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Integral equations with infinite delay are considered as functional equations in a Banach space. Two types of Hyers–Ulam stability criteria are established. First, it is shown that a linear autonomous equation is Hyers–Ulam stable if and only if it has no characteristic value with zero real part. Second, it is proved that the Hyers–Ulam stability of a linear autonomous equation is preserved under sufficiently small nonlinear perturbations. The proofs are based on a recently developed decomposition theory of linear integral equations with infinite delay. [ABSTRACT FROM AUTHOR]

The present research deals with generalizations of the Salem function with arguments defined in terms of certain alternating expansions of real numbers. Special attention is given to modelling such functions by systems of functional equations. [ABSTRACT FROM AUTHOR]

Let G be a topological group, and let C(G) denote the algebra of continuous, complex valued functions on G. We find the solutions f , g , h ∈ C (G) of the Levi-Civita equation f (x y) = f (x) h (y) + g (x) f (y) , x , y ∈ G , which is an extension of the sine addition law. Representations of G on C 2 play an important role. As a corollary we get the solutions f , g ∈ C (G) of the sine subtraction law f (x y ∗) = f (x) g (y) - g (x) f (y) , x , y ∈ G , in which x ↦ x ∗ is a continuous involution, meaning that (x y) ∗ = y ∗ x ∗ and x ∗ ∗ = x for all x , y ∈ G . [ABSTRACT FROM AUTHOR]

Imposing some conditions on derivatives of the known functions, using the Fiber Contraction Theorem we prove the existence of C 1 solutions of a class of iterative functional equations which involves iterates of the unknown functions and a nonlinear term. [ABSTRACT FROM AUTHOR]

The note is concerned with the functional equation λ 1 H 1 (f (x)) + λ 2 H 2 (f 2 (x)) + ⋯ + λ n H n (f n (x)) = F (x) , which is a generalised form of the so-called polynomial-like iterative equation. We investigate the existence of nondecreasing convex (both usual and higher order) solutions to this equation on open intervals using the Schauder fixed point theorem. The results supplement those proved by Trif (Aquat Math, 79:315–327, 2010) for the polynomial-like iterative equation by generalising them to a greater extent. This assertion is supported by some examples illustrating their applicability. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to show that in certain classes of real (or complex) functions, Bernoulli polynomials are essentially the only ones satisfying the Raabe functional equation. For the class of real 1-periodic functions which are expandable as Fourier series, we point out new solutions of the Raabe functional equation, not related to Bernoulli polynomials. Furthermore, we will give for the considered classes various proofs, making the mathematical content of the paper quite rich. [ABSTRACT FROM AUTHOR]

An open problem posed by G. Maksa and Z. Páles is to find the general solution of the functional equation { ‖ f (x) - β f (y) ‖ : β ∈ T n } = { ‖ x - β y ‖ : β ∈ T n } (x , y ∈ H) where f : H → K is between two complex normed spaces and T n : = { e i 2 k π n : k = 1 , ⋯ , n } is the set of the nth roots of unity. With the aid of the celebrated Wigner's unitary-antiunitary theorem, we show that if n ≥ 3 and H and K are complex inner product spaces, then f satisfies the above equation if and only if there exists a phase function σ : H → T n such that σ · f is a linear or anti-linear isometry. Moreover, if the solution f is continuous, then f is a linear or anti-linear isometry. [ABSTRACT FROM AUTHOR]

The functional equation f (x + y) = g (x y) is investigated with unknown functions f : A + A → Y , g : A · A → Y in the following cases: A : = α , β ⊆ F + where F is an Archimedean ordered field; A is the set of all positive integers; A is the set of all positive dyadic rational numbers. The set Y is an arbitrarily fixed (infinite) set. The main result of the paper shows that there exists a set A ⊆ R + that is closed under addition and multiplication and there exist functions f, g : A → Y which satisfy the equation f (x + y) = g (x y) for all x , y ∈ A such that the range of the function f is infinite. Finally, some application of the above results is also given. [ABSTRACT FROM AUTHOR]

