Aequationes Math Research Articles

Homomorphisms from Functional Equations: The Goldie Equation, II

AbstractThis first of three sequels to Homomorphisms from Functional equations: The Goldie equation (Ostaszewski in Aequationes Math 90:427–448, 2016) by the second author—the second of the resulting quartet—starts from the Goldie functional equation arising in the general regular variation of our joint paper (Bingham et al. in J Math Anal Appl 483:123610, 2020). We extend the work there in two directions. First, we algebraicize the theory, by systematic use of certain groups—the Popa groups arising in earlier work by Popa, and their relatives the Javor groups . Secondly, we extend from the original context on the real line to multi-dimensional (or infinite-dimensional) settings.

Further remarks on local K-boundedness of K-subadditive set-valued maps

AbstractLet X be an abelian metric group with an invariant metric, Y be a real normed space and K be a convex cone in Y. We prove that a K-subadditive (K-superadditive) compact- and convex-valued map $$F:X\rightarrow \mathcal{C}\mathcal{C}(Y)$$ F : X → C C ( Y ) , for which the functionals $$f_{y^*}(x)=\inf y^*(F(x))$$ f y ∗ ( x ) = inf y ∗ ( F ( x ) ) are lower (upper, resp.) semicontinuous for any real continuous and non-negative on K functional $$y^*$$ y ∗ , has to be locally K-bounded on X. Our results refer to the papers Banakh and Jabłońska (Israel J Math 230:361–386, 2019), Jabłońska and Nikodem (Math Inequal Appl 22:1081–1089, 2019) and Nikodem (Aequationes Math 62:175–183, 2001).

Remarks on a linearization of Koopmans recursion

Let X be a metric space and U:X^{infty }rightarrow mathbb {R} be a continuous function satisfying the Koopmans recursion U(x_0,x_1,x_2,ldots )=varphi (x_0, U(x_1,x_2,ldots )), where varphi :Xtimes I rightarrow I is a continuous function and I is an interval. Denote by succeq a preference relation defined on the product X^{infty } represented by a function U:X^{infty }rightarrow mathbb {R}, called a utility function, that means (x_0,x_1,ldots )succeq (y_0,y_1,ldots )Leftrightarrow U(x_0,x_1,ldots )ge U(y_0,y_1,ldots ). We consider a problem when the preference relation succeq can be represented by another utility function V satisfying the affine recursion V(x_0,x_1,x_2,ldots ) = alpha (x_0)V(x_1,x_2,ldots )+ beta (x_0). Under suitable assumptions on relation succeq we determine the form of the functions varphi defining the utility functions possessing the above property. The problem is reduced to solving a system of simultaneous functional equations. The subject is strictly connected to a problem of preference in economics. In this note we extend the results obtained in Zdun (Aequ Math 94, 2020).

ON THE DIVISIBILITY OF SUMS OF q-SUPER CATALAN NUMBERS

AbstractThe integrality of the numbers $A_{n,m}={(2n)!(2m)!}/{n!m!(n+m)!}$ was observed by Catalan as early as 1874 and Gessel named $A_{n,m}$ the super Catalan numbers. The positivity of the q-super Catalan numbers (q-analogue of the super Catalan numbers) was investigated by Warnaar and Zudilin [‘A q-rious positivity’, Aequationes Math.81 (2011), 177–183]. We prove the divisibility of sums of q-super Catalan numbers, which establishes a q-analogue of Apagodu’s congruence involving super Catalan numbers.

Further results on a new class of functional equations satisfied by polynomial functions

