functional equation which was instrumental in the characterization of symmetric divergence of degree α in [3]. This equation arises in the characterization of symmetric weighted divergence of degree α and symmetric inset divergence of degree α.

1994-09-30



Journal: Annales Mathematicae Silesianae

A functional equation arising from an asymptotic formula for iterates

Detlef Gronau , Maciej Sablik

Language: EN | Published: 30-09-1994 | Abstract | pp. 173-187

We consider the real solutions of the functional equation
(⚹)       ϕ^m(x) = 1/m ϕ(mx),   ϕ(0) = 0,
where m∈ℕ and ϕ^m denotes the m-th iterate of the unknown function ϕ. We will handle this functional equation for a fixed m, but also for all naturals m, and give a representation of all C²-solutions (even weaker, see Theorem 2.1) of (⚹), but also treat the case of other solutions of this equation. In the introduction we will show the origin of this equation.

1994-09-30



Journal: Annales Mathematicae Silesianae

On a functional equation for Dido's problem

Peter Kahlig , Janusz Matkowski

Language: EN | Published: 30-09-1999 | Abstract | pp. 167-180

By some geometrical considerations we formulate a functional equation which is related to the ancient isoperimetric problem of Dido. The continuous solution of this Dido functional equation depends on an arbitrary function. However, we show that in a class of functions of suitable asymptotic behavior at infinity, the Dido functional equation has a one-parameter family of "principal" solutions. Some applications are given.

1999-09-30



Journal: Annales Mathematicae Silesianae

A Levi–Civita equation on monoids, two ways

Bruce Ebanks

Language: EN | Published: 12-05-2022 | Abstract | pp. 151-166

We consider the Levi–Civita equation
f(xy) = g₁(x)h₁(y) + g₂(x)h₂(y)
for unknown functions f, g_1, g_2, h_1, h₂ : S→ℂ, where S is a monoid. This functional equation contains as special cases many familiar functional equations, including the sine and cosine addition formulas. In a previous paper we solved this equation on groups and on monoids generated by their squares under the assumption that f is central. Here we solve the equation on monoids by two different methods. The first method is elementary and works on a general monoid, assuming only that the function f is central. The second way uses representation theory and assumes that the monoid is commutative. The solutions are found (in both cases) with the help of the recently obtained solution of the sine addition formula on semigroups. We also find the continuous solutions on topological monoids.

2022-05-12



Journal: Annales Mathematicae Silesianae

Stability of functional equations in dislocated quasimetric spaces

Beata Hejmej

Language: EN | Published: 14-05-2018 | Abstract | pp. 215-225

We present a result on the generalized Hyers–Ulam stability of a functional equation in a single variable for functions that have values in a complete dislocated quasi-metric space. Next, we show how to apply it to prove stability of the Cauchy functional equation and the linear functional equation in two variables, also for functions taking values in a complete dislocated quasimetric space. In this way we generalize some earlier results proved for classical complete metric spaces.

2018-05-14



Journal: Annales Mathematicae Silesianae

On some extensions of the Gołąb-Schinzel functional equation

Peter Kahlig , Janusz Matkowski

Language: EN | Published: 30-09-1994 | Abstract | pp. 13-31

Composite functional equations (arising in applications) are presented that may be interpreted as extensions of the Gołąb-Schinzel equation and as modifications of d'Alembert's equation. Depending on the type of the considered equation, continuous, and finite rate of growth solutions are discussed. Geometric interpretations are given.

1994-09-30



Journal: Annales Mathematicae Silesianae

Convex solutions of an iterative functional differential equation

Li Lin Huang , Hou Yu Zhao

Language: EN | Published: 05-03-2025 | Abstract

In this paper, by Schauder's fixed point theorem and the Banach contraction principle, we consider the existence, uniqueness, and stability of convex solutions of a nonhomogeneous iterative functional differential equation. Finally, some examples were considered by our results.

2025-03-05



Journal: Annales Mathematicae Silesianae

Stability of the Pexider functional equation

Roman Badora , Barbara Przebieracz , Peter Volkmann

Language: EN | Published: 30-09-2010 | Abstract | pp. 7-13

A stability result for the Pexider equation will be derived from a stability theorem published in [9] for the Cauchy functional equation. Then we discuss the quality of some constants occuring in this context; as a model case we consider functions defined on the multiplicative semigroup {1, 0}.

2010-09-30



Journal: Annales Mathematicae Silesianae

On a functional equation connected to Gauss quadrature rule

Barbara Koclęga-Kulpa , Tomasz Szostok

Language: EN | Published: 30-09-2008 | Abstract | pp. 27-40

We consider the functional equation
F(y)−F(x) = (y−x)[f(αx+βy)+f(βx+αy)]
stemming from Gauss quadrature rule. In previous results equations of this type with rational only coefficients α and β were considered. In this paper we allow these numbers to be irrational. We find all solutions of this equation for functions acting on ℝ. However, some results are valid also on integral domains.

