Remarks on hyperstability of the Cauchy functional equation

We present some simple observations on hyperstability for the Cauchy equation on a restricted domain. Namely, we show that (under some weak natural assumptions) functions that satisfy the equation approximately (in some sense), must be actually solutions to it. In this way we demonstrate in particular that hyperstability is not a very exceptional phenomenon as it has been considered so far. We also provide some simple examples of applications of those results.     

On a fixed point theorem in 2-Banach spaces and some of its applications

The aim of this article is to prove a fixed point theorem in 2-Banach spaces and show its applications to the Ulam stability of functional equations. The obtained stability results concern both some single variable equations and the most important functional equation in several variables, namely, the Cauchy equation. Moreover, a few corollaries corresponding to some known hyperstability outcomes are presented.     

Remark on hyperstability of the general linear equation

We present a result concerning the hyperstability of the general linear equation. Namely, we show that a function satisfying the equation approximately must be actually a solution to it.     

Hyperstability of the Jensen functional equation

We present some observations on hyperstability for the Jensen equation on restricted domain. Namely, we show, under some weak natural assumptions, that functions satisfying the equation approximately (in some sense) must be actually solutions to it.     

ON HYPERSTABILITY OF THE BIADDITIVE FUNCTIONAL EQUATION

We present results on approximate solutions to the biadditive equation f(x+y,z-w) + f(x-y,z+w) = 2f(x,z)-2f(y,w)on a restricted domain. The proof is based on a quite recent fixed point theorem in some function spaces. Our main results state that,under some weak natural assumptions,functions satisfying the equation approximately(in some sense) must be actually solutions to it. In this way we obtain inequalities characterizing biadditive mappings and inner product spaces.Our outcomes are connected with the well known issues of Ulam stability and hyperstability.     

On the hyperstability of {\sigma}-Drygas functional equation on semigroups

In this paper, we obtain hyperstability results for the \({\sigma}\)-Drygas functional equation and the inhomogeneous \({\sigma}\)-Drygas functional equation on semigroups. Namely, we show that a function satisfying the \({\sigma}\)-Drygas equation approximately must be exactly the solution of it.     

Arithmetically homogeneous functions: characterizations,stability and hyperstability

A function between two Abelian semigroups is arithmetically homogeneous of degree n if \( h(ix)=i^{n}h(x)\) for all positive integers i. We give new functional characterizations for arithmetically homogeneous functions, for polynomials of arithmetically homogeneous functions, and for Fréchet monomials. We also give large classes of control functions that provide generalized stability, or hyperstability for a homogeneous equation, and, consequently, new similar classes for classical monomial equations, so extending some known stability results.     

ON APPROXIMATELY(p,q)-WRIGHT AFFINE FUNCTIONS AND INNER PRODUCT SPACES

We prove, using the fixed point approach, some results on hyperstability(in normed spaces) of the equation that defines the generalization of p-Wright affine functions and show that they yield a simple characterization of the complex inner product spaces.     

Hyperstability results for a generalized radical cubic functional equation related to additive mapping in non-Archimedean Banach spaces

The aim of this paper is to introduce and solve the following generalized radical cubic functional equation

$$\begin{aligned} f\left( \root 3 \of {\sum _{i=1}^{k}x_{i}^{3}}\right) =\sum _{i=1}^{k}f(x_{i}),\quad k\in \mathbb {N}_{2}. \end{aligned}$$

We also investigate some hyperstability results for this equation in non-Archimedean Banach spaces.

     

Asymptotic aspect of Drygas,quadratic and Jensen functional equations in metric abelian groups

The asymptotic stability behavior of Drygas, quadratic and Jensen functional equations is investigated. Indeed, we show that if these equations hold approximately for large arguments with an upper bound \({\varepsilon}\), then they are also valid approximately everywhere with a new upper bound which is a constant multiple of \({\varepsilon}\). These results will be applied to the study of asymptotic properties of Drygas, quadratic and Jensen functional equations. We also obtain some results of hyperstability character for these functional equations.     