In its additive version, Bohr–Mollerup's remarkable theorem states that the unique (up to an additive constant) convex solution f(x) to the equation Δ f (x) = ln x on the open half-line (0 , ∞) is the log-gamma function f (x) = ln Γ (x) , where Δ denotes the classical difference operator and Γ (x) denotes the Euler gamma function. In a recently published open access book, the authors provided and illustrated a far-reaching generalization of Bohr–Mollerup's theorem by considering the functional equation Δ f (x) = g (x) , where g can be chosen from a wide and rich class of functions that have convexity or concavity properties of any order. They also showed that the solutions f(x) arising from this generalization satisfy counterparts of many properties of the log-gamma function (or equivalently, the gamma function), including analogues of Bohr–Mollerup's theorem itself, Burnside's formula, Euler's infinite product, Euler's reflection formula, Gauss' limit, Gauss' multiplication formula, Gautschi's inequality, Legendre's duplication formula, Raabe's formula, Stirling's formula, Wallis's product formula, Weierstrass' infinite product, and Wendel's inequality for the gamma function. In this paper, we review the main results of this new and intriguing theory and provide an illustrative application. [ABSTRACT FROM AUTHOR]

In this paper we determine the solutions (φ , f 1 , f 2) of the Pexider functional equation φ ( x + y 2) (f 1 (x) - f 2 (y)) = 0 , (x , y) ∈ I 1 × I 2 , where I 1 and I 2 are nonempty open subintervals. Special cases of the above equation regularly arise in problems with two-variable means. We show that, under a rather weak regularity condition, the coordinate-functions of a typical solution of the equation are constant over several subintervals of their domain. The regularity condition in question will be that the set of zeros of φ is closed. We also discuss particular solutions where this condition is not met. [ABSTRACT FROM AUTHOR]

This present paper is inspired by one of the questions posed by Okeke (Results Math 78(96):1-30, 2023, see Remark 2.10b). In particular, we aim to develop a robust computer code based on the theoretical results obtained in Okeke (2023), which determines the polynomial solutions of the following functional equation, 0.1 ∑ i = 1 n γ i F (a i x + b i y) = ∑ j = 1 m (α j x + β j y) f (c j x + d j y) , for all x , y ∈ R , γ i , α j , β j ∈ R , and a i , b i , c j , d j ∈ Q , and their special forms. The primary motivation for writing such a computer code is that solving even simple equations belonging to class (0.1) needs long and tiresome calculations. Therefore, one of the advantages of such a computer code is that it allows us to solve complicated problems quickly, easily, and efficiently. Additionally, the computer code will significantly improve the level of accuracy in calculations. Along with that, there is also the factor of speed. We point out that the computer code will operate with symbolic calculations provided by the programming language Python, which means that it does not contain any numerical or approximate methods, and it yields the exact solutions of the equations considered. [ABSTRACT FROM AUTHOR]

We study functions related to the classical Brjuno function, namely k-Brjuno functions and the Wilton function. Both appear in the study of boundary regularity properties of (quasi) modular forms and their integrals. We consider various possible versions of them, based on the α -continued fraction developments. We study their BMO regularity properties and their behaviour near rational numbers of their finite truncations. We then complexify the functional equations which they fulfill and we construct analytic extensions of the k-Brjuno and Wilton functions to the upper half-plane. We study their boundary behaviour using an extension of the continued fraction algorithm to the complex plane. We also prove that the harmonic conjugate of the real k-Brjuno function is continuous at all irrational numbers and has a decreasing jump of π / q k at rational points p/q. [ABSTRACT FROM AUTHOR]

For ν ∈ [ 0 , 1 ] and a complex parameter σ , R e σ > 0 , we discuss a linear inhomogeneous functional difference equation with variable coefficients on a complex plane z ∈ C : (a 1 σ + a 2 σ ν) Y (z + β , σ) - Ω (z) Y (z , σ) = F (z , σ) , β ∈ R , β ≠ 0 , where Ω (z) and F (z) are given complex functions, while a 1 and a 2 are given real non-negative numbers. Under suitable conditions on the given functions and parameters, we construct explicit solutions of the equation and describe their asymptotic behavior as | z | → + ∞ . Some applications to the theory of functional difference equations and to the theory of boundary value problems governed by subdiffusion in nonsmooth domains are then discussed. [ABSTRACT FROM AUTHOR]

In this research paper, we investigate a generalization of Vincze's type functional equations involving several (up to four) unknown functions in connection with the maximum functional equation as max { ψ (x y) , ψ (x y - 1) } = ψ (x) η (y) + ψ (y) , max { ψ (x y) , ψ (x y - 1) } = ψ (x) η (y) + χ (y) , max { ψ (x y) , ψ (x y - 1) } = ϕ (x) η (y) , max { ψ (x y) , ψ (x y - 1) } = ϕ (x) η (y) + χ (y) , where G is an arbitrary group, x , y ∈ G , and ψ , η , χ , ϕ : G → R are unknown functions. [ABSTRACT FROM AUTHOR]