2008-09-30



Journal: Annales Mathematicae Silesianae

A functional equation with biadditive functions

Radosław Łukasik

Language: EN | Published: 02-05-2022 | Abstract | pp. 193-205

Let S, H, X be groups. For two given biadditive functions A:S²→X, B:H²→X and for two unknown mappings T:S→H, g:S→S we will study the functional equation
B(T(x),T(y)) = A(x,g(y)), x,y∈S,
which is a generalization of the orthogonality equation in Hilbert spaces.

2022-05-02



Journal: Annales Mathematicae Silesianae

The stability of the Pexider equation

Kazimierz Nikodem

Language: EN | Published: 30-09-1991 | Abstract | pp. 91-93

A stability theorem for the Pexider functional equation (1) is proved. It is an analogue of the classical theorem of Hyers [5] relating to the stability of the Cauchy functional equation.

1991-09-30



Journal: Annales Mathematicae Silesianae

On a functional equation appearing on the margins of a mean invariance problem

Justyna Jarczyk , Witold Jarczyk

Language: EN | Published: 09-07-2020 | Abstract | pp. 96-103

Given a continuous strictly monotonic real-valued function α, defined on an interval I, and a function ω:I→(0,+∞) we denote by B_ω^α the Bajraktarević mean generated by α and weighted by ω:
B_ω^α(x,y) = α^-1(\frac{ω(x)}{ω(x)+ω(y)}α(x) + \frac{ω(y)}{ω(x)+ω(y)}α(y)), x,y∈I.
We find a necessary integral formula for all possible three times differentiable solutions (ϕ,ψ) of the functional equation
r(x)B_s^ϕ(x,y) + r(y)B_t^ψ{t}(x,y) = r(x)x + r(y)y,
where r, s,t:I→(0,+∞) are three times differentiable functions and the first derivatives of ϕ,ψ and r do not vanish. However, we show that not every pair (ϕ,ψ) given by the found formula actually satisfies the above equation.

2020-07-09



Journal: Annales Mathematicae Silesianae

A Kannappan-cosine functional equation on semigroups

Ahmed Jafar , Omar Ajebbar , Elhoucien Elqorachi

Language: EN | Published: 27-04-2024 | Abstract | pp. 143-154

In this paper we determine the complex-valued solutions of the Kannappan-cosine functional equation g(xyz₀) = g(x)g(y) − f(x)f(y), x,y∈S, where S is a semigroup and z₀ is a fixed element in S.

2024-04-27



Journal: Annales Mathematicae Silesianae

The cosine-sine functional equation on semigroups

Bruce Ebanks

Language: EN | Published: 05-10-2021 | Abstract | pp. 30-52

The primary object of study is the “cosine-sine” functional equation f(xy) = f(x)g(y)+g(x)f(y)+h(x)h(y) for unknown functions f,g,h:S→ℂ, where S is a semigroup. The name refers to the fact that it contains both the sine and cosine addition laws. This equation has been solved on groups and on semigroups generated by their squares. Here we find the solutions on a larger class of semigroups and discuss the obstacles to finding a general solution for all semigroups. Examples are given to illustrate both the results and the obstacles.
We also discuss the special case f(xy) = f(x)g(y)+g(x)f(y)–g(x)g(y) separately, since it has an independent direct solution on a general semigroup.
We give the continuous solutions on topological semigroups for both equations.

2021-10-05



Journal: Annales Mathematicae Silesianae

New results about quadratic functional equation on semigroups

Ahmed Akkaoui , Brahim Fadli

Language: EN | Published: 19-11-2024 | Abstract

Let S be a semigroup, let (H, +) be a uniquely 2-divisible, abelian group and let ϕ, ψ be two endomorphisms of S that need not be involutive. In this paper, we express the solutions f : S→H of the following quadratic functional equation
f(xϕ(y)) + f(ψ(y)x) = 2f(x) + 2f(y),   x,y ∈ S,
in terms of bi-additive maps and solutions of the symmetrized additive Cauchy equation. Some applications of this result are presented.

2024-11-19



Journal: Annales Mathematicae Silesianae

A parametric functional equation originating from number theory

Aziz Mouzoun , Driss Zeglami , Youssef Aissi

Language: EN | Published: 17-01-2022 | Abstract | pp. 71-91

Let S be a semigroup and α,β∈ℝ. The purpose of this paper is to determine the general solution f:ℝ²→S of the following parametric functional equation
f(x₁x₂+αy₁y₂,x₁y₂+x₂y₁+βy₁y₂) = f(x₁,y₁)f(x₂,y₂),
for all (x₁,y₁), (x₂,y₂)∈ℝ², that generalizes some functional equations arising from number theory and is connected with the characterizations of the determinant of matrices.