Whenever a numerical method produces accurate results, it creates an interesting functional equation, and because regularities is not assumed, unexpected solutions can emerge. Thus, this paper is mainly devoted to finding solutions to a generalized functional equation constructed in this spirit; namely, we solve the generalized form of the functional equation considered in Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012), then considered in Nadhomi et al. (Aequationes Math 95:1095–1117, 2021) and continued in Okeke and Sablik (Results Math 77:125, https://doi.org/10.1007/s00025-022-01664-x, 2022), that is we find the polynomial functions satisfying the following functional equation, 0.1∑i=1nγiF(aix+biy)=∑j=1m(αjx+βjy)f(cjx+djy),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sum _{i=1}^n \\gamma _i F(a_i x + b_i y)= \\sum _{j=1}^m(\\alpha _j x + \\beta _j y) f(c_j x + d_j y), \\end{aligned}$$\\end{document}for every x,yin mathbb R, gamma _i,alpha _j,beta _j in mathbb R, and a_i,b_i,c_j,d_j in mathbb Q, and its special forms. Thus we continue investigations presented in Nadhomi et al. (Aequationes Math 95:1095–1117, 2021) where we generalized the left hand side of Fechner–Gselmann equation and those from Okeke and Sablik (Results Math 77:125, https://doi.org/10.1007/s00025-022-01664-x, 2022) where the right hand side of the Fechner–Gselmann equation was studied. It turns out that under some assumptions on the parameters involved, the pair (F, f) solving Eq. (0.1) happens to be a pair of polynomial functions.

Limit Theorems for Deviation Means of Independent and Identically Distributed Random Variables

We derive a strong law of large numbers, a central limit theorem, a law of the iterated logarithm and a large deviation theorem for so-called deviation means of independent and identically distributed random variables. (For the strong law of large numbers, we suppose only pairwise independence instead of (total) independence.) The class of deviation means is a special class of M-estimators or more generally extremum estimators, which are well studied in statistics. The assumptions of our limit theorems for deviation means seem to be new and weaker than the known ones for M-estimators in the literature. In particular, our results on the strong law of large numbers and on the central limit theorem generalize the corresponding ones for quasi-arithmetic means due to de Carvalho (Am Stat 70(3):270–274, 2016) and the ones for Bajraktarević means due to Barczy and Burai (Aequ Math 96(2):279–305, 2022).

Non-periodic solutions of the Goła̧b–Schinzel type functional equation

We determine the solutions of the Goła̧b-Schinzel type functional equation in the class of non-periodic functions. Applying this result we give a positive answer to the problem raised by E. Jabłońska (Aequationes Math 87:125–133, 2014).

Linear functional equations and their solutions in Lorentz spaces

Assume that $$\varOmega \subset {\mathbb {R}}^k$$ is an open set, V is a separable Banach space over a field $${\mathbb {K}}\in \{{\mathbb {R}},{\mathbb {C}}\}$$ and $$f_1,\ldots ,f_N :\varOmega \rightarrow \varOmega $$ , $$g_1,\ldots , g_N:\varOmega \rightarrow {\mathbb {K}}$$ , $$h_0:\varOmega \rightarrow V$$ are given functions. We are interested in the existence and uniqueness of solutions $$\varphi :\varOmega \rightarrow V$$ of the linear functional equation $$\varphi =\sum _{k=1}^{N}g_k\cdot (\varphi \circ f_k)+h_0$$ in Lorentz spaces. The investigation of the existence of the solutions in the case $$L^1([0,1],{\mathbb {R}})$$ , that is in the Lorentz space $$L^{1,1}([0,1],{\mathbb {R}})$$ , was motivated by Nikodem’s paper On $$\epsilon $$ -invariant measures and a functional equation published in Czechoslovak Math. J., 41(116)(4):565–569, 1991. We have studied it in Some classes of linear operators involved in functional equations published in Ann. Funct. Anal., 10(3):381–394, 2019. Inspired by Matkowski’s Dissertationes Math. paper Integrable solutions of functional equations from 1975, we studied the Orlicz case in our paper Linear functional equations and their solutions in generalized Orlicz spaces published in Aequationes Mathematicae in 2021 in volume 95, number 6. Here, we build on the ideas from that paper, using the connection between Lorentz and Orlicz spaces as detailed in the paper Fine behavior of functions whose gradients are in an Orlicz space authored by Malý, Swanson, and Ziemer and published in Studia Math. 190(1), 33–71 in 2009. However, we do not only look at real-valued functions, but allow separable Banach spaces as target spaces. As in the paper about Orlicz spaces, an important tool is a result by Hajłasz, Change of variables formula under minimal assumptions as published in Colloq. Math. 64(1), 93–101 (1993), concerning an optimal change of variable formula. It is used for proving that we can apply Banach’s fixed point theorem.

