# Generalized Hyers-Ulam stability

## Top PDF Generalized Hyers-Ulam stability: ### Generalized Hyers-Ulam Stability of the Pexiderized Cauchy Functional Equation in Non-Archimedean Spaces

for all x ∈ X. Rassias 3 succeeded in extending the result of Hyers by weakening the condition for the Cauchy diﬀerence to be unbounded. A number of mathematicians were attracted to this result of Rassias and stimulated to investigate the stability problems of functional equations. The stability phenomenon that was introduced and proved by Rassias is called the generalized Hyers-Ulam stability. Forti 4 and G˘avrut¸a 5 have generalized the result of Rassias, which permitted the Cauchy diﬀerence to become arbitrary unbounded. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem. A large list of references can be found, for example, in 3, 6–30. ### Generalized Hyers-Ulam stability of second order linear ordinary differential equation with initial condition

To this end, several works has been done in the direction of differential equations as first credited to Alisina and Gar , who were the first to considered the Hyers-Ulam stability of differential equation. They proved the Hyers-Ulam stability of the differential equation y 0 = y: That if given ε > 0, f be a differentiable function on an open interval I into R , where R is the real number field, with | f 0 (t) − f (t)| ≤ ε for all t ∈ I, then there exist a differentiable function g : I → R such that g 0 (t ) = g(t) and | f (t) − g(t )| ≤ 3ε for all t ∈ I. ### Generalized Hyers-Ulam Stability of a System of Bi-Reciprocal Functional Equations

 E. Movahednia. Fixed point and generalized Hyers-Ulam-Rassias stability of a quadratic functional equation. Journal of Mathematics and Computer Science, 6:72–78, 2013.  A. Pietsch. Nuclear locally convex spaces. Ergebnisse der Mathematik 66, Springer, 1972.  Th.M. Rassias. On the stability of the linear mapping in banach spaces. Proceedings of ### Isomorphisms in unital \$C^*\$-algebras

for all x ∈ X. The above inequality has provided a a lot of influence in the de- velopment of what is called Hyers-Ulam-Rassias stability or generalized Hyers-Ulam stability of functional equations. Beginning around the year 1980 the topic of ap- proximate homomorphisms, or the stability of the equation of homomorphism, was ### On stability of functional equations related to quadratic mappings in fuzzy Banach spaces

is called a quadratic functional equation. Quadratic functional equations were used to characterize inner product spaces. In particular, every solution of the quadratic equation is said to be a quadratic mapping. The generalized Hyers-Ulam stability problem for the quadratic functional equation (.) was proved by Skof []. Recently, the stability problem of the radical quadratic functional equations in various spaces was proved in the papers [–]. ### A general theorem on the stability of a class of functional equations including monomial equations

We combine Lemmas . and . in the following theorem to formulate the main theo- rem which is easily applicable. We here notice that conditions () and () are satisﬁed in usual cases provided we apply the direct method for proving the (generalized) Hyers-Ulam stability of various functional equations of the form Df (x  , x  , . . . , x n ) = . ### Fuzzy Stability of Generalized Square Root Functional Equation in Several Variables: A Fixed Point Approach

ρ i 6= 1 and s : X → R with X as space of non-negative real numbers and investigated generalized Hyers-Ulam stability of equation (2). It is easy to verify that the function f : X → R such that f (x) = √ x is a solution of the functional equation (2). ### Functional Inequalities Associated with Jordan von Neumann Type Additive Functional Equations

G˘avrut¸a  provided a further generalization of Rassias’ theorem. During the last two decades, a number of papers and research monographs have been published on vari- ous generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [11–14]). ### Approximate homomorphisms and derivations on random Banach algebras

The above theorem had a lot of inﬂuence on the development of the generalization of the Hyers-Ulam stability concept during the last three decades. This new concept is known as generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equa- tions (see [, ]). Furthermore, Gˇavruta [] provided a generalization of Rassias’ theorem which allows the Cauchy diﬀerence to be controlled by a general unbounded function. ### Hyers-Ulam-Rassias stability of nth order linear ordinary differential equations with initial conditions

Stability of nonhomogeneous second order Linear Differential Equations of the form y 00 + p ( x ) y 0 + q ( x ) y + r ( x ) = 0 under some special conditions. Pasc Gavruta, Jung, Li , investigated the Hyers-Ulam stability for second order linear differential equations with boundary conditions of the form y 00 + β ( x ) y ( x ) = 0. Jinghao Huang, Qusuay H. Alqifiary, and Yongjin Li , proved the generalized superstability of nth order linear differential equations with initial conditions of the form y (n) ( x ) + β ( x ) y ( x ) = 0. Recently, M.I. Modebei, O.O. Olaiya, I. Otaide , investigated generalized Hyers-Ulam stability of second order linear ordinary differential equation y 00 + β ( x ) y = f ( x ) with initial condition. ### Fixed Points, Inner Product Spaces, and Functional Equations