Explicit forms of invariant means for all mean-type mappings generated by the bivariable classical arithmetic, geometric and harmonic means (complementary to the Gauss AGM -theorem) are given. A generalization for higher dimension mean-type mappings as well as two open problems and an application in determining implicit solutions of some functional equations, are presented. [ABSTRACT FROM AUTHOR]

In this paper, a computer program developed in the computer algebra system Maple is presented, which investigates alienness and strong alienness of linear functional equations. [ABSTRACT FROM AUTHOR]

Motivated by the well known fact that any nonzero solution of the fundamental Cauchy functional equation may arbitrarily be prescribed on a Hamel basis, we deal with the following problem: given a functional equation * E 1 (φ) = E 2 (φ) with the unknown function φ : X ⟶ Y , what must the set ∅ ≠ Z ⊂ X be like in order to ensure that an arbitrary function φ 0 : Z ⟶ Y admits a unique function φ : X ⟶ Y solving equation ( ∗ ) and such that φ | Z = φ 0 ; if such a set does exist it is termed to be a basic set. We discuss the problem of existence of basic sets for exponential functions, d'Alembert's functions, sine functions, Cuculière's functions and hyperbolic tangent type functions, among others. [ABSTRACT FROM AUTHOR]

Aichinger's equation is used to give simple proofs of several well-known characterizations of polynomial functions as solutions of certain functional equations. Concretely, we use that Aichinger's equation characterizes polynomial functions to solve, for arbitrary commutative groups, Ghurye–Olkin's functional equation, Wilson's functional equation, the Kakutani–Nagumo–Walsh functional equation, and a general version of Fréchet's unmixed functional equation. [ABSTRACT FROM AUTHOR]

In this paper we give necessary conditions for quadratic functions f : R → R that satisfy the additional equation y 2 f (x) = x 2 f (y) under the condition x y = 1 . [ABSTRACT FROM AUTHOR]

It is known that the zero utility principle under Cumulative Prospect Theory can be uniquely extended from the family of all ternary risks. On the other hand, the extension from the family of all binary risks need not be unique. We establish a characterization of those zero utility principles which coincide on the family of binary risks. The characterization is expressed in terms of relations between the systems of generators of the principles. [ABSTRACT FROM AUTHOR]

Let X be a metric space and U : X ∞ → R be a continuous function satisfying the Koopmans recursion U (x 0 , x 1 , x 2 , ...) = φ (x 0 , U (x 1 , x 2 , ...)) , where φ : X × I → I is a continuous function and I is an interval. Denote by ⪰ a preference relation defined on the product X ∞ represented by a function U : X ∞ → R , called a utility function, that means (x 0 , x 1 , ...) ⪰ (y 0 , y 1 , ...) ⇔ U (x 0 , x 1 , ...) ≥ U (y 0 , y 1 , ...) . We consider a problem when the preference relation ⪰ can be represented by another utility function V satisfying the affine recursion V (x 0 , x 1 , x 2 , ...) = α (x 0) V (x 1 , x 2 , ...) + β (x 0) . Under suitable assumptions on relation ⪰ we determine the form of the functions φ 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). [ABSTRACT FROM AUTHOR]

Maciej Sablik is a mathematician who has made significant contributions to the field of functional equations and inequalities. He began his mathematical journey at the Institute of Mathematics of the Silesian University of Katowice, where he studied under Professor Marek Kuczma. After completing his doctoral dissertation in 1980, Sablik's career progressed steadily, and he eventually became a professor of mathematics in 2011. He has participated in numerous international conferences and established scientific collaborations around the world. Sablik is also known for his involvement in various mathematical organizations and his proficiency in multiple languages. [Extracted from the article]