On optimal inequalities between three-point quadratures

We examine the family of all (at most) three-point symmetric quadratures on [-1,1] which are exact on polynomials of order 3 to find all possible inequalities between them in the class of 3-convex functions. Next we optimise them by using convex combinations of the quadratures considered. We find the optimal quadrature and use it to construct the adaptive method of approximate integration. An effective method to estimate the error of this method is also given. It needs a considerably fewer number of subdivisions of the interval of integration than the classical adaptive methods as well as the method developed by the second-named author in his recent paper (Wa̧sowicz in Aequ Math 94(5):887–898, 2020).

Interview with Ludwig Reich

On the occasion of his 80th birthday, Ludwig Reich, editor-in-chief of Aequationes Mathematicae from 1996 to 2008, gave an interview conducted via Zoom. This paper presents his answers to several questions regarding his education and academic life.

Complementary means with respect to a nonsymmetric invariant mean

It is known that if a bivariate mean K is symmetric, continuous and strictly increasing in each variable, then for every mean M there is a unique mean N, such that K is invariant with respect to the mean-type mapping left( M,Nright) ,which means that Kcirc left( M,Nright) =K and N is called a K-complementary mean for M (Matkowski in Aequ Math 57(1):87–107, 1999). This paper extends this result for a large class of nonsymmetric means. As an application, the limits of the sequences of iterates of the related mean-type mappings are determined, which allows us to find all continuous solutions of some functional equations.

Some remarks on the stability of the Cauchy equation and completeness

It was proved in Forti and Schwaiger (C R Math Acad Sci Soc R Can 11(6):215–220, 1989), Schwaiger (Aequ Math 35:120–121, 1988) and with different methods in Schwaiger (Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Bȩdlewo, Poland, May 17–23, 2015, Springer, Cham, pp 275–295, 2017) that under the assumption that every function defined on suitable abelian semigroups with values in a normed space such that the norm of its Cauchy difference is bounded by a constant (function) is close to some additive function, i.e., the norm of the difference between the given function and that additive function is also bounded by a constant, the normed space must necessarily be complete. By Schwaiger (Ann Math Sil 34:151–163, 2020) this is also true in the non-archimedean case. Here we discuss the situation when the bound is a suitable non-constant function.

On Subadditive Functions Bounded Above on\xa0a\xa0\u201cLarge\u201d Set

It is well known that boundedness of a subadditive function need not imply its continuity. Here we prove that each subadditive function f:Xrightarrow {mathbb {R}} bounded above on a shift–compact (non–Haar–null, non–Haar–meagre) set is locally bounded at each point of the domain. Our results refer to results from Kuczma’s book (An Introduction to the theory of functional equations and inequalities. Cauchy’s equation and Jensen’s inequality, 2nd edn, Birkhäuser Verlag, Basel, 2009, Chapter 16) and papers by Bingham and Ostaszewski [Proc Am Math Soc 136(12):4257–4266, 2008, Aequationes Math 78(3):257–270, 2009, Dissert Math 472:138pp., 2010, Indag Math (N.S.) 29:687–713, 2018, Aequationes Math 93(2):351–369, 2019).

On the functional equation $$G\left( x,G\left( y,x\right) \right) = G\left( y,G\left( x,y\right) \right) $$ and means

We consider the functional equation $$G\left( x,G\left( y,x\right) \right) =G\left( y,G\left( x,y\right) \right) $$, posed in Jarczyk and Jarczyk (Aequ Math 72:198–200, 2006). We show that every continuous and reducible solution generates a mean resembling the weighted quasi-arithmetic mean, but no weighted quasi-arithmetic mean is a solution of this equation. This fact particularly implies that the equation is not a direct consequence of the bisymmetry equation and the reflexivity condition. The closedness of the family of solutions with respect to conjugacy is noted. Finally, the translative solutions, homogeneous solutions, and suitable iterative composite functional equation for single variable functions are discussed.

Some monotonicity properties in F-normed Musielak\u2013Orlicz spaces

Strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity and their orthogonal counterparts are considered in the case of Musielak–Orlicz function spaces L^Phi (mu ) endowed with the Mazur–Orlicz F-norm as well as in the case of their subspaces E^Phi (mu ) with the F-norm induced from L^Phi (mu ). The presented results generalize some of the results from Cui et al. (Aequ Math 93:311–343, 2019) and Hudzik et al. (J Nonlinear Convex Anal 17(10):1985–2011, 2016), obtained only for Orlicz spaces as well as their subspaces of order continuous elements equipped with the Mazur–Orlicz F-norm.