General criteria for hyperstability of Cauchy-type equations

Aequationes mathematicae - We provide three large classes of control functions that ensure the hyperstability of the Cauchy equation on restricted domains included in various types of commutative...     

HYERS-ULAM-RASSIAS STABILITY OF FUNCTIONAL EQUATIONS IN MENGER PROBABILISTIC NORMED SPACES

We introduce the approximately quadratic functional equation in Menger probabilistic normed spaces. More precisely, we show under some suitable conditions that an approximately quadratic functional equation can be approximated by a quadratic function in above mentioned spaces. Also we consider the stability problem for approximately pexiderized functional equation in Menger probabilistic normed spaces.     

The hyperstability of general linear equation via that of Cauchy equation

Aequationes mathematicae - 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...     

Differential games with a given value function

The set of solutions of a differential game with a terminal payoff functional is investigated. A method is obtained that allows us to establish whether a given function is a value of some differential game with a terminal payoff functional. The condition obtained is in fact the condition for the given function to be a minimax (viscosity) solution of some Hamilton-Jacobi equation with Hamiltonian homogeneous in the third variable. We also obtain a sufficient condition for a function to belong to the set of values of differential games with a terminal payoff function.     

The properties of functional inclusions and Hyers–Ulam stability

We prove that a set-valued function satisfying some functional inclusions admits, in appropriate conditions, a unique selection satisfying the corresponding functional equation. As a consequence we obtain the result on the Hyers–Ulam stability of that functional equation.     

Numerical analysis of an inverse problem for the eikonal equation

We are concerned with the inverse problem for an eikonal equation of determining the speed function using observations of the arrival time on a fixed surface. This is formulated as an optimisation problem for a quadratic functional with the state equation being the eikonal equation coupled to the so-called Soner boundary condition. The state equation is discretised by a suitable finite difference scheme for which we obtain existence, uniqueness and an error bound. We set up an approximate optimisation problem and show that a subsequence of the discrete mimina converges to a solution of the continuous optimisation problem as the mesh size goes to zero. The derivative of the discrete functional is calculated with the help of an adjoint equation which can be solved efficiently by using fast marching techniques. Finally we describe some numerical results.     

Hyperstability of the Jensen functional equation in ultrametric spaces

In this paper, we present hyperstability results of Jensen functional equations in ultrametric Banach spaces.     

Asymptotic stability of the Cauchy and Jensen functional equations

The aim of this note is to investigate the asymptotic stability behaviour of the Cauchy and Jensen functional equations. Our main results show that if these equations hold for large arguments with small error, then they are also valid everywhere with a new error term which is a constant multiple of the original error term. As consequences, we also obtain results of hyperstability character for these two functional equations.     

Hyers-Ulam stability for linear equations of higher orders

We show that, in the classes of functions with values in a real or complex Banach space, the problem of Hyers-Ulam stability of a linear functional equation of higher order (with constant coefficients) can be reduced to the problem of stability of a first order linear functional equation. As a consequence we prove that (under some weak additional assumptions) the linear equation of higher order, with constant coefficients, is stable in the case where its characteristic equation has no complex roots of module one. We also derive some results concerning solutions of the equation.     

Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation

We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied earlier by the author and present two classes of special functions, namely, ultraexponential and infralogarithm f -type functions. As a result of this investigation, we obtain a general solution of the Abel equation α(f(x)) = α (x) + 1 under some conditions on a real function f and prove a new completely different uniqueness theorem for the Abel equation stating that an infralogarithm f -type function is its unique solution. We also show that an infralogarithm f -type function is an essentially unique solution of the Abel equation. Similar theorems are proved for ultraexponential f -type functions and their functional equation β(x) = f(β(x − 1)), which can be considered as dual to the Abel equation. We also solve a certain problem unsolved before and study some properties of two considered functional equations and some relations between them.