is called a quadratic functional equation. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof 6 for mappings f : X → Y , where X is a normed space and Y is a Banach space. Cholewa 7 noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik 8 proved the generalized Hyers-Ulam stability of the quadratic functional equation. The generalized Hyers-Ulam stability of the above quadratic functional equation and of two functional equations of quadratic type was obtained by C˘adariu and Radu 9. ### Hyers-Ulam and Hyers-Ulam-Rassias stability of nonlinear integral equations with delay

then a homomorphism h : G −→ G 0 exists with ρ(f (x), h(x)) < ε for all x ∈ G?. Hyers  gave a first affirmative partial answer to the question of Ulam for Banach spaces, he proved that each solution of the inequality kf (x + y) − f(x) − f (y)k ≤ ε, for all x and y, can be approximated by an exact solution, say an additive function (Hyer-Ulam stability ). Hyers’s theorem was generalized by Aoki  for additive map- pings and by Rassias  for linear mappings by considering an unbounded Cauchy difference. More precisely, he attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows: kf(x + y) − f (x) − f (y)k ≤ ε(kxk p + kyk p ) ### Superstability of Generalized Derivations

The well-known problem of stability of functional equations started with a question of Ulam 1 in 1940. In 1941, Ulam’s problem was solved by Hyers 2 for Banach spaces. Aoki 3 provided a generalization of Hyers’ theorem for approximately additive mappings. In 1978, Rassias 4 generalized Hyers’ theorem by obtaining a unique linear mapping near an approximate additive mapping. ### A Fixed Point Approach to the Stability of Pexider Quadratic Functional Equation with Involution

A basic question in the theory of functional equations is as follows. “When is it true that a function, which approximately satisfies a functional equation must be close to an exact solution of the equation?” The first stability problem concerning group homomorphisms was raised by Ulam 1 in 1940 and aﬃrmatively answered by Hyers in 2. Subsequently, the result of Hyers was generalized by Aoki 3 for additive mappings and by Rassias 4 for linear mappings by considering an unbounded Cauchy diﬀerence. The paper of Rassias has provided a lot of influence in the development of what we now call Hyers-Ulam-Rassias stability of functional equations. For more information, see 5–7. Specially, Maligranda 8 and Moszner 9 provided a very interesting discussion on the definition of functional equations’ stability. ### On the generalized Hyers Ulam Rassias stability problem of radical functional equations

In , the stability problem of functional equations originated from the question of Ulam [, ] concerning the stability of group homomorphisms. The famous Ulam sta- bility problem was partially solved by Hyers [] in Banach spaces. Later, Aoki [] and Bourgin [] considered the stability problem with unbounded Cauchy diﬀerences. Ras- sias [–] provided a generalization of Hyers’ theorem by proving the existence of unique linear mappings near approximate additive mappings. On the other hand, Rassias [, ] considered the Cauchy diﬀerence controlled by a product of diﬀerent powers of norm. The above results have been generalized by Forti [] and Gˇavruta [], who permitted the Cauchy diﬀerence to become arbitrary unbounded. Gajda and Ger [] showed that one can get analogous stability results for subadditive multifunctions. Gruber [] remarked that Ulam’s problem is of particular interest in probability theory and in the case of func- tional equations of diﬀerent types. Recently, Baktash et al. [], Cho et al. [–], Gordji et al. [–], Lee et al. [, ], Najati et al. [, ], Park et al. [], Saadati et al. [] and Savadkouhi et al. [] have studied and generalized several stability problems of a large variety of functional equations. ### Hyers Ulam stability of a generalized additive set valued functional equation

The stability problem of functional equations originated from a question of Ulam [] concerning the stability of group homomorphisms. Hyers [] gave the ﬁrst aﬃrmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was general- ized by Aoki [] for additive mappings and by Th.M. Rassias [] for linear mappings by considering an unbounded Cauchy diﬀerence. A generalization of the Th.M. Rassias the- orem was obtained by Găvruta [] by replacing the unbounded Cauchy diﬀerence with a general control function in the spirit of Th.M. Rassias’ approach. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [–]). ### A new method for the generalized Hyers-Ulam-Rassias stability

On the other hand, in 1991 J.A.Baker used the Banach fixed point theorem to give Hyers-Ulam stability results for a nonlinear functional equation. Following this idea, V.Radu  applied the fixed point alternative theorem for Hyers-Ulam- Rassias stability, D. Mihet¸  applied the Luxemburg-Jung fixed point theorem in generalized metric spaces to study the Hyers-Ulam stability for two functional equations in a single variable, L.G˘ avrut¸a  used the Matkowski’s fixed point theorem to obtain a new general result concerning the Hyers-Ulam stability of a functional equation in a single variable. ### Hyers Ulam stability of functional equations in matrix normed spaces

is called the Cauchy additive functional equation. In particular, every solution of the Cauchy additive functional equation is said to be an additive mapping. Hyers [] gave the ﬁrst aﬃrmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki [] for additive mappings and by Rassias [] for linear mappings by considering an unbounded Cauchy diﬀerence. A generalization of the Rassias theorem was obtained by Găvruta [] by replacing the unbounded Cauchy diﬀerence by a general control function in the spirit of Rassias’ approach.  