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.

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To this end, several works has been done in the direction of differential equations as first credited to Alisina and Gar [1], 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.

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[16] 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. [17] A. Pietsch. Nuclear locally convex spaces. Ergebnisse der Mathematik 66, Springer, 1972. [18] Th.M. Rassias. On the **stability** of the linear mapping in banach spaces. Proceedings of

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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

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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 [–].

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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 ) = .

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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).

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G˘avrut¸a [10] 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]).

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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.

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.

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then a homomorphism h : G −→ G 0 exists with ρ(f (x), h(x)) < ε for all x ∈ G?. **Hyers** [10] 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 [1] for additive map- pings and by Rassias [13] 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 )

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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.

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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**.

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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.

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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 [–]).

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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 [40] applied the fixed point alternative theorem for **Hyers**-**Ulam**- Rassias **stability**, D. Mihet¸ [34] 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 [13] 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.

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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.

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[13] Y. Li and Y. Shen, “**Hyers**-**Ulam** **Stability** of Nonhomo- geneous Linear Differential Equations of Second Order,” International Journal of Mathematics and Mathematical Sciences, Vol. 2009, 2009, Article ID: 576852, p 7. [14] P. Gavruta, S. Jung and Y. Li, “**Hyers**-**Ulam** **Stability** for

The above result by Alsina and Ger was **generalized** by Miura, Takahasi and Choda [19], by Miura [16], also by Takahasi, Miura and Miyajima [26]. Indeed, they dealt with the **Hyers**-**Ulam** **stability** 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 [18] proved the **Hyers**-**Ulam** **stability** of the first-order linear differential equations y 0 (t) + g(t)y(t) = 0, where g(t) is a continuous function, while Jung [12] proved the **Hyers**-**Ulam** **stability** of differential equations of the form ϕ(t)y 0 (t) = y(t).

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