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Integrable solutions of functional equations

100%
Matkowski J.
Instytut Matematyczny Polskiej Akademii Nauk
EN
Contents Introduction............................................................................................................................................................................... 5 0. Explanatory notes, definitions and a lemma................................................................................................................. 5 1. Some fixed point theorems............................................................................................................................................... 7 2. Integrable solutions of a linear functional equation of order 1................................................................................... 15 3. Integrable solutions of a non-linear functional equation of order 1.......................................................................... 28 4. Integrable solutions of systems of functional equations............................................................................................ 40 5. Integrable solutions of equations of higher orders...................................................................................................... 46 6. The case of general measures........................................................................................................................................ 53 References............................................................................................................................................................................... 63
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Generalized weighted quasi-arithmetic means and the Kolmogorov-Nagumo theorem

95%
Matkowski J.
Colloquium Mathematicum
|
2013
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tom 133
|
nr 1
35-49
EN
A generalization of the weighted quasi-arithmetic mean generated by continuous and increasing (decreasing) functions $f₁,...,f_{k}:I → ℝ$, k ≥ 2, denoted by $A^{[f₁,...,f_{k}]}$, is considered. Some properties of $A^{[f₁,...,f_{k}]}$, including "associativity" assumed in the Kolmogorov-Nagumo theorem, are shown. Convex and affine functions involving this type of means are considered. Invariance of a quasi-arithmetic mean with respect to a special mean-type mapping built of generalized means is applied in solving a functional equation. For a sequence of continuous strictly increasing functions $f_{j}:I → ℝ$, j ∈ ℕ, a mean $A^{[f₁,f₂,...]}: ⋃_{k=1}^{∞} I^{k} → I$ is introduced and it is observed that, except symmetry, it satisfies all conditions of the Kolmogorov-Nagumo theorem. A problem concerning a generalization of this result is formulated.
3
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Fixed points and iterations of mean-type mappings

95%
Matkowski J.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2215-2228
EN
Let (X, d) be a metric space and T: X → X a continuous map. If the sequence (T n)n∈ℕ of iterates of T is pointwise convergent in X, then for any x ∈ X, the limit $$\mu _T (x) = \mathop {\lim }\limits_{n \to \infty } T^n (x)$$ is a fixed point of T. The problem of determining the form of µT leads to the invariance equation µT ○ T = µT, which is difficult to solve in general if the set of fixed points of T is not a singleton. We consider this problem assuming that X = I p, where I is a real interval, p ≥ 2 a fixed positive integer and T is the mean-type mapping M =(M 1,...,M p) of I p. In this paper we give the explicit forms of µM for some classes of mean-type mappings. In particular, the classical Pythagorean harmony proportion can be interpreted as an important invariance equality. Some applications are presented. We show that, in general, the mean-type mappings are not non-expansive.
4
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Invariance identity in the class of generalized quasiarithmetic means

95%
Matkowski J.
Colloquium Mathematicum
|
2014
|
tom 137
|
nr 2
221-228
EN
An invariance formula in the class of generalized p-variable quasiarithmetic means is provided. An effective form of the limit of the sequence of iterates of mean-type mappings of this type is given. An application to determining functions which are invariant with respect to generalized quasiarithmetic mean-type mappings is presented.
5
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On subadditive functions and ψ-additive mappings

95%
Matkowski J.
Open Mathematics
|
2003
|
tom 1
|
nr 4
435-440
EN
In [4], assuming among others subadditivity and submultiplicavity of a function ψ: [0, ∞)→[0, ∞), the authors proved a Hyers-Ulam type stability theorem for “ψ-additive” mappings of a normed space into a normed space. In this note we show that the assumed conditions of the function ψ imply that ψ=0 and, consequently, every “ψ-additive” mapping must be additive
6
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A functional equation related to an equality of means problem

95%
Matkowski J.
Colloquium Mathematicum
|
2011
|
tom 122
|
nr 2
289-298
EN
The functional equation (F(x)-F(y))/(x-y) = (G(x)+G(y))(H(x)+H(y)) where F,G,H are unknown functions is considered. Some motivations, coming from the equality problem for means, are presented.
7
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Affine and convex functions with respect to the logarithmic mean

95%
Matkowski J.
Colloquium Mathematicum
|
2003
|
tom 95
|
nr 2
217-230
EN
The class of all functions f:(0,∞) → (0,∞) which are continuous at least at one point and affine with respect to the logarithmic mean is determined. Some related results concerning the functions convex with respect to the logarithmic mean are presented.
8
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The converse of the Hölder inequality and its generalizations