2022-01-17



Journal: Annales Mathematicae Silesianae

Functional equations with an anti-endomorphism for functions with multidimensional codomains

Ayoub Ouhabi , Driss Zeglami , Mohamed Ayoubi

Language: EN | Published: 25-12-2024 | Abstract

Let S be a semigroup, ℍ be the skew field of quaternions, and ψ: S→S be an anti-endomorphism. We determine the general solution of the functional equation
g(xy) - g(xψ(y)) = 2g(x)g(y),    x,y ∈ S,
where g: S→ℂ is the unknown function. And when S = M is a monoid, we solve the functional equation
g(xy) + g(xψ(y)) = 2g(x)g(y),   x,y ∈ M,
where g: M→ℍ is the unknown function.

2024-12-25



Journal: Annales Mathematicae Silesianae

Alienation of Drygas' and Cauchy's functional equations

Youssef Aissi , Driss Zeglami , Brahim Fadli

Language: EN | Published: 13-04-2021 | Abstract | pp. 131-148

Inspired by the papers [2, 10] we will study, on 2-divisible groups that need not be abelian, the alienation problem between Drygas’ and the exponential Cauchy functional equations, which is expressed by the equation
f(x + y) + g(x + y) + g(x - y) = f(x)f(y) + 2g(x) + g(y) + g(-y).
We also consider an analogous problem for Drygas’ and the additive Cauchy functional equations as well as for Drygas’ and the logarithmic Cauchy functional equations. Interesting consequences of these results are presented.

2021-04-13



Journal: Annales Mathematicae Silesianae

Solutions and stability of generalized Kannappan’s and Van Vleck’s functional equations

Elhoucien Elqorachi , Ahmed Redouani

Language: EN | Published: 05-08-2017 | Abstract | pp. 169-200

We study the solutions of the integral Kannappan’s and Van Vleck’s functional equations
∫_Sf(xyt)dμ(t) + ∫_Sf(xσ(y)t)dμ(t) = 2f(x)f(y), x,y∈S;
∫_Sf(xσ(y)t)dμ(t) - ∫_Sf(xyt)dμ(t) = 2f(x)f(y), x,y∈S,
where S is a semigroup, is an involutive automorphism of S and μ is a linear combination of Dirac measures (δ_{z_i})_i∈I, such that for all i∈I, z_i is in the center of S. We show that the solutions of these equations are closely related to the solutions of the d’Alembert’s classic functional equation with an involutive automorphism. Furthermore, we obtain the superstability theorems for these functional equations in the general case, where σ is an involutive morphism.

2017-08-05



Journal: Annales Mathematicae Silesianae

On some conditional functional equations

Tomasz Szostok

Language: EN | Published: 30-01-2003 | Abstract | pp. 65-77

Let X, Y be real linear spaces. We are looking for a function G : X²→ℝ such that the equation
f(x+y) = G(x,y)[f(x)+f(y)]
is equivalent to the orthogonal Cauchy equation
x⟂y ⇒ f(x+y)=f(x)+f(y).
Several kinds of orthogonalities are considered. The quotient ‖(x-y)/(x+y)‖ closely connected with the James orthogonality plays here a distinguished role. Similar problems are considered for the Ptolemaic equation
x⟂y ⇒ f(x+y)f(x-y) = f(x)²+f(y)².
As a result a characterization of inner product spaces is obtained.

2003-01-30



Journal: Annales Mathematicae Silesianae

Alienation of the Jensen, Cauchy and d’Alembert equations

Barbara Sobek

Language: EN | Published: 23-09-2016 | Abstract | pp. 181-191

Let (S,+) be a commutative semigroup, σ: S→S be an endomorphism with σ² = id and let K be a field of characteristic different from 2. Inspired by the problem of strong alienation of the Jensen equation and the exponential Cauchy equation, we study the solutions f,g: S→K of the functional equation
f(x+y) + f(x+σ(y)) + g(x+y) = 2f(x) + g(x)g(y)   for x,y∈S.
We also consider an analogous problem for the Jensen and the d’Alembert equations as well as for the d’Alembert and the exponential Cauchy equations.

2016-09-23



Journal: Annales Mathematicae Silesianae

On a Jensen–Hosszú equation I

Zygfryd Kominek

Language: EN | Published: 30-09-2009 | Abstract | pp. 57-60

We solve functional equation of the form
f(x+y−xy) + f(xy) = 2f((x+y)/2)
in the class of functions transforming the space of all reals into itself. We also prove that this equation is stable in the Hyers-Ulam’s sense.

2009-09-30



Journal: Annales Mathematicae Silesianae

On the Cauchy equation on spheres

Roman Ger , Justyna Sikorska

Language: EN | Published: 30-09-1997 | Abstract | pp. 89-99

We deal with a conditional functional equation
ϕ(x) = ϕ(y)  implies  f(x+y) = f(x)+f(y).
Under some assumptions imposed on function ϕ, the domains and ranges of ϕ and f we derive the additivity of f. As a consequence we obtain two Hyers-Ulam stability results related to the equation considered.

