Stability of a Quadratic Functional Equation

Let G, H be uniquely 2-divisible Abelian groups. We study the solutions f, g: G → H of Pexider type functional equation (*) $$f(x+y)+f(x-y)+g(x+y)=2f(x)+2f(y)+g(x)+g(y),$$ resulting from summing up the well known quadratic functional equation and additive Cauchy functional equation side by side. We show that modulo a constant equation (*) forces f to be a quadratic function, and g to be an additive one (alienation phenomenon). Moreover, some stability result for equation (*) is also presented.

So far, we have discussed the stability problems of functional equations in connection with additive or linear functions. In this chapter, the Hyers–Ulam–Rassias stability of quadratic functional equations will be proved. Most mathematicians may be interested in the study of the quadratic functional equation since the quadratic functions are applied to almost every field of mathematics. In Section 8.1, the Hyers–Ulam–Rassias stability of the quadratic equation is surveyed. The stability problems for that equation on a restricted domain are discussed in Section 8.2, and the Hyers–Ulam–Rassias stability of the quadratic functional equation will be proved by using the fixed point method in Section 8.3. In Section 8.4, the Hyers–Ulam stability of an interesting quadratic functional equation different from the “original” quadratic functional equation is proved. Finally, the stability problem of the quadratic equation of Pexider type is discussed in Section 8.5.

In Park et al. (J. Chungcheong Math. Soc. 21:455–466, 2008) considered the following Jensen additive and quadratic type functional equation $$2 f \biggl(\frac{x+y}{2} \biggr) + f \biggl( \frac{x-y}{2} \biggr ) + f \biggl(\frac{y-x}{2} \biggr) = f(x) + f(y) . $$ In this paper, we investigate the following additive and quadratic functional equation $$ 2 f(x+y) + f(x-y) + f(y-x) = 3f(x) + f(-x) + 3f(y) + f(-y) . $$ (34.1) Furthermore, we prove the generalized Hyers–Ulam stability of the functional equation (34.1) in Banach spaces.

In this paper, we introduce the functional equations f(2x−y)+f(x+2y)=5[f(x)+f(y)],f(2x−y)+f(x+2y)=5f(x)+4f(y)+f(−y),f(2x−y)+f(x+2y)=5f(x)+f(2y)+f(−y),f(2x−y)+f(x+2y)=4[f(x)+f(y)]+[f(−x)+f(−y)].\\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} $$\\begin{aligned} f(2x-y)+f(x+2y)&=5\\bigl[f(x)+f(y)\\bigr], \\\\ f(2x-y)+f(x+2y)&=5f(x)+4f(y)+f(-y), \\\\ f(2x-y)+f(x+2y)&=5f(x)+f(2y)+f(-y), \\\\ f(2x-y)+f(x+2y)&=4\\bigl[f(x)+f(y)\\bigr]+\\bigl[f(-x)+f(-y)\\bigr]. \\end{aligned}$$ \\end{document} We show that these functional equations are quadratic and apply them to characterization of inner product spaces. We also investigate the stability problem on restricted domains. These results are applied to study the asymptotic behaviors of these quadratic functions in complete β-normed spaces.

Linear complex vector functional equations: general classes of cyclic functional equations functional equations with operations between arguments functional equations with constant parameters functional equations with constant coefficients systems of linear functional equations. Nonlinear complex vector functional equations: quadratic functional equations modified quadratic functional equations expanded quadratic functional equations higher order functional equations systems of nonlinear functional equations.

We approximate a fuzzy almost quadratic function by a quadratic function in a fuzzy sense. More precisely, we establish a fuzzy Hyers--Ulam--Rassias stability of the quadratic functional equation $f(x+y)+f(x-y)=2f(x)+2f(y)$. Our result can be regarded as a generalization of the stability phenomenon in the framework of normed spaces. We also prove a generalized version of fuzzy stability of the Pexiderized quadratic functional equation $f(x+y)+f(x-y)=2g(x)+2h(y)$.

The quadratic functional equation$$ f\left( {x + y} \right) + f\left( {x - y} \right) - 2f\left( x \right) - 2f\left( y \right) = 0$$ (3.1) clearly has f(x) = cx 2 as a solution with c an arbitrary constant when f is a real function of a real variable. We define any solution of (3.1) to be a quadratic function, even in more general contexts. We shall be interested in functions f: E 1 → E 2 where both E 1 and E 2 are real vector spaces, and we need a few facts concerning the relation between a quadratic function and a biadditive function sometimes called its polar. This relation is explained in Proposition 1, p. 166, of the book by J. Aczél and J. Dhombres (1989) for the case where E 2 = R, but the same proof holds for functions f: E 1 → E 2. It follows then that f: E 1 → E 2 is quadratic if and only if there exists a unique symmetric function B: E 1 × E 1 → E 2, additive in x for fixed y, such that f (x) = B(x, x). The biadditive function B, the polar of f, is given by$$B\left( {x,y} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ - \\ 4 \end{array}} \right)\left( {f\left( {x + y} \right) - f\left( {x - y} \right)} \right)$$

We prove a general uniqueness theorem that can be easily applied to the (generalized) Hyers-Ulam stability of the Cauchy additive functional equation, the quadratic functional equation, and the quadratic-additive type functional equations. This uniqueness theorem can replace the repeated proofs for uniqueness of the relevant solutions of given equations while we investigate the stability of functional equations.

