Stability Of Functional Equation Research Articles

A Note on the Stability of Functional Equations via a Celebrated Direct Method

More than ten years after Justyna Sikorska [8] attempted to solve the Heyers-Ulam stability of a single variable equation by using direct method. In this paper, we will improve the results of Justyna Sikorska by using a more efficient approach. Relations between the generalized functional equation, the dependence of their different parameters and several properties are also further explored. To achieve the problem, we try to develop some new techniques to overcome the fundamental difficulties caused by the different properties of the function and the presence of several variables in the equation. Furthermore, we continue to construct and study a couple of functional equations by making a new direct method.

Stability Problem for the Nonlinear Functional Equation

We discuss on the stability for functional equations in a single variable. We will prove the theorem about the stability of the nonlinear functional equation from which we can derive the result obtained by T. Trif (Publ Math Debrecen 60:47–61, 2002) on the stability of the linear functional equation.

On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey II

Ulam stability is motivated by the following issue: how much an approximate solution of an equation differs from the exact solutions to the equation. It is connected to some other areas of investigation, e.g., optimization, approximation theory and shadowing. In this paper, we present and discuss the published results on such stability for functional equations in the classes of function-taking values in 2-normed spaces. In particular, we point to several pitfalls they contain and provide possible simple improvements to some of them. Thus we show that the easily noticeable symmetry between them and the analogous results proven for normed spaces is, in fact, mainly apparent. Our article complements the earlier similar review published in Symmetry (13(11), 2200) because it concerns the outcomes that have not been discussed in this earlier publication.

On Asymptotic Behavior of a 2-Linear Functional Equation

In this paper, we deal with a 2-linear functional equation. The Hyers-Ulam stability of this functional equation is shown on some restricted unbounded domains, and the obtained results are applied to get several hyperstability consequences. Moreover, some asymptotic behaviors of 2-linear functions are investigated. We also study the Hyers-Ulam stability and superstability of the 2-linear functional equation in 2-Banach spaces.

Hyers-Ulam-Rassias Stability for Functional Equation in Neutrosophic Normed Spaces

In Neutrosophic Normed spaces, we investigate a unique quadratic function and a unique additive quadratic function of the Hyers-Ulam-Rassias stability for the functional equation which is said to be a functional equation associated with inner products space.

Well-posedness and Ulam’s stability of functional equations in F-metric space with an application

In this paper, we consider a fixed point problem related to some contraction mappings and introduce new classes of Picard operators for such mappings in the framework of F-metric space, yielding some interesting and novel results. As application of the obtained results, we investigate the Hyers-Ulam stability of a fixed point problem, a Cauchy functional equation, and an integral equation. Also, we present the well-posedness of the fixed point problem and integral equation. Some illustrative examples are also provided to support the new findings.

Ulam stability of functional equations in 2-Banach spaces via the fixed point method

Using the fixed point method, we prove the Ulam stability of two general functional equations in several variables in 2-Banach spaces. As corollaries from our main results, some outcomes on the stability of a few known equations being special cases of the considered ones will be presented. In particular, we extend several recent results on the Ulam stability of functional equations in 2-Banach spaces.

Stability of functional equations in a single variable via JS-metrics

Some stability results concerning functional equations in a single variable are proved, using the Banach Contraction Principle in a complete JS–metric space. It is also shown that certain functional equations possess a unique solution.

3-Quadratic Functions in Lipschitz Spaces

Stability of functional equations is a classical problem proposed by Ulam. In this paper, we prove the stability of
 the 3-quadratic functional equations in Lipschitz spaces.

A Note on the Stability of Some Functional Equations on Certain Groupoids

In this paper, we show that the stability of Cauchy set-valued functional equations and of Jensen set-valued functional equations can be derived from the stability of the corresponding equations in single-valued version.

Stability of Functional Equations and Properties of Groups

Abstract Investigating Hyers–Ulam stability of the additive Cauchy equation with domain in a group G, in order to obtain an additive function approximating the given almost additive one we need some properties of G, starting from commutativity to others more sophisticated. The aim of this survey is to present these properties and compare, as far as possible, the classes of groups involved.

On stability of the functional equation of p-Wright affine functions in (2, \u03b1)-Banach spaces

Stability of functional equations has recent applications in many fields. We show that the stability results obtained by J. Brzdęk and concerning the functional equation of the p-Wright affine function:f(px1+(1−p)x2)+f((1−p)x1+px2)=f(x1)+f(x2),can be proved also in (2, α)-Banach spaces, for some real number α∈(0,1). This is done using some fixed-point theorem.

