Type Functional Equation Research Articles

A functional equation related to symmetry of operators

In this note we determine the unique solution to the functional equation \(f(x + y) ( x- y) = \left( f(x) -f (y) \right) (x + y)\). We require no additional assumptions on the function \(f\). Moreover we solve this functional equation if \(f\) is only defined on a finite interval. The interest in this type of functional equation is motivated by the study of symmetrizing measures for (the generator of) a Levy-driven Ornstein–Uhlenbeck process.

On the stability of a class of cosine type functional equations

The aim of this paper is to investigate the stability problem for the pexiderized trigonometric functional equation f₁(xy)+f₂(xσ(y))=2g₁(x)g₂(y), x,y∈G, where G is an arbitrary group, f₁,f₂,g₁ and g₂ are complex valued functions on G and σ is an involution of G. Results of this paper also can be extended to the setting of monoids (that is, a semigroup with identity) that need not be abelian.

Solution and stability of a mixed type functional equation

In this paper, we obtain the general solution and investigate the generalized Hyers-Ulam-Rassias stability for the new mixed type additive and cubic functional equation 3f(x+3y) - f(3x+y)=12[f(x+y)+f(x-y)] - 16[f(x)+f(y)]+12f(2y)-4f(2x). As some corollaries, we show that the stability of this equation can be controlled by the sum and product of powers of norms.

Non-Archimedean hyperstability of a Cauchy-Jensen type functional equation

Non-Archimedean hyperstability of a Cauchy-Jensen type functional equation

On the superstability of the cosine and sine type functional equations

AbstractIn this paper, we study the superstablity problem of the cosine and sine type functional equations: $$f(x\sigma (y)a) + f(xya) = 2f(x)f(y)$$ and $$f(x\sigma (y)a) - f(xya) = 2f(x)f(y),$$ where f : S → ℂ is a complex valued function; S is a semigroup; σ is an involution of S and a is a fixed element in the center of S.

Modelling the Additive Functional Equations through RSM Matrices

This paper suggests one possible method to model additive type of functional equations using eigenvalues and eigenvectors of matrices with suitable numerical examples. The authors have defined a new type of Row Sum Matrix(RSM) and have discussed its eigenvalues and eigenvectors in order to model functional equations. The famous additive cauchy functional equation and Logical functional equation have also been modelled using identity matrix and Logical matrix in this study.

REFINED HYERS-ULAM STABILITY OF MODULE LEFT (m,n)-DERIVATIONS IN METRIC SPACES

In this paper, we consider the Hyers-Ulam stability of Cauchy type functional equations of module left (m,n)-derivations and generalized module left (m,n)-derivations in complete metric spaces.

A numerical approach for solving Volterra type functional integral equations with variable bounds and mixed delays

In this paper, the Taylor collocation method has been used the integro functional equation with variable bounds. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points. We have introduced the method to solve the functional integral equations with variable bounds. We have also improved error analysis for this method by using the residual function to estimate the absolute errors. To illustrate the pertinent features of the method numeric examples are presented and results are compared with the other methods. All numerical computations have been performed on the computer algebraic system Maple 15.

On the probabilistic stability of the 2-variable k-AC-mixed type functional equation

In this paper, we obtain the general solution and the generalized Ulam-Hyers stability of the 2-variable $k$-AC mixed type functional equation$$\begin{aligned}& f(x+k y, z+k w)+f(x-k y, z-k w) \\& \quad=k^2[f(x+y, z+w)+f(x-y, z-w)]+2\left(1-k^2\right) f(x, z) .\end{aligned}$$for any $k \in Z-\{0, \pm 1\}$ in $\alpha$-Šerstnev Menger Probabilistic normed spaces.

Hyperstability of general linear functional equation

Our purpose is to investigate criteria for hyperstability of linear type functional equations. We prove that a function satisfying the equation approximately in some sense, must be a solution of it. We give some conditions on coefficients of the functional equation and a control function which guarantee hyperstability. Moreover, we show how our outcomes may be used to check whether the particular functional equation is hyperstable. Some relevant examples of applications are presented.

Hyperstability of an n-dimensional Jensen type functional equation

In this paper, we will investigate some hyperstability results of an n-dimensional Jensen type functional equation $$\begin{aligned} \sum _{i=1}^{n} p_{i} f(x_{i})=f\left( \sum _{i=1}^{n} p_{i} x_{i}\right) , \end{aligned}$$ where \(n>1\) is an integer, and \(p_{1},\ldots ,p_{n}\) are positive rational numbers with $$\begin{aligned} \sum _{i=1}^{n} p_{i}=1. \end{aligned}$$

Integral functional equations on locally compact groups with involution

Our main goal is to introduce some integral-type generalizations of the cosine and sine equations for complex-valued functions defined on a group G that need not be abelian. These equations provide a joint generalization of many trigonometric type functional equations such as d’Alembert’s, Cauchy’s, Gajda’s, Kannappan’s and Van Vleck’s equations. We prove that the continuous solutions for the first type and the central continuous solutions for the second one of these equations can be expressed in terms of characters, additive maps and matrix elements of irreducible, 2-dimensional representations of the group G. So the theory is part of harmonic analysis on groups.