Given a probability space (Ω , A , P) , a complete separable Banach space X with the σ -algebra B (X) of all its Borel subsets, an operator Λ : Ω → L (X , X) and ξ : Ω → X we consider the B (X) ⊗ A -measurable function f : X × Ω → X given by f (x , ω) = Λ (ω) x + ξ (ω) and investigate the continuous dependence of a weak limit π f of the sequence of iterates (f n (x , ·)) n ∈ N of f, defined by f 0 (x , ω) = x , f n + 1 (x , ω) = f (f n (x , ω) , ω n + 1) for x ∈ X and ω = (ω 1 , ω 2 , ⋯) . Moreover for X taken as a Hilbert space we characterize π f via the functional equation φ f (u) = ∫ Ω φ f (Λ (ω) u) φ ξ (u) P (d ω) with the aid of its characteristic function φ f . We also indicate the continuous dependence of a solution of that equation. [ABSTRACT FROM AUTHOR]

The author, together with M. Veerapazham, S. Wang and W. Zhang, considered the existence and uniqueness of continuous solutions to an iterative equation involving the multiplication of iterates in Gopalakrishna et al. (Sci. China Math., to appear). In this paper we continue to investigate this equation for differentiable solutions. Similarly to continuous solutions until Gopalakrishna et al. (Sci. China Math., to appear), no result on differentiable solutions of this equation on non-compact intervals of R has been obtained until now. Although our strategy here is to use the logarithmic conjugation to reduce the equation to the well-known polynomial-like iterative equation, all known results on differentiable solutions of the latter are given on compact intervals. We revisit polynomial-like iterative equations on the whole of R and use the Banach contraction principle to prove the existence and uniqueness of differentiable solutions to our equation on R + . We then technically extend our results on R + to R - and show that none of the pairs of solutions obtained on R + and R - can be combined at the origin to obtain even a continuous solution of the equation on the whole R . [ABSTRACT FROM AUTHOR]

Our main result is that we describe the solutions g , f : S → C of the functional equation g (x σ (y)) = g (x) g (y) - f (x) f (y) + α f (x σ (y)) , x , y ∈ S , where S is a semigroup, α ∈ C is a fixed constant and σ : S → S an involutive automorphism. [ABSTRACT FROM AUTHOR]

We study the functional equation for unknown functions f , g , h , k , ℓ : S → C , where S is a semigroup and m 1 , m 2 : S → C are multiplicative functions. The study is divided into two main parts: m 1 = m 2 and m 1 ≠ m 2 . In some cases we assume that one or more of the unknown functions is central and/or that S is a monoid. The solutions are found in all cases under the umbrella assumption that S is a commutative monoid. The continuous solutions on topological semigroups are also found. [ABSTRACT FROM AUTHOR]

There is an asymptotic correspondence between the multiplicative relations among multinomial coefficients and the (additive) recursive property of Shannon entropy known as the chain rule. We show that both types of identities are manifestations of a unique algebraic construction: a 1-cocycle condition in information cohomology, an algebraic invariant of presheaves of modules on certain categories of observables. Depending on the coefficients, the 1-cocycles can be information measures (Shannon entropy, Tsallis α -entropy) or generalized (Fontené-Ward) multinomial coefficients. In each case the 1-cocycle condition encodes a system of functional equations. We obtain in particular a combinatorial analogue of the "fundamental equation of information theory": a simple functional equation that uniquely characterizes the generalized binomial coefficients. The asymptotic correspondence mentioned above extends to any α -entropy and certain multinomial coefficients with compatible asymptotic behavior, shedding new light on the meaning of the chain rule and its deformations. [ABSTRACT FROM AUTHOR]

The equality problem of two-variable Bajraktarević means can be expressed as the functional equation f g - 1 ( f (x) + f (y) g (x) + g (y) ) = h k - 1 ( h (x) + h (y) k (x) + k (y) ) (x , y ∈ I) , where I is a nonempty open real interval, f , g , h , k : I → R are continuous functions, g, k are positive and f/g, h/k are strictly monotone. This functional equation, for the first time, was solved by Losonczi in 1999 under 6th-order continuous differentiability assumptions. Additional and new characterizations of this equality problem have been found recently by Losonczi, Páles and Zakaria under the same regularity assumptions in 2021. In this paper it is shown that the same conclusion can be obtained under substantially weaker regularity conditions, namely, assuming only first-order differentiability. [ABSTRACT FROM AUTHOR]

We consider a functional equation which is a generalization of Whitehead's functional equation on groups. We present its general solution on free groups. Solutions on other selected groups are also given. [ABSTRACT FROM AUTHOR]