1997-09-30



Journal: Annales Mathematicae Silesianae

The alienation phenomenon and associative rational operations

Roman Ger

Language: EN | Published: 30-09-2013 | Abstract | pp. 75-88

The alienation phenomenon of ring homomorphisms may briefly be described as follows: under some reasonable assumptions, a map f between two rings satisfies the functional equation
(*)       f(x + y) + f(xy) = f(x) + f(y) + f(x)f(y)
if and only if f is both additive and multiplicative. Although this fact is surprising for itself it turns out that that kind of alienation has also deeper roots. Namely, observe that the right hand side of equation (*) is of the form Q(f(x), f(y)) with the map Q(u, v) = u+v+uv being a special rational associative operation. This gives rise to the following question: given an abstract rational associative operation Q does the equation
f(x + y) + f(xy) = Q(f(x), f(y))
force f to be a ring homomorphism (with the target ring being a field)?
Plainly, in general, that is not the case. Nevertheless, the 2-homogeneity of f happens to be a necessary and sufficient condition for that effect provided that the range of f is large enough.

2013-09-30



Journal: Annales Mathematicae Silesianae

A variant of d'Alembert's functional equation on semigroups with endomorphisms

Ahmed Akkaoui , Mohamed El Fatini , Brahim Fadli

Language: EN | Published: 22-03-2022 | Abstract | pp. 1-14

Let S be a semigroup, and let ϕ,ψ:S→S be two endomorphisms (which are not necessarily involutive). Our main goal in this paper is to solve the following generalized variant of d’Alembert’s functional equation
f(xϕ(y))+f(ψ(y)x) = 2f(x)f(y), x,y∈S,
where f:S→ℂ is the unknown function by expressing its solutions in terms of multiplicative functions. Some consequences of this result are presented.

2022-03-22



Journal: Annales Mathematicae Silesianae

Differentiable solutions of functional equations in Banach spaces

Maciej Sablik

Language: EN | Published: 30-09-1993 | Abstract | pp. 17-55

We deal with the functional equation
ϕ(F(x)) = g(x,ϕ(x))
where functions F and g are given, defined in open subsets of Banach spaces and taking values in Banach spaces as well. We prove theorems on the existtence and uniqueness of solutions of the equation in classes of differentiable functions. As corollaries we get some results on the conjugacy of diffeomorphisms. Analogous results have been known in the finite dimensional case only.

1993-09-30



Journal: Annales Mathematicae Silesianae

Mixed stability of the d'Alembert functional equation

Maciej Przybyła

Language: EN | Published: 30-09-2005 | Abstract | pp. 53-57

In the present paper we will prove the theorem concerning the mixed stability of the d'Alembert functional equation, i.e. we will show that if ɛ > 0,
s ≥ 1, δ = [2^s + \sqrt{2^2s+16ɛ+8}]/4, X is a real normed space and f: X→ℂ satisfies the inequality
|f(x+y) + f(x-y) - 2f(x)f(y)| ≤ ɛ(∥x∥^s + ∥y∥^s)
for all x,y∈X, then |f(x)| ≤ δ∥x∥^s for all x∈X such that ∥x∥ ≥ 1, or f(x+y) + f{x-y) = 2f(x)f(y) for all x,y∈X.

2005-09-30



Journal: Annales Mathematicae Silesianae

On approximate solutions of an iterative functional equation

Karol Baron , Alice Simon

Language: EN | Published: 30-09-1994 | Abstract | pp. 79-84

We consider approximate solutions of the functional equation (1) in the class of functions which satisfy on compact sets the condition (2) with an increasing, subadditive, continuous at zero and vanishing at zero function γ: [0,+∞)→[0.+∞).

1994-09-30



Journal: Annales Mathematicae Silesianae

A variant of d’Alembert’s matrix functional equation

Youssef Aissi , Driss Zeglami , Mohamed Ayoubi

Language: EN | Published: 14-12-2020 | Abstract | pp. 21-43

The aim of this paper is to characterize the solutions Φ:G→M₂(ℂ) of the following matrix functional equations
\frac{Φ(xy)+Φ(σ(y)x)}{2} = Φ(x)Φ(y), x,y∈G,
and
\frac{Φ(xy)-Φ(σ(y)x)}{2} = Φ(x)Φ(y), x,y∈G,
where G is a group that need not be abelian, and σ:G→G is an involutive automorphism of G. Our considerations are inspired by the papers [13, 14] in which the continuous solutions of the first equation on abelian topological groups were determined.

2020-12-14



Journal: Annales Mathematicae Silesianae

On a functional equation

Andrzej Smajdor

Language: EN | Published: 30-09-1994 | Abstract | pp. 217-226

The functional equation
ϕ(x+y) + ψ(x) + ψ(y) = ψ(x+y) + ϕ(x) + ϕ(y)
is studied for set-valued functions.

1994-09-30



Journal: Annales Mathematicae Silesianae

Corrigendum to "A functional equation characterizing homographic functions'' [Annales Mathematicae Silesianae 24 (2010), 61–70]

Janusz Matkowski

Language: EN | Published: 30-09-2011 | Abstract | pp. 101-102

Corrigendum to "A functional equation characterizing homographic functions'' [Annales Mathematicae Silesianae 24 (2010), 61–70].

