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 generalizedHyers-Ulamstability. 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.
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-Ulamstability of differential equation. They proved the Hyers-Ulamstability 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.
 E. Movahednia. Fixed point and generalizedHyers-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
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 generalizedHyers-Ulamstability of functional equations. Beginning around the year 1980 the topic of ap- proximate homomorphisms, or the stability of the equation of homomorphism, was
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 generalizedHyers-Ulamstability 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 [–].
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-Ulamstability of various functional equations of the form Df (x , x , . . . , x n ) = .
ρ i 6= 1 and s : X → R with X as space of non-negative real numbers and investigated generalizedHyers-Ulamstability 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).
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 generalizedHyers-Ulamstability to a number of functional equations and mappings (see [11–14]).
The above theorem had a lot of inﬂuence on the development of the generalization of the Hyers-Ulamstability concept during the last three decades. This new concept is known as generalizedHyers-Ulamstability 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.
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-Ulamstability 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 generalizedHyers-Ulamstability of second order linear ordinary differential equation y 00 + β ( x ) y = f ( x ) with initial condition.
is called a quadratic functional equation. A generalizedHyers-Ulamstability 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 generalizedHyers-Ulamstability of the quadratic functional equation. The generalizedHyers-Ulamstability of the above quadratic functional equation and of two functional equations of quadratic type was obtained by C˘adariu and Radu 9.
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-Ulamstability ). 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 )
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 generalizedHyers’ theorem by obtaining a unique linear mapping near an approximate additive mapping.
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.
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.
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 [–]).
On the other hand, in 1991 J.A.Baker used the Banach fixed point theorem to give Hyers-Ulamstability 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-Ulamstability 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-Ulamstability of a functional equation in a single variable.
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.
 Y. Li and Y. Shen, “Hyers-UlamStability of Nonhomo- geneous Linear Differential Equations of Second Order,” International Journal of Mathematics and Mathematical Sciences, Vol. 2009, 2009, Article ID: 576852, p 7.  P. Gavruta, S. Jung and Y. Li, “Hyers-UlamStability for
The above result by Alsina and Ger was generalized by Miura, Takahasi and Choda , by Miura , also by Takahasi, Miura and Miyajima . Indeed, they dealt with the Hyers-Ulamstability of the differential equation y 0 (t) = λy(t), while Alsina and Ger investigated the differential equation y 0 (t) = y(t). Miura et al  proved the Hyers-Ulamstability of the first-order linear differential equations y 0 (t) + g(t)y(t) = 0, where g(t) is a continuous function, while Jung  proved the Hyers-Ulamstability of differential equations of the form ϕ(t)y 0 (t) = y(t).