In the paper there are two kinds of functional symmetry considered, i.e., (1)-power symmetry and (- 1) -power symmetry, in a rich subfamily B of the family of all complex valued functions on a symmetric set G ⊂ C . The first one defines the values f (- z) as identical with f(z) and the other as the inverse of the f(z) in G . The announced notions allow a unique decomposition of functions f ∈ B into a product of two factors f 1 , f - 1 having the (1)- and the (- 1) -power symmetry property, respectively. In the paper there are also given examples of such partitions f = f 1 · f - 1 , for various f ∈ B , a solution of the problem of invariance of the above functional symmetries with respect to some one- and two-argument function operations and two applications of the mentioned function decomposition. [ABSTRACT FROM AUTHOR]

Translativity of the beta-type function B f : I 2 → 0 , ∞ , B f x , y : = f x f y f x + y , where f is a single variable function defined either on I = R or I = [ 0 , ∞) , or I = 0 , ∞ , is considered. In each of these three cases a complete solution is given. [ABSTRACT FROM AUTHOR]

In this paper we attempt to define axiomatic measures of non-compactness for Sobolev spaces of integer order W n , p (Ω) , where Ω ⊂ R d (which is equivalent to Ω being any set of infinite measure). We consider two cases, one with Ω being an open subset with finite measure, and another when Ω = R d , and discuss basic features of the measure of non-compactness in each case. Next we define a partial measure of non-compactness on space L p (Ω ; B) , where B is a Banach space. Furthermore, we give some application to solve the functional integro-differential equation in W 1 , p (Ω). [ABSTRACT FROM AUTHOR]

Line conjugates and (B)line conjugates are defined. Associated points, lines, conics, and cubics are introduced and investigated in the context of functional equations. Triangle centers and central lines are treated as functions of the sidelengths a, b, c of a variable reference triangle ABC. Relationships usually regarded as geometric are regarded as algebraic (e.g. collinearity of points, concurrence of lines, perspectivity of triangles). The methods depend on homogeneous barycentric (or trilinear) coordinates. [ABSTRACT FROM AUTHOR]

In this paper we prove the following result: Let R be a prime ring with c h a r (R) ≠ 2 , 3 , 5 and let T : R → R be an additive mapping satisfying the relation T (x 4) = x T (x 2) x for all x ∈ R . In this case T is of the form T (x) = λ x for all x ∈ R and some fixed element λ ∈ C , where C is the extended centroid of R. [ABSTRACT FROM AUTHOR]

In this paper we introduce the concept of translation invariant functions: considering an arbitrary set ∅ ≠ S ⊂ R n , the function F : S ⟶ R is translation invariant if F (x) = F (y) implies F (x + t) = F (y + t) for any vectors x , y , t ∈ R n such that x , y , x + t , y + t ∈ S . In our main results we shall consider an open, connected set ∅ ≠ D ⊂ R n . We prove that if F : D ⟶ R is a translation invariant, continuous function, then there exists a vector a = (a 1 , ⋯ , a n) ∈ R n and a strictly monotone, continuous function f such that F (x 1 , ⋯ , x n) = f (a 1 x 1 + ⋯ + a n x n) holds for all (x 1 , ⋯ , x n) ∈ D . Using this result we also show that continuous solutions F : D ⟶ R of the system of functional equations F (x 1 , ⋯ , x j + t j , ⋯ , x n) = Ψ j (F (x 1 , ⋯ , x j , ⋯ , x n) , t j) (j = 1 , ⋯ , n) can be represented as the composition of a strictly monotone, continuous function and a linear functional as well. Applying the latter theorem, we give a characterization of Cobb–Douglas type utility functions. [ABSTRACT FROM AUTHOR]

In the present paper we consider the problem of characterization of maps which can be expressed as an affine difference i.e. a three-place map of the form t f (x) + (1 - t) f (y) - f (t x + (1 - t) y). We give a general solution of a functional equation associated with this problem. [ABSTRACT FROM AUTHOR]

Let S be a monoid (= semigroup with identity), and let σ : S → S be a homomorphism such that σ ∘ σ = i d . In an earlier paper we solved the Pexiderized d'Alembert functional equation (PDFE) f (x y) + g (σ (y) x) = h (x) k (y) for unknown f , g , h , k : S → C , assuming that S is either regular or generated by its squares and that one of the unknown functions is central. The present paper has two main results. The first describes the solutions of PDFE on a general monoid in terms of multiplicative functions, solutions of a special case of the sine subtraction law, and solutions of other functional equations with just one unknown function. The second main result uses the first one to give a more detailed solution of PDFE on a larger class of monoids than has been treated previously. We also find the continuous solutions on topological monoids. Examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