2011-09-30



Journal: Annales Mathematicae Silesianae

On dependence of Lipschitzian solution of non-linear functional equation on an arbitrary function

Katarzyna Jakowska-Suwalska

Language: EN | Published: 30-09-1985 | Abstract | pp. 116-119

We shall deal with the existence and dependence on an arbitrary function of solutions of the functional equation
(1)     ϕ(f(x)) = g(x,ϕ(x))
in the class of functions fulfilling a Lipschitz condition.

1985-09-30



Journal: Annales Mathematicae Silesianae

On Popoviciu-Ionescu functional equation

Jose M. Almira

Language: EN | Published: 23-09-2016 | Abstract | pp. 5-15

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.

2016-09-23



Journal: Annales Mathematicae Silesianae

A generalized a-Wright convexity and related functional equation

Janusz Matkowski , Małgorzata Wróbel

Language: EN | Published: 30-09-1996 | Abstract | pp. 7-12

Let I be an interval and M,N: I×I→I some means with the strict internality property. Suppose that ϕ: I→ℝ is a non-constant and continuous solution of the functional equation
ϕ(M(x,y)) + ϕ(N(x,y)) = ϕ(x) + ϕ(y).
Then ϕ is one-to-one; moreover for every lower semicontinuous function f: I→ℝ satisfying the inequality
f(M(x,y)) + f(N(x,y)) ≤ f(x) + f(y),
the function f◦ϕ^-1 is convex on ϕ(I). This is a generalization of an earlier result of Zs. Páles. An application to the a-Wright convex function is given.

1996-09-30



Journal: Annales Mathematicae Silesianae

On a functional equation connected with Ptolemaic inequality

Irena Fidytek

Language: EN | Published: 28-09-2001 | Abstract | pp. 27-44

In the first part of the present paper we consider Ptolemaic inequality and give some necessary conditions for its solutions. The other part is devoted to solution of some functional equation which, simultaneously, satisfies the Ptolemaic inequality.

2001-09-28



Journal: Annales Mathematicae Silesianae

On stability of the Cauchy functional equation in groupoids

Imke Toborg , Peter Volkmann

Language: EN | Published: 18-12-2016 | Abstract | pp. 155-164

We give some stability results for the functional equation a(xy) = a(x) + a(y), where a: G→E, G being a groupoid and E a Banach space.

2016-12-18



Journal: Annales Mathematicae Silesianae

Remarks connected with the weak limit of iterates of some random-valued functions and iterative functional equations

Karol Baron

Language: EN | Published: 08-05-2020 | Abstract | pp. 36-44

The paper consists of two parts. At first, assuming that (Ω,A,P) is a probability space and (X,ρ) is a complete and separable metric space with the σ-algebra B of all its Borel subsets we consider the set R_c of all B ⊗A measurable and contractive in mean functions f:X×Ω→X with finite integral ∫_Ωρ(f(x,ω),x)P(dω) for x∈X, the weak limit π^f of the sequence of iterates of f∈R_c, and investigate continuity-like property of the function f↦π^f, f∈?R_c, and Lipschitz solutions ϕ that take values in a separable Banach space of the equation
ϕ(x) = ∫_Ωϕ(f(x,ω))P(dω) + F(x).
Next, assuming that X is a real separable Hilbert space, Λ:X→X is linear and continuous with ‖Λ‖<1, and μ is a probability Borel measure on X with finite first moment we examine continuous at zero solutions ϕ:X→ℂ of the equation
ϕ(x) = μ^{^}(x)ϕ(Λx)
which characterizes the limit distribution π^f for some special f∈R_c.

2020-05-08



Journal: Annales Mathematicae Silesianae

Addition formulae with singularities

Katarzyna Domańska , Roman Ger

Language: EN | Published: 30-09-2004 | Abstract | pp. 7-20

We deal with functional equations of the form
f(x+y) = F(f(x),f(y))
(so called addition formulas) assuming that the given binary operation F is associative but its domain of definition is disconnected (admits "singularities"). The function
Flu,v) := (u+v)/(1+uv)
serves here as a good example; the corresponding equation characterizes the hyperbolic tangent. Our considerations may be viewed as counterparts of L. Losonczi's [4] and K. Domańska's [2] results on local solutions of the functional equation
f(F(x,y)) = f(x) + f(y)
with the same behaviour of the given associative operation F.
Our results exhibit a crucial role of 1 that turns out to be the critical value towards the range of the unknown function. What concerns the domain we admit fairly general structures (groupoids, groups, 2-divisible groups). In the case where the domain forms a group admitting subgroups of index 2 the family of solutions enlarges considerably.

2004-09-30



Journal: Annales Mathematicae Silesianae

On a system of simultaneous iterative functional equations

Janusz Matkowski

Language: EN | Published: 29-09-1995 | Abstract | pp. 123-135

A system of two simultaneous functional equations in a single variable, related to a generalized Gołąb-Schinzel functional equation, is considered.