General uniqueness theorem concerning the stability of additive, quadratic, and cubic functional equations

We prove a general uniqueness theorem that can easily be applied to the proof of (generalized) Hyers-Ulam stability of the additive, quadratic, cubic, or the cubic-quadratic-additive type functional equation. By using this uniqueness theorem, we can omit the repeated proof for uniqueness of the relevant solutions of those equations.

A general theorem on the stability of a class of functional equations including quadratic-additive functional equations.

We prove a general stability theorem of an n-dimensional quadratic-additive type functional equation \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} Df(x_1, x_2, \\ldots , x_n) = \\sum _{i=1}^m c_i f \\big ( a_{i1}x_1 + a_{i2}x_2 + \\cdots + a_{in}x_n \\big ) = 0 \\end{aligned}$$\\end{document}Df(x1,x2,…,xn)=∑i=1mcif(ai1x1+ai2x2+⋯+ainxn)=0by applying the direct method.

STABILITY OF (α, β, γ)-DERIVATIONS ON LIE C*-ALGEBRA ASSOCIATED TO A PEXIDERIZED QUADRATIC TYPE FUNCTIONAL EQUATION

In this article, we considered the stability of the following (, , )-derivation and homomorphisms associated to the quadratic type functional equation , where is an involution of the Lie -algebra A and k is a fixed positive integer. The Hyers-Ulam stability on unbounded domains is also studied. Applications of the results for the asymptotic behavior of the generalized quadratic functional equation are provided.

Function Spaces, Approximation Theory, and Their Applications

The purpose of this special issue was to present new developments in the theory of function spaces, along with the deep interconnections with approximation theory and the applications in various fields of pure and applied mathematics. The reaction of the mathematical community was very satisfactory. We collected thirty-five submissions, covering a wide range of mathematical topics, ten of which were found to be suitable for publications in this issue. The major part of the accepted papers treats function spaces and their applications. In this respect, in the article by X Yang et al. a new class of function spaces, named “multi-βnormed spaces”, is introduced, in connection with stability properties of certain type of functional equations, while, in the paper by A. A. Bakery, sequential spaces of Orlicz type are studied and connected with the theory of summability. In the review paper by L. Angeloni and G. Vinti, the approximation theory in the space of functions with bounded variation is developed, in view of applications to signal processing. Different notions of variation are considered and several approximation theorems for families of integral or discrete type operators are given. In the more theoretical article by S. Wulede et al., a new class of Banach spaces which generalizes the class of uniformly extremely convex Banach spaces is introduced, and some characterizations of these spaces are given. Another paper by N. Khan treats the convergence of new type of double sequences, here introduced, in n-normed spaces. An interesting abstract approach to the theory of filter convergence is given in the article by A. Boccuto and X. Dimitriou, in which the links with function spaces and approximation theory are also dealt with. Other aspects of the theory of function spaces and their interconnections with calculus of variations, numerical analysis, complex variables, and stochastic processes are discussed, respectively, in the articles by T. Ma and Y. Feng, H. Wang et al., S. Wang and T. Zhan, and finally P. Duan.These four papers point out how certain methods of general approximation theory in function spaces can be employed in order to solve problems coming from a large variety of mathematical fields. We think that these contributions may represent starting points for new researches in the field of function spaces and approximation theory.

A Fixed Point Approach to the Stability of an Additive-Quadratic-Cubic-Quartic Type Functional Equation

By applying a fixed point approach, we investigate the stability problems for an AQCQ functional equation of the formfx+2y+fx-2y-4fx+y-4fx-y+6fx-f2y-f-2y+4fy+4f-y=0.

STABILITY OF THE GENERALIZED QUADRATIC AND QUARTIC TYPE FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN FUZZY NORMED SPACES

In this paper, we prove some stability results concerning the generalized quadratic and quartic type functional equation in the context of nonArchimedean fuzzy normed spaces in the spirit of Hyers-Ulam-Rassias. As applications, we establish some results of approximately generalized quadratic and quartic type mapping in non-Archimedean normed spaces. Also, we show that the assumption of the non-Archimedean absolute value of 2 is less than 1 cannot be omitted in our corollaries. The results improve and extend some recent results.

On the stability of α−Cauchy-Jensen type functional equation in Banach algebras

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in Banach algebras for the following $\alpha$-Cauchy-Jensen functional equation:$$f\left(\frac{x+y}{\alpha}+z\right)+f\left(\frac{x-y}{\alpha}+z\right)=\frac{2}{\alpha} f(x)+2 f(z),$$where $\alpha \in \mathbb{N}_{\geq 2}$.

General Quadratic-Additive Type Functional Equation and Its Stability

We investigate the general functional equation of the formf(ax+by+cz)-abf(x+y)-bcf(y+z)−acf(x+z)-a((a+1)/2-b-c)f(x)-b((b+1)/2-a-c)f(y)-c((c+1)/2-a-b)f(z)-(a(a-1)/2)f(-x)−(b(b-1)/2)f(-y)-(c(c-1)/2)f(-z)=0, whose solutions are quadratic-additive mappings in connection with stability problems.