Let $$(S,+)$$ be a commutative semigroup, $$(H,+)$$ a uniquely 2-divisible commutative group, and $$\sigma $$ an involution of S. First, we find the general solution $$g:S\rightarrow H$$ of the functional equation $$\begin{aligned}&g(x+y+z)+g(x+\sigma (y))+g(y+\sigma (z))\\&\quad =g(x+\sigma (y)+z)+g(x+y)+g(y+z), \quad x,y,z \in S. \end{aligned}$$ Second, we study the solutions $$f,g:S\rightarrow H$$ of the following functional equation $$\begin{aligned} f(x+y)+g(x+y)+g(x+\sigma (y))=f(x)+f(y)+2g(x)+2g(y), \quad x,y \in S, \end{aligned}$$ resulting from summing up the additive Cauchy functional equation $$f(x+y)=f(x)+f(y)$$ and the quadratic functional equation $$g(x+y)+g(x+\sigma (y))=2g(x)+2g(y)$$ side by side.

In this paper, we prove the generalized Hyers-Ulam stability of the quadratic functionalequation$$f(x+y)+f(x-y)=2f(x)+2f(y)$$in non-Archimedean $mathcal{L}$-fuzzy normed spaces.

Quadratic functional equations, bilinear forms equivalent to the quadratic equation, and some generalizations are treated in this chapter. Among the normed linear spaces (n.l.s.), inner product spaces (i.p.s.) play an important role. The interesting question when an n.l.s. is an i.p.s. led to several characterizations of i.p.s. starting with Frechet [291], Jordan and von Neumann [398], etc. Functional equations are instrumental in many characterizations. One of the objectives of the next chapter is to bring out the involvement of functional equations in various characterizations of i.p.s.

We study the stability of functional equations in quasi-Banach spaces where the quasi-norm is not assumed to be a p-norm. To overcome the modulus of concavity greater than 1 and the discontinuity of quasi-norms we use the squeeze inequality presented in an explicit revision of Aoki–Rolewicz Theorem [13, Theorem 1]. As illustrations, we prove an extension of the stability of a mixed additive and quadratic functional equation in p-Banach spaces to quasi-Banach spaces with better approximation. The technique may be used to prove extensions of other results on the stability of functional equations in p-Banach spaces to quasi-Banach spaces.

In this paper, we prove a general uniqueness theorem that can easily be applied to the (generalized) Hyers-Ulam stability of a large class of functional equations, which includes monomial functional equations (e.g. the Cauchy additive functional equation, the quadratic functional equation, and the cubic functional equation, etc.). This uniqueness theorem can save us much trouble in proving the uniqueness of relevant solutions repeatedly appearing in the stability problems for functional equations in fuzzy spaces.

We obtain in terms of additive and multi-additive functions the general solution f : S → H of each of the functional equations $$\displaystyle \sum _{\lambda \in \varPhi } f(x+\lambda y+a_{\lambda })=Nf(x),\ x,y\in S, $$ $$\displaystyle \sum _{\lambda \in \varPhi }f(x+\lambda y+a_{\lambda })=Nf(x)+Nf(y),\ x,y\in S, $$ where (S, +) is an abelian monoid, Φ is a finite group of automorphisms of S, \(N=\left \vert \varPhi \right \vert \) designates the number of its elements, \( \left \{ a_{\lambda },\lambda \in \varPhi \right \} \) are arbitrary elements of S, and (H, +) is an abelian group. In addition, some applications are given. These equations provide a common generalization of many functional equations (Cauchy’s, Jensen’s, quadratic, Φ-quadratic equations, …).

Rassias(2001) introduced the pioneering cubic functional equation in the history of mathematical analysis: and solved the pertinent famous Ulam stability problem for this inspiring equation. This Rassias cubic functional equation was the historic transition from the following famous Euler-Lagrange-Rassias quadratic functional equation: to the cubic functional equations. In this paper, we prove the Ulam-Hyers stability of the cubic functional equation: in fuzzy normed linear spaces. We use the definition of fuzzy normed linear spaces to establish a fuzzy version of a generalized Hyers-Ulam-Rassias stability for above equation in the fuzzy normed linear space setting. The fuzzy sequentially continuity of the cubic mappings is discussed.