1995-09-29



Journal: Annales Mathematicae Silesianae

On the existence of two solutions of functional boundary value problems

Svatoslav Staněk

Language: EN | Published: 23-07-2022 | Abstract | pp. 93-109

The functional differential equation (g(x'(t)))' = (Fx)(t) is considered. Here g is an increasing homomorphism on ℝ, g(0) = 0 and F: C¹(J)→L₁(J) is a continuous operator satisfying a growth condition with respect to x. A class of nonlinear functional boundary conditions is considered and sufficient conditions for the existence at least one positive and one negative solutions of the boundary value problems are given. Results are proved by the homotopy theory, the Leray-Schauder degree and the Borsuk theorem.

2022-07-23



Journal: Annales Mathematicae Silesianae

New Pexiderizations of Drygas’ functional equation on abelian semigroups

Youssef Aissi , Driss Zeglami

Language: EN | Published: 29-08-2023 | Abstract | pp. 169-184

Let (S,+) be an abelian semigroup, let (H,+) be an abelian group which is uniquely 2-divisible, and let ϕ be an endomorphism of S. We find the solutions f,h: S→H of each of the functional equations
f(x+y) + f(x+ϕ(y)) = h(x) + f(y) + f◦ϕ(y), x,y∈S,
f(x+y) + f(x+ϕ(y)) = h(x) + 2f(y), x,y∈S,
in terms of additive and bi-additive maps. Moreover, as applications, we determine the solutions of some related functional equations.

2023-08-29



Journal: Annales Mathematicae Silesianae

On analytic solutions of the equation ϕ(f(x)) = g(x,ϕ(x)) (III)

Joanna Ger

Language: EN | Published: 30-09-1985 | Abstract | pp. 93-102

In the present paper we deal with local analytic solutions of the functional equation announced in the title. This paper is a continuation of [2] and [3].

1985-09-30



Journal: Annales Mathematicae Silesianae

Sur les lois de sortie des semi-groupes de convolution

Imed Bachar

Language: EN | Published: 28-09-2001 | Abstract | pp. 7-15

Let μ = (μ_t)_t>o be a convolution semigroup on ℝ^d. An exit law for μ is a positive measurable function ϕ: ]0,∞[ × ℝ^d → [0,∞] which verifies the functional equation (by putting ϕ_t:=ϕ(t,·))
∀_s,t>0 : μ_s ⚹ ϕ_t = ϕ_s+t   λ.a.e.
where λ is the Lebesgue measure on ℝ^d. Following [1], we prove in this paper that the solutions of this equation are on the form
ϕ_t = d(μ^{^}_t ⚹ β)/dλ   λ.a.e.
where μ^{^} := (μ^{^}_t)_t>o is the reflected convolution semigroup of μ, β is a positive measure on ℝ^d such that μ^{^}_t ⚹ β ≪ λ, for evry t > 0.
Moreover, we study the global solutions and their interpretations in terms of the negative definite function associated to μ.

2001-09-28



Journal: Annales Mathematicae Silesianae

A note on bifurcation of equilibrium forms of a gas baloon parachute

Anita Zgorzelska , Hanna Guze

Language: EN | Published: 20-05-2025 | Abstract

We will be concerned with deformations of a free elastic top rim of a parachute of a gas balloon. The top rim is connected with the circular deflation port of the balloon envelope by heavy duty flexible load tapes. The inside part of the balloon is filled with compressed gas. Equilibrium forms of the parachute may be found as solutions of a certain nonlinear functional-differential equation with two physical parameters: an elasticity coefficient of tapes and a physical parameter describing compressed gas. This equation possesses radially symmetric solutions corresponding to circular shapes of the top rim. Our goal is to study the existence of symmetry breaking bifurcation of the top rim of parachute.

2025-05-20



Journal: Annales Mathematicae Silesianae

Some remarks on the Daróczy equation

Lech Bartłomiejczyk

Language: EN | Published: 29-09-1995 | Abstract | pp. 47-63

The general solution of the functional equation
f(x) = f(x+1) + f(x(x+1)),
considered both on (0,+∞) and ℝ, are studied. Constructions of odd and even solutions are given.

1995-09-29



Journal: Annales Mathematicae Silesianae

Families of commuting formal power series, semicanonical forms and iterative roots

Ludwig Reich

Language: EN | Published: 30-09-1994 | Abstract | pp. 189=201

We consider for a given formal power series F(x) = ρx + c₂x² + ... , ρ ≠ 0, with complex coefficients and for a given integer N > 1 the functional equation G^N = F where G is again a formal series (G is called an iterative root of F). If ρ is a root of 1 and F is not conjugate to its linear part we derive a criterion for the existence of solutions G and describe the general solution. Representations of the coefficients of G by means of universal polynomials are given, also in the case where ρ is not a root of 1 (where existence is almost trivial). Our main tools are maximal families of commuting series (i.e. the Aczél-Jabotinsky equation of third type) and semicanonical forms.

