Fixed Point Approach to the Estimation of Approximate General Quadratic Mappings

Transcription

1 Int. Journal of Math. Analysis, Vol. 7, 013, no. 6, Fixed Point Approach to the Estimation of Approximate General Quadratic Mappings Kil-Woung Jun Department of Mathematics, Chungnam National University 79 Daehangno, Yuseong-gu, Daejeon, , Korea Yang-Hi Lee Department of Mathematics Education Gongju National University of Education, Gongju, , Korea Hark-Mahn Kim Department of Mathematics, Chungnam National University 79 Daehangno, Yuseong-gu, Daejeon, , Korea Abstract The purpose of this paper is to investigate the stability of a mixed functional equation with quadratic and Jensen additive mappings by using the fixed point method without dividing an approximate mapping by two approximate even and odd parts of the approximate mapping. Mathematics Subject Classification: 39B8, 39B7 Keywords: Fixed Point Method, Generalized Hyers-Ulam stability, Approximate General Quadratic Mappings 1 Introduction In 1940, S. M. Ulam [7] raised a question concerning the stability of homomorphisms: Given a group G 1, a metric group G with the metric d(, ), and a positive number ε, does there exist a δ>0 such that if a mapping f : G 1 G

2 76 K. Jun, Y. Lee and H. Kim satisfies the inequality d(f(xy),f(x)f(y)) <δfor all x, y G 1, then there exists a homomorphism h : G 1 G with d(f(x),h(x)) <εfor all x G 1? When this problem has a solution, we say that the homomorphisms from G 1 to G are stable. Thus, for very general functional equations, the stability question of functional equations is that a functional equation F 1 (f) =F (f) is stable if any function f satisfying approximately the equation F 1 (f) =F (f) is near to an exact solution h of the equation F 1 (f) =F (f). In 1941, D. H. Hyers [11] considered the case of approximately additive mappings between Banach spaces and proved the following result. Suppose that E 1 and E are Banach spaces and f : E 1 E satisfies the following condition: there is a constant ɛ 0 such that f(x + y) f(x) f(y) ɛ for all x, y E 1. Then the it h(x) = n f( n x) n exists for all x E 1 and it is a unique additive mapping h : E 1 E such that f(x) h(x) ɛ. The method which was provided by Hyers, and which produces the additive mapping h, was called a direct method. This method is the most important and most powerful tool for studying the stability of various functional equations. Hyers Theorem was generalized by T. Aoki [1] and D.G. Bourgin [3] for additive mappings by considering an unbounded Cauchy difference. In 1978, Th.M. Rassias [] also provided a generalization of Hyers Theorem for linear mappings which allows the Cauchy difference to be unbounded like this x p + y p. Let E 1 and E be two Banach spaces and f : E 1 E be a mapping such that f(tx) is continuous in t R for each fixed x. Assume that there exists ɛ>0 and 0 p<1 such that f(x + y) f(x) f(y) ɛ( x p + y p ), x, y E 1. Then there exists a unique R-linear mapping T : E 1 E, defined by T (x) = n f( n x) n such that f(x) T (x) ɛ p x p for all x E 1. A generalized result of Th.M. Rassias theorem was obtained by P. Gǎvruta in [9] and S. Jung in [14]. In 1990, Th.M. Rassias [3] during the 7th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for p 1. In 1991, Z. Gajda [8] following the same approach as in [], gave an affirmative solution to this question for p>1. It was shown by Z. Gajda [8], as well as by Th.M. Rassias and P. Semrl [4], that one cannot prove a Th.M. Rassias type theorem when p = 1. The counterexamples of Z. Gajda [8], as well as of Th.M. Rassias and P. Semrl [4], have stimulated several mathematicians to invent new approximately additive or approximately linear mappings. In particular, J.M. Rassias