On the equality problem of generalized Bajraktarevi\u0107 means

The purpose of this paper is to investigate the equality problem of generalized Bajraktarević means, i.e., to solve the functional equation *f(-1)(p1(x1)f(x1)+⋯+pn(xn)f(xn)p1(x1)+⋯+pn(xn))=g(-1)(q1(x1)g(x1)+⋯+qn(xn)g(xn)q1(x1)+⋯+qn(xn)),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} f^{(-1)}\\bigg (\\frac{p_1(x_1)f(x_1)+\\dots +p_n(x_n)f(x_n)}{p_1(x_1)+\\dots +p_n(x_n)}\\bigg )=g^{(-1)}\\bigg (\\frac{q_1(x_1)g(x_1)+\\dots +q_n(x_n)g(x_n)}{q_1(x_1)+\\dots +q_n(x_n)}\\bigg ), \\end{aligned}$$\\end{document}which holds for all (x_1,dots ,x_n)in I^n, where nge 2, I is a nonempty open real interval, the unknown functions f,g:Irightarrow {mathbb {R}} are strictly monotone, f^{(-1)} and g^{(-1)} denote their generalized left inverses, respectively, and p=(p_1,dots ,p_n):Irightarrow {mathbb {R}}_{+}^n and q=(q_1,dots ,q_n):Irightarrow {mathbb {R}}_{+}^n are also unknown functions. This equality problem in the symmetric two-variable (i.e., when n=2) case was already investigated and solved under sixth-order regularity assumptions by Losonczi (Aequationes Math 58(3):223–241, 1999). In the nonsymmetric two-variable case, assuming the three times differentiability of f, g and the existence of iin {1,2} such that either p_i is twice continuously differentiable and p_{3-i} is continuous on I, or p_i is twice differentiable and p_{3-i} is once differentiable on I, we prove that (*) holds if and only if there exist four constants a,b,c,din {mathbb {R}} with adne bc such that cf+d>0,g=af+bcf+d,andqℓ=(cf+d)pℓ(ℓ∈{1,⋯,n}).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} cf+d>0,\\qquad g=\\frac{af+b}{cf+d},\\qquad \\text{ and }\\qquad q_\\ell =(cf+d)p_\\ell \\qquad (\\ell \\in \\{1,\\dots ,n\\}). \\end{aligned}$$\\end{document}In the case nge 3, we obtain the same conclusion with weaker regularity assumptions. Namely, we suppose that f and g are three times differentiable, p is continuous and there exist i,j,kin {1,dots ,n} with ine jne kne i such that p_i,p_j,p_k are differentiable.

Weighted sharing and uniqueness for the shifts of meromorphic functions

In this paper, we deal with the uniqueness problem for $q$-shifts of meromorphic functions. The results obtained in this paper are the $q$-shift analogues of theorems given by Yang and Zhang [Non-existence of meromorphic solution of a Fermat type functional equation, Aequationes Math. 76(2008), 140-150], Chen, Chen and Li [Uniqueness of difference operators of meromorphic functions, J. Ineq. Appl. 2012(2012), Art 48].

The hyperstability of general linear equation via that of Cauchy equation

In this paper, we show that the hyperstability of the general linear equation recently proved by Piszczek (Aequationes Math 88:163–168, 2014) is a direct consequence of that of the Cauchy equation proved earlier by Brzdȩk (Acta Math Hung 141:58–67, 2013).

Fejér and Hermite–Hadamard type results for H-invex functions with applications

We investigate the class of H-invex functions including, e.g., the subclasses of convex, c-strongly convex, \( \varphi \)-uniformly convex and superquadratic functions. For H-invex functions we prove Fejer and Hermite–Hadamard type inequalities. We generalize some results by Abramovich et al. (J Math Anal Appl 344:1048–1556, 2008), Banic et al. (J Korean Math Soc 45:513–525, 2008), and Merentes et al. (Aequ Math 80:193–199, 2010).

Vertex-edge domination in graphs

We establish that for any connected graph G of order $$n \ge 6$$ , a minimum vertex-edge dominating set of G has at most n/3 vertices, thus affirmatively answering the open question posed by Boutrig et al. (Aequ Math 90(2):355–366, 2016).