95%
Matkowski J.
Studia Mathematica
|
1994
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tom 109
|
nr 2
171-182
EN
Let (Ω,Σ,μ) be a measure space with two sets A,B ∈ Σ such that 0 < μ (A) < 1 < μ (B) < ∞ and suppose that ϕ and ψ are arbitrary bijections of [0,∞) such that ϕ(0) = ψ(0) = 0. The main result says that if $ʃ_Ω xydμ ≤ ϕ^{-1} (ʃ_{Ω} ϕ∘x dμ) ψ^{-1} (ʃ_{Ω} ψ∘x dμ)$ for all μ-integrable nonnegative step functions x,y then ϕ and ψ must be conjugate power functions. If the measure space (Ω,Σ,μ) has one of the following properties: (a) μ (A) ≤ 1 for every A ∈ Σ of finite measure; (b) μ (A) ≥ 1 for every A ∈ Σ of positive measure, then there exist some broad classes of nonpower bijections ϕ and ψ such that the above inequality holds true. A general inequality which contains integral Hölder and Minkowski inequalities as very special cases is also given.
9
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Uniformly bounded set-valued Nemytskij operators acting between generalized Hölder function spaces

60%
Matkowski J., Wróbel M.
Open Mathematics
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2012
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tom 10
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nr 2
609-618
EN
We show that the generator of any uniformly bounded set-valued Nemytskij composition operator acting between generalized Hölder function metric spaces, with nonempty, bounded, closed, and convex values, is an affine function.
10
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A pair of linear functional inequalities and a characterization of $L^{p}$-norm

60%
Krassowska D., Matkowski J.
Annales Polonici Mathematici
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2005
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tom 85
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nr 1
1-11
EN
It is shown that, under some general algebraic conditions on fixed real numbers a,b,α,β, every solution f:ℝ → ℝ of the system of functional inequalities f(x+a) ≤ f(x)+α, f(x+b) ≤ f(x)+β that is continuous at some point must be a linear function (up to an additive constant). Analogous results for three other similar simultaneous systems are presented. An application to a characterization of $L^{p}$-norm is given.
11
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Uniformly bounded Nemytskij operators generated by set-valued functions between generalized Hölder function spaces

60%
Matkowski J., Wróbel M.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2011
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tom 31
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nr 2
183-198
EN
We prove that the generator of any uniformly bounded set-valued Nemytskij operator acting between generalized Hölder function metric spaces, with nonempty compact and convex values is an affine function with respect to the function variable.
12
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On the Bellman equation for asymptotics of utility from terminal wealth

60%
Matkowski J., Stettner Ł.
Applicationes Mathematicae
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2010
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tom 37
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nr 1
89-97
EN
The asymptotics of utility from terminal wealth is studied. First, a finite horizon problem for any utility function is considered. To study a long run infinite horizon problem, a certain positive homogeneity (PH) assumption is imposed. It is then shown that assumption (PH) is practically satisfied only by power and logarithmic utility functions.
13
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Convex-like inequality, homogeneity, subadditivity, and a characterization of $L^p$-norm

60%
Matkowski J., Pycia M.
Annales Polonici Mathematici
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1994-1995
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tom 60
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nr 3
221-230
EN
Let a and b be fixed real numbers such that 0 < min{a,b} < 1 < a + b. We prove that every function f:(0,∞) → ℝ satisfying f(as + bt) ≤ af(s) + bf(t), s,t > 0, and such that $limsup_{t → 0+} f(t) ≤ 0$ must be of the form f(t) = f(1)t, t > 0. This improves an earlier result in [5] where, in particular, f is assumed to be nonnegative. Some generalizations for functions defined on cones in linear spaces are given. We apply these results to give a new characterization of the $L^p$-norm.
14
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Invariance in the class of weighted quasi-arithmetic means

60%
Jarczyk J., Matkowski J.
Annales Polonici Mathematici
|
2006
|
tom 88
|
nr 1
39-51
EN
Under the assumption of twice continuous differentiability of some of the functions involved we determine all the weighted quasi-arithmetic means M,N,K such that K is (M,N)-invariant, that is, K∘(M,N) = K. Some applications to iteration theory and functional equations are presented.
15
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A differential equation related to the $l^{p}$-norms

49%
Bojarski J., Małolepszy T., Matkowski J.
Annales Polonici Mathematici
|
2011
|
tom 101
|
nr 3
251-265
EN
Let p ∈ (1,∞). The question of existence of a curve in ℝ₊² starting at (0,0) and such that at every point (x,y) of this curve, the $l^{p}$-distance of the points (x,y) and (0,0) is equal to the Euclidean length of the arc of this curve between these points is considered. This problem reduces to a nonlinear differential equation. The existence and uniqueness of solutions is proved and nonelementary explicit solutions are given.
16
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On subaddidive and Ψ-additive mappings

48%
Matkowski J.
Open Mathematics
|
2004
|
tom 2
|
nr 3
493-493
17
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A Generalization of m-Convexity and a Sandwich Theorem

43%
Lara T., Matkowski J., Merentes N., Quintero R., Wróbel M.
Annales Mathematicae Silesianae
|
2017
|
tom 31
|
nr 1
107-126
EN
Functional inequalities generalizing m-convexity are considered. A result of a sandwich type is proved. Some applications are indicated.
18
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On a characterization of $L^p$-norm

42%
Matkowski J.
Annales Polonici Mathematici
|
1989-1990
|
tom 50
|
nr 2
137-144
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