172 research outputs found
On a problem of Janusz Matkowski and Jacek Wesołowski
No full textWe study the problem of the existence of increasing and continuous solutions φ :[0 , 1] → [0 , 1] such that φ (0) = 0 and φ (1) = 1 of the functional equation
φ ( x )= N ∑ n =0 φ ( f n ( x )) − N ∑ n =1 φ ( f n (0)) ,
where N ∈ N and f 0 ,...,f N :[0 , 1] → [0 , 1] are strictly increasing contractions satisfying the following condition 0 = f 0 (0) <f 0 (1) = f 1 (0) < ··· <f N − 1 (1) = f N (0) <f N (1) = 1. In particular, we give an answer to the problem posed in Matkowski (Aequationes Math. 29:210–213, 1985 ) by Janusz Matkowski concerning a very special case of that equation
A functional equation characterizing homographic functions Janusz Matkowski
No full textSome functional equations related to homographic functions and their characterization are presented
On the commutation of generalized means on probability spaces
No full textLet f and g be real-valued continuous injections defined on a non-empty real interval I, and let (X,L,λ) and (Y,M,μ) be probability spaces in each of which there is at least one measurable set whose measure is strictly between 0 and 1. We say that (f,g) is a (λ,μ)-switch if, for every L⊗M-measurable function h:X×Y→R for which h[X×Y] is contained in a compact subset of I, it holds f−1(∫Xf(g−1(∫Yg∘hdμ))dλ)=g−1(∫Yg(f−1(∫Xf∘hdλ))dμ), where f−1 is the inverse of the corestriction of f to f[I], and similarly for g−1. We prove that this notion is well-defined, by establishing that the above functional equation is well-posed (the equation can be interpreted as a permutation of generalized means and raised as a problem in the theory of decision making under uncertainty), and show that (f,g) is a (λ,μ)-switch if and only if f=ag+b for some a,b∈R, a≠0
Persistently optimal policies in stochastic dynamic programming with generalized discounting
No full textIn this paper we study a Markov decision process with a non-linear discount function. Our approach is in spirit of the von Neumann-Morgenstern concept and is based on the notion of expectation. First, we define a utility on the space of trajectories of the process in the finite and infinite time horizon and then take their expected values. It turns out that the associated optimization problem leads to a non-stationary dynamic programming and an infinite system of Bellman equations, which result in obtaining persistently optimal policies. Our theory is enriched by examples.Stochastic dynamic programming, Persistently optimal policies, Variable discounting, Bellman equation, Resource extraction, Growth theory
A new approach with new solutions to the Matkowski and Wesołowski problem
No full textBased on a result of de Rham, we give a family of functions solving the Matkowski and Wesołowski problem. This family consists of Holder continuous functions, and it coincides cfwith the whole family of solutions to the Matkowski and Wesołowski problem found earlier by a different method. Moreover, applying some results due to Hata and Yamaguti and due to Berg and Kruppel, we prove that there are functions solving the Matkowski and Wesołowski problem that are not H¨older continuous
On a generalized conjecture by Alzer and Matkowski
No full textWe study a recent conjecture proposed by Horst Alzer and Janusz Matkowski concerning a bilinearity property of the Cauchy exponential difference for real-to-real functions. The original conjecture was affirmatively resolved by Tomasz Małolepszy. We deal with generalizations for real or complex mappings acting on a linear space
On extension of solutions of a simultaneous system of iterative functional equations
No full textSome sufficient conditions which allow to extend every local solution of a simultaneous system of equations in a single variable of the form to a global one are presented. Extensions of solutions of functional equations, both in single and in several variables, play important role (cf. for instance [M. Kuczma, Functional equations in a single variable, Monografie Mat. 46, Polish Scientific Publishers, Warsaw, 1968, M. Kuczma, B. Choczewski, R. Ger, Iterative functional equations, Encyclopedia of Mathematics and Its Applications v. 32, Cambridge, 1990, J. Matkowski, Iteration groups, commuting functions and simultaneous systems of linear functional equations, Opuscula Math. 28 (2008) 4, 531-541])
Another look at the Matkowski and Weso{\l}owski problem yielding a new class of solutions
No full textThe following MW--problem was posed independently by Janusz Matkowski and
Jacek Weso{\l}owski in different forms in 1985 and 2009, respectively: Are
there increasing and continuous functions ,
distinct from the identity on , such that ,
and
for every ? By now, it is known that each of the de Rham functions
, where , is a solution of the MW--problem, and for any Borel
probability measure concentrated on the formula
defines a solution
of this problem as well. In this paper, we give a
new family of solutions of the MW--problem consisting of Cantor-type functions.
We also prove that there are strictly increasing solutions of the MW--problem
that are not of the above integral form with any Borel probability measure
The Pexider type generalization of the Minkowski inequality
No full textAbstractLet (Ω,Σ,μ) be a measure space such that 0<μ(A)<1<μ(B)<∞ for some A,B∈Σ. The following converse Minkowski inequality theorem is proved in Matkowski (2008) [4]. If φ,ψ,γ:(0,∞)→(0,∞) are bijective, φ is increasing, and φ−1(∫Ω(x+y)φ∘(x+y)dμ)≤ψ−1(∫Ω(x)ψ∘xdμ)+γ−1(∫Ω(y)γ∘ydμ) for all nonnegative μ-integrable simple functions x,y :Ω→R (where Ω(x) stands for the support of x), then there exists a real p≥1 such that φ(t)φ(1)=ψ(t)ψ(1)=γ(t)γ(1)=tp. In the present paper we show that if, in the basic measure space, there is no A∈Σ such that either 1<μ(A)<∞ or 0<μ(A)<1, then there are some broad classes of non-power functions which satisfy the above Minkowski type inequality. Moreover we prove that, in the converse of the Minkowski inequality theorem, the assumption of the increasing monotonicity of φ is essential
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