Stability of mixed type functional equation in normed spaces using fuzzy concept

ABSTRACTS. M. Ulam once addressed the problem ‘when is it true that a mathematical object satisfying a certain property approximately must be close to an object satisfying the property exactly?’. This problem was solved by D. H. Hyers in 1941 using the functional equation and thereafter numerous research papers and monographs have been published for various types of functional equations in different spaces. The solution proposed by D. H. Hyers (1941) later developed into the famous generalized Hyers-Ulam-Rassias stability of functional equations. In this paper, we intend to attain the general solution to a new mixed type functional equation and interrogate the generalized Hyers–Ulam–Rassias stability in fuzzy normed spaces. Also, we seek to provide its application for generating secret keys in client–server environment.

The Ulam stability of functional equation on matrix random normed spaces

The Ulam stability of functional equation on matrix random normed spaces

Stability of functional equations arising from number theory and determinant of matrices

In this paper, we consider the Ulam–Hyers stability of the functional equations f(ux−vy,uy−vx)=f(x,y)f(u,v), f(ux+vy,uy−vx)=f(x,y)f(u,v), f(ux+vy,uy+vx)=f(x,y)f(u,v), f(ux−vy,uy+vx)=f(x,y)f(u,v) for all x,y,u,v∈R, where f:R2→R, which arise from number theory and are connected with the characterizations of the determinant and permanent of two-by-two matrices.

Stability of functional equations of n-Apollonius type in fuzzy ternary Banach algebras

Using the fixed point method, we investigate the generalized Hyers–Ulam stability of the ternary homomorphisms and ternary derivations between fuzzy ternary Banach algebras for the additive functional equation of n-Apollonius type, namely $${\sum_{i=1}^{n} f(z-x_{i}) = -\frac{1}{n} \sum_{1 \leq i < j \leq n} f(x_{i}+x_{j}) + n f (z-\frac{1}{n^{2}} \sum_{i=1}^{n}x_{i}),}$$ where \({n \geq 2}\) is a fixed positive integer.

Stability of functional equations connected with quadrature rules

We study the stability properties of the equation $$F(y) - F(x) = (y - x) \sum_{i=1}^{n}a_{i}f(\alpha_i x + \beta_{i}y)$$ (0.1) which is motivated by numerical integration. In Szostok and Wasowicz (Appl Math Lett 24(4):541–544, 2011) the stability of the simplest equation of the type (0.1) was investigated thus the inequality $$\left|F(y) - F(x) - (y - x)f\left(x + y\right)\right| \leq \varepsilon$$ was studied. In the current paper we present a somewhat different approach to the problem of stability of (0.1). Namely, we deal with the inequality $$\left|\frac{F(y) - F(x)}{y - x} - \sum_{i=1}^{n}a_{i}f(\alpha_{i}x + \beta_{i}y)\right| \leq \varepsilon.$$

Erratum to: Stability of functional equations on hypergroups

In this paper the following changes should be made. Theorem 2 should read as follows: Theorem 2. Let K be an amenable discrete hypergroup and let f, g, h, k : K→C be functions, where f is unbounded. Then the function (x, y) → f(x ∗ y)− g(x)h(y)− k(y) is bounded if and only if we have f(x) = cm(x) + φ(x) + b1(x), g(x) = cm(x) + φ(x) + b2(x), h(x) = m(x), k(x) = φ(x) + b3(x), where m : K → C is an exponential, b1, b2, b3 : K → C are bounded functions, c is a complex number, and φ : K → C satisfies the functional equation φ(x ∗ y) = φ(x)m(y) + φ(y) (1) for each x, y in K, further, if b2 is nonzero, then m is bounded.

STABILITY OF QUARTIC SET-VALUED FUNCTIONAL EQUATIONS

We will show the general solution of the functional equation <TEX>$$\begin{eqnarray}f(x+ay)+f(x-ay)+2(a^2-1)f(x)\\=a^2f(x+y)+a^2f(x-y)+2a^2(a^2-1)f(y)\end{eqnarray}$$</TEX> and investigate the Hyers-Ulam stability of the quartic set-valued functional equation.

Stability of functional equations in ( n , β ) -normed spaces

In this paper, we first introduce the notions of $(n,\beta)$ -normed space and non-Archimedean $(n,\beta)$ -normed space, then we study the Hyers-Ulam stability of the Cauchy functional equation and the Jensen functional equation in non-Archimedean $(n,\beta)$ -normed spaces and that of the pexiderized Cauchy functional equation in $(n,\beta)$ -normed spaces.