1994-09-30



Journal: Annales Mathematicae Silesianae

Note on an iterative functional equation

Karol Baron , Janusz Morawiec

Language: EN | Published: 10-01-2024 | Abstract | pp. 12-17

We study the problem of solvability of the equation
ϕ(x) = ∫_Ωg(ω)ϕ(f(x,ω))P(dω) + F(x)
where P is a probability measure on a σ-algebra of subsets of Ω, assuming Hölder continuity of F on the range of f.

2024-01-10



Journal: Annales Mathematicae Silesianae

Transforming the functional equation of Gołąb-Schinzel into one of Cauchy

Peter Kahlig , Jens Schwaiger

Language: EN | Published: 30-09-1994 | Abstract | pp. 33-38

It is shown that Gołąb-Schinzel's equation may be transformed into one of Cauchy's equations by an embedding and limit process concerning the general continuous solution.

1994-09-30



Journal: Annales Mathematicae Silesianae

On analytic solutions of the non-linear functional equation

Joanna Ger

Language: EN | Published: 30-09-1985 | Abstract | pp. 103-115

In the present paper we consider the problem of the existence and uniqueness of local analytic solutions of the equation ϕ(f(x)) = V(ϕ(x))+U(x). This paper is a continuation of [1], [2] and [3].

1985-09-30



Journal: Annales Mathematicae Silesianae

A functional equation characterizing homographic functions Janusz Matkowski

Janusz Matkowski

Language: EN | Published: 30-09-2010 | Abstract | pp. 61-70

Some functional equations related to homographic functions and their characterization are presented.

2010-09-30



Journal: Annales Mathematicae Silesianae

Boundary value problems for one-parameter second-order differential equations

Svatoslav Staněk

Language: EN | Published: 30-09-1993 | Abstract | pp. 89-98

The paper establishes sufficient conditions for the existence of solutions of a one-parameter differential equation x" = f(t,x,x',λ) satisfying some of the following boundary conditions:
γ(x) = 0,  x'(a) = x'(b) = 0,
x'(a) = x'(b) = 0,  x(c)-x(d) = 0
and
x'(a) = x'(t₀) = x'(b) = 0.
Here γ is a functional. The application is given for a class of one-parameter functional boundary value problems.

1993-09-30



Journal: Annales Mathematicae Silesianae

Mixed type of additive and quintic functional equations

Abasalt Bodaghi , Pasupathi Narasimman , Krishnan Ravi , Behrouz Shojaee

Language: EN | Published: 30-09-2015 | Abstract | pp. 35-50

In this paper, we investigate the general solution and Hyers–Ulam–Rassias stability of a new mixed type of additive and quintic functional equation of the form
f(3x + y) - 5f(2x + y) + f(2x - y) + 10f(x + y) - 5f(x - y) = 10f(y) + 4f(2x) - 8f(x)
in the set of real numbers.

2015-09-30



Journal: Annales Mathematicae Silesianae

On some conditional functional equations of Gołąb-Schinzel type

Janusz Brzdęk

Language: EN | Published: 29-09-1995 | Abstract | pp. 65-80

We consider equation (6) in the class of continuous functions f: I→ℝ satisfying (7), where f is a non-trivial real interval and n,k are fixed positive integers. The obtained results are applied to get the solutions of the system of functional equations (3)-(5) in the class of pairs of functions f,g: I→ℝ such that f is continuous. Some connections between solutions of the equations and a class of subsemigroups of some Lie groups are established as well.

1995-09-29



Journal: Annales Mathematicae Silesianae

Steklov type operators and functional equations

Gabriela Motronea , Dorian Popa , Ioan Raşa

Language: EN | Published: 15-02-2024 | Abstract | pp. 57-63

We consider sequences of Steklov type operators and an associated functional equation. For a suitable sequence, we establish asymptotic formulas.

2024-02-15



Journal: Annales Mathematicae Silesianae

Two functional equations on groups

Zsolt Ádám , Károly Lajkó , Gyula Maksa , Fruzsina Mészáros

Language: EN | Published: 28-09-2007 | Abstract | pp. 7-13

In this note we give the general solution of the functional equation
f(x) f(x+y) = f(y)² f(x−y)² g(y),  x,y∈G,
and all the solutions of
f(x) f(x+y) = f(y)² f(x−y)² g(x) ,  x,y∈G,
with the additional supposition g(x) ≠ 0 for all x∈G. In both cases G denotes an arbitrary group written additively and f,g: G→ℝ are the unknown functions.

2007-09-28



Journal: Annales Mathematicae Silesianae

On a factorization of mappings with a prescribed behaviour of the Cauchy difference

Roman Ger

Language: EN | Published: 30-09-1994 | Abstract | pp. 141-155

We deal with functional congruences of the form
f(x+y) - f(x) - f(y) ∈ U + V
where U and V are given sets subjected to satisfy some "separability" conditions essentially weaker than that occurring in [4] which proved to be pretty useful especially while investigating various types of Hyers-Ulam stability problems. The goal is to factorize f into a sum of two functions whose Cauchy differences remain in U and V, respectively, or, at least to obtain an approximation of f by such a sum. An application of the newly established result in that spirit is given. Moreover, a stability result for the celebrated cocycle equation is presented and, finally, the behaviour of mappings whose Cauchy differences fall into a given Hamel basis is described.