Let S be a semigroup that need not be abelian, σ : S → S be an involutive endomorphism and let (H , +) be a uniquely 2-divisible abelian group. We study the solutions f , g : S → H of each of the functional equations f (x y) + g (x y) + g (x σ (y)) = f (x) + f (y) + 2 g (x) + 2 g (y) , and f (x y) + g (x y) + g (x σ (y)) = f (x) + f (y) + 2 g (x) + g (y) + g (σ (y)). Moreover, we show that the Jensen type equation f (x y) + f (x σ (y)) = 2 f (x) and the generalized quadratic functional equation g (x y) + g (x σ (y)) = 2 g (x) + 2 g (y) are strongly alien in the sense of Dhombres. [ABSTRACT FROM AUTHOR]

The document summarizes the 60th International Symposium on Functional Equations held in Poland, featuring talks on topics like convex sequences and separation problems. It includes abstracts of talks, problems, and a list of participants from various countries. The symposium serves as a platform for scholars to share research on mathematical concepts and engage in discussions on solutions to specific problems. [Extracted from the article]

Iteration is one of the most important topics in the field of functional equations, which is also concerned with dynamical systems and operator theory. In this paper we discuss a class of multi-plateau mappings with finitely many P-intervals whose range crosses the diagonals and classifies them into four classes. For each class the properties of all the orbits of the two endpoints of the P-intervals are investigated in a similar way as for symbolic dynamical systems, and finally the iteration of these multi-plateau mappings are given. [ABSTRACT FROM AUTHOR]

By the well-known I. Kotlarski lemma, if ξ1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\xi _1$$\end{document}, ξ2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\xi _2$$\end{document}, and ξ3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\xi _3$$\end{document} are independent real-valued random variables with nonvanishing characteristic functions, L1=ξ1-ξ3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_1=\xi _1-\xi _3$$\end{document} and L2=ξ2-ξ3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_2=\xi _2-\xi _3$$\end{document}, then the distribution of the random vector (L1,L2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(L_1, L_2)$$\end{document} determines the distributions of the random variables ξj\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\xi _j$$\end{document} up to shift. Siran Li and Xunjie Zheng generalized this result for the linear forms L1=ξ1+a2ξ2+a3ξ3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_1=\xi _1+a_2\xi _2+a_3\xi _3$$\end{document} and L2=b2ξ2+b3ξ3+ξ4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L_2=b_2\xi _2+b_3\xi _3+\xi _4$$\end{document} assuming that all ξj\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\xi _j$$\end{document} have first and second moments, ξ2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\xi _2$$\end{document} and ξ3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\xi _3$$\end{document} are identically distributed, and aj\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a_j$$\end{document}, bj\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$b_j$$\end{document} satisfy some conditions. In the article, we give a simpler proof of this theorem. In doing so, we also prove that the condition of existence of moments can be omitted. Moreover, we prove an analogue of the Li–Zheng theorem for independent random variables with values in the field of p-adic numbers, in the field of integers modulo p, where p≠2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p\ne 2$$\end{document}, and in the discrete field of rational numbers. [ABSTRACT FROM AUTHOR]

A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We emphasize two methods of proofs, one elementary (relying significantly on functional equations) and the other based on modular forms. We close by proving analogous results for generalized overcubic partitions. [ABSTRACT FROM AUTHOR]

For a given p-variable mean M:Ip→I\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M :I^p \rightarrow I$$\end{document} (I is a subinterval of R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}$$\end{document}), following (Horwitz in J Math Anal Appl 270(2):499–518, 2002) and (Lawson and Lim in Colloq Math 113(2):191–221, 2008), we can define (under certain assumptions) its (p+1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(p+1)$$\end{document}-variable β\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta $$\end{document}-invariant extension as the unique solution K:Ip+1→I\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K :I^{p+1} \rightarrow I$$\end{document} of the functional equation K(M(x2,⋯,xp+1),M(x1,x3,⋯,xp+1),⋯,M(x1,⋯,xp))=K(x1,⋯,xp+1),for allx1,⋯,xp+1∈I\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned}&K\big (M(x_2,\dots ,x_{p+1}),M(x_1,x_3,\dots ,x_{p+1}),\dots ,M(x_1,\dots ,x_p)\big )\\&\quad =K(x_1,\dots ,x_{p+1}), \text { for all }x_1,\dots ,x_{p+1} \in I \end{aligned}$$\end{document}in the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions. [ABSTRACT FROM AUTHOR]