1994-09-30



Journal: Annales Mathematicae Silesianae

A note on the exponential distribution

Wolfgang Sander

Language: EN | Published: 30-01-1990 | Abstract | pp. 70-71

A characterization of the exponential distribution via a functional equation on a restricted domain is given.

1990-01-30



Journal: Annales Mathematicae Silesianae

On approximately additive functions

Roman Badora

Language: EN | Published: 30-09-1994 | Abstract | pp. 111-126

In the present paper we find a linear operator on a function space, essentially larger than the space of all bounded functions on an amenable semigroup, which behaves like an invariant mean. This leads to an extension of the Hyers-Ulam stability theorem for Cauchy's functional equation in the case of vector-valued mappings defined on amenable semigroups.

1994-09-30



Journal: Annales Mathematicae Silesianae

On pseudoadditive mappings

Zygfryd Kominek

Language: EN | Published: 30-09-2008 | Abstract | pp. 41-44

We solve the following functional equation
f(x+y+z) + g(x+y) = q(z) + p(y) + h(x),
where f, g, q, p, h are unknown functions transforming a semigroup into a group.

2008-09-30



Journal: Annales Mathematicae Silesianae

Remark on homomorphisms of groups

Ivica Gusić

Language: EN | Published: 23-07-2022 | Abstract | pp. 81-83

Assume that (X,+) are commutative groups, and C is a subset of X fulfilling the conditions C+C ⊆ C and C-C = X. A function f: X→Y is called C-additive function if it satisfies functional equation f(x+y) = f(x) + f{y), for all x,y∈X such that y-x∈C∪(-C)∪{0}. In [1, Theorem 8.4] has been proved that every C-additive function f: X→Y is additive. In the proof the comutativity has been essentially used. Here we present a simple proof of an analogous statement in the case of arbitrary groups.

2022-07-23



Journal: Annales Mathematicae Silesianae

Stability of functional equations and properties of groups

Gian Luigi Forti

Language: EN | Published: 09-05-2019 | Abstract | pp. 77-96

Investigating Hyers–Ulam stability of the additive Cauchy equation with domain in a group G, in order to obtain an additive function approximating the given almost additive one we need some properties of G, starting from commutativity to others more sophisticated. The aim of this survey is to present these properties and compare, as far as possible, the classes of groups
involved.

2019-05-09



Journal: Annales Mathematicae Silesianae

Regular solutions of some functional equations in the indeterminate case

Małgorzata Rozmus-Chmura

Language: EN | Published: 30-09-1985 | Abstract | pp. 120-129

The paper deals with the existence and uniqueness of regular solutions of the equation ϕ(x) = h(x,ϕ(f(x))). Also in the indeterminate case the existence of solutions of ϕ(f(x)) = g(x,ϕ(x)) is studied.

1985-09-30



Journal: Annales Mathematicae Silesianae

On a functional equation related to two-sided centralizers

Irena Kosi-Ulbl

Language: EN | Published: 31-01-2018 | Abstract | pp. 227-235

The main aim of this manuscript is to prove the following result. Let n>2 be a fixed integer and R be a k-torsion free semiprime ring with identity, where k∈{2, n−1, n}. Let us assume that for the additive mapping T:R→R
3T(xⁿ) = T(x)x^n-1 + xT(x^n-2)x + x^n-1T(x),  x∈R,
is also fulfilled. Then T is a two-sided centralizer.

2018-01-31



Journal: Annales Mathematicae Silesianae

Superstability of the d'Alembert functional equation in L_p^+ spaces

Maciej J. Przybyła

Language: EN | Published: 30-09-2004 | Abstract | pp. 39-47

Let (X,+,-,0,Σ,μ) be an abelian complete measurable group with μ(X)>0. Let f: X→ℂ be a function. We will show that if A(f)∈L_p⁺(X×X,ℂ) where
A{f)(x,y) = f{x+y) + f(x-y) - 2f(x)f(y),   x,y∈X,
then f∈L_p⁺(X,ℂ) or there exists exactly one function g: X→ℂ with
g{x+y) + g(x-y) - 2g(x)g(y),   x,y∈X
such that f is equal to g almost everywhere with respect to the measure μ.
L_p⁺ denotes the space of all functions for which the upper integral of ∥f∥^p is finite.

2004-09-30



Journal: Annales Mathematicae Silesianae

Quasi-Jensen functions

Jacek Chmieliński

Language: EN | Published: 30-09-1992 | Abstract | pp. 30-41

There is defined quasi-Jensen function as a solution of a certain functional inequality which generalizes the classical Jensen equation: f((x+y)/2) = (f(x)+f(y))/2. The introduced inequality is analogous to the inequality which defines J. Tabor's quasi-additive functions. The main result of this paper is to show strong relationship between quasi-Jensen and quasi-additive functions.

1992-09-30