Three functional inequalities are shown to uniquely characterize the exponential function. Each of the three inequalities is indispensable in the sense that no two of the three suffice. [ABSTRACT FROM AUTHOR]

The main result of the present paper is about the solutions of the functional equation F ( x + y 2) + f 1 (x) + f 2 (y) = G (g 1 (x) + g 2 (y)) , x , y ∈ I , derived originally, in a natural way, from the invariance problem of generalized weighted quasi-arithmetic means, where F , f 1 , f 2 , g 1 , g 2 : I → R and G : g 1 (I) + g 2 (I) → R are the unknown functions assumed to be continuously differentiable with 0 ∉ g 1 ′ (I) ∪ g 2 ′ (I) , and the set I stands for a nonempty open subinterval of R . In addition to these, we will also touch upon solutions not necessarily regular. More precisely, we are going to solve the above equation assuming first that F is affine on I and g 1 and g 2 are continuous functions strictly monotone in the same sense, and secondly that g 1 and g 2 are invertible affine functions with a common additive part. [ABSTRACT FROM AUTHOR]

In this paper it is proved that, for a function f : X → E mapping from a normed linear space X into an inner product space E, the functional inequality ‖ 2 f (x) + 2 f (y) - f (x - y) ‖ ⩽ ‖ f (x + y) ‖ , ‖ x ‖ + ‖ y ‖ ⩾ d for some d > 0 , implies f is quadratic. Other types of functional inequalities related to the quadratic functional equation have also been investigated. Besides we establish the Hyers–Ulam stability on restricted domains, and we improve the bounds and thus the stability results obtained in Jung (J Math Anal Appl 222:126–137, 1998) and Rassias (J Math Anal Appl 276: 747–762, 2002). Finally we apply our recent results to the asymptotic behavior of quadratic functional equations of different types. [ABSTRACT FROM AUTHOR]

We show that, for a (not necessarily continuous) weakly contractive mean-type mapping M : I p → I p (where I is an interval and p ∈ N ), the functional equation K ∘ M = K has at most one solution in the family of continuous means K : I p → I . Some general approach to this equation is also given. [ABSTRACT FROM AUTHOR]

Let S be a semigroup, let M be a monoid with neutral element e, and let K be an algebraically closed field of characteristic ≠ 2 with identity element 1. Inspired by Stetkær's procedure [19] we describe, in terms of multiplicative functions and characters of 2-dimensional representations of S, the solutions g : S → K of the functional equation g (x y) + μ (y) g (ψ (y) x) = 2 g (x) g (y) , x , y ∈ S , where ψ : S → S is an anti-endomorphism that need not be involutive and μ : S → K is a multiplicative function such that μ (x ψ (x)) = 1 for all x ∈ S . This enables us to find the solutions g : M → K of the new functional equation g (x σ (y)) + g (ψ (y) x) = 2 g (x) g (y) , x , y ∈ M , where σ : M → M is an involutive endomorphism. [ABSTRACT FROM AUTHOR]

We observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space ℓ 2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness. [ABSTRACT FROM AUTHOR]

The study of functional equations in which the unknown functions are assumed to be additive has a long history and continues to be an active area of research. Here we discuss methods for solving functional equations of the form (∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$*$$\end{document}) ∑j=1kxpjfj(xqj)=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sum _{j=1}^{k} x^{p_j}f_j(x^{q_j}) = 0$$\end{document}, where the pj,qj\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p_j,q_j$$\end{document} are non-negative integers, the fj:R→S\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f_j:R \rightarrow S$$\end{document} are additive functions, S is a commutative ring, and R is a sub-ring of S. This area of research has ties to commutative algebra since homomorphisms and derivations satisfy equations of this type. Methods for solving all homogeneous equations of the form (∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$*$$\end{document}) can be found in Ebanks (Aequ Math 89(3):685-718, 2015), Ebanks (Results Math 73(3):120, 2018) and Gselmann et al. (Results Math 73(2):27, 2018). It seems that this fact may have been overlooked, judging by some results about a particular case of (∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$*$$\end{document}) in recent publications. We also present a new method for the homogeneous case by combining the results above with [6], and we show how to solve non-homogeneous equations of the form (∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$*$$\end{document}). [ABSTRACT FROM AUTHOR]