3 Approximate general quadratic mappings 77 [0, 1] proved a similar stability theorem in which he replaced the unbounded Cauchy difference by this factor x p y q for p, q R with p + q 1. In 003, L. Cădariu and V. Radu [4] observed that the existence of the solution F for a Cauchy functional equation and the estimation of the mapping F with the approximate mapping f of the equation can be obtained from the alternative fixed point theorem. This method is called a fixed point method. In addition, they applied this method to prove the stability theorems of the Jensen s functional equation: ( ) x + y f f(x) f(y) =0 f(x) f(x + y) f(x y) =0. (1) Furthermore, they [4] obtained the stability of the quadratic functional equation: f(x + y)+f(x y) f(x) f(y) = 0 () by using the fixed point method. There are many interesting stability results concerning the stability problems obtained by using the fixed point method. A large list of references can be found in the papers [5, 6, 7, 10, 1, 13, 16, 17, 19] and references therein. If we consider the functions f 1,f : R R defined by f 1 (x) =ax + b and f (x) =cx, respectively, where a, b and c are real constants, then we notice that f 1 satisfies the equation (1) and f is a solution of the equation (), respectively. In 008, A. Najati and M. Moghimi [18] obtained stability results of the following functional equation f(x + y)+f(x y)+3f(y) 3f(x) f( y) f(y) =0, (3) of which the general solution function f between linear spaces is of the form f(x) = Q(x) + A(x) + f(0), where Q(x) is a quadratic mapping satisfying the equation () and A(x) + f(0) is a Jensen mapping satisfying the equation (1). Thus, we say that the equation (3) a general quadratic functional equation and a solution of (3) is called a general quadratic mapping. In their processing [18], by splitting to handle the odd part and the even part of the given mapping f, respectively, they could have an additive mapping A which is close to the odd part f(x) f( x) of f and a quadratic mapping Q which is approximate to the even part f(x)+f( x) f(0) of it, and then they could prove the existence of a general quadratic mapping F which is close to the given mapping f by combining A and Q + f(0). Now, without splitting the given approximate mapping f : X Y of the equation (3) into two approximate even and odd parts, we are going to derive the desired solution F near the approximate mapping f in this paper. Precisely, we introduce a strictly contractive mapping with Liptshitz constant

4 78 K. Jun, Y. Lee and H. Kim 0 <L<1. And then, using the fixed point method in the sense of L. Cădariu and V. Radu together with suitable conditions, we can show that the contractive mapping has the fixed point F in a generalized metric function space. Actually, the fixed point F yields the precise solution of the functional equation (3) near f. In Section, we prove several stability results of the functional equation (3) using the fixed point method under suitable conditions. In Section 3, we use the results in Section to get stability result of the Jensen s functional equation (1) and to get that of the quadratic functional equation (), respectively. generalized Hyers-Ulam stability of (3) In this section, we prove the generalized Hyers Ulam stability of the general quadratic functional equation (3). We recall the following fundamental result of the fixed point theory [15, 6]. Theorem.1 Suppose that a complete generalized metric space (X, d), which means that the metric d may assume infinite values, and a strictly contractive mapping Λ:X X with the Lipschitz constant 0 <L<1 are given. Then, for each given element x X, either d(λ n x, Λ n+1 x)=+, n N {0}, or there exists a nonnegative integer k such that: d(λ n x, Λ n+1 x) < + for all n k; the sequence {Λ n x} is convergent to a fixed point y of Λ; y is the unique fixed point of Λ in X 1 := {y X, d(λ k x, y) < + }; d(y, y ) 1 1 L d(y, Λy) for all y X 1. Throughout this paper, let V be a linear space and Y a Banach space. For a given mapping f : V Y, we use the following abbreviation Df(x, y) :=f(x + y)+f(x y)+3f(y) 3f(x) f( y) f(y) for all x, y V. In the following theorem, we can prove the stability of the general quadratic functional equation (3) using the fixed point method.

5 Approximate general quadratic mappings 79 Theorem. Let f : V Y be a mapping for which there exists a mapping ϕ : V [0, ) such that Df(x, y) ϕ(x, y) (4) for all x, y V. If ϕ(x, y) =ϕ( x, y) for all x, y V and there exists a constant 0 <L<1 such that ϕ(x, y) Lϕ(x, y), (5) for all x, y V, then there exists a unique mapping F : V Y such that DF(x, y) =0for all x, y V and f(x) F (x) ϕ(0,x) 4(1 L) for all x V. In particular, F is represented by F (x) =f(0) + n [ f( n x)+f( n x) + f(n x) f( n ] x) 4 n n+1 (6) (7) for all x V. Proof. It follows from (5) that ϕ( n x, n y) n n n L n ϕ(x, y) =0 for all x, y V. If we consider the mapping f := f f(0), then f : V Y satisfies f(0) = 0 and D f(x, y) = Df(x, y) ϕ(x, y) (8) for all x, y V. Let S be the set of all mappings g : V Y with g(0) = 0, and then we introduce a generalized metric d on S by d(g, h) := inf {K R + g(x) h(x) Kϕ(0,x) x V }. (9) It is easy to show that (S, d) is a generalized complete metric space. Now, we consider the mapping Λ : S S defined by Λg(x) := g(x) g( x) 4 + g(x)+g( x) 8 (10) for all g S and all x V. Then we notice that Λ n g(x) = g(n x) g( n x) n+1 + g(n x)+g( n x) 4 n (11)

6 80 K. Jun, Y. Lee and H. Kim for all n N and x V. First, we show that Λ is a strictly contractive self-mapping of S with the Lipschitz constant L. Let g, h S and let K [0, ] be an arbitrary constant with d(g, h) K. From the definition of d, we have Λg(x) Λh(x) = 3 8 g(x) h(x) + 1 g( x) h( x) 8 (1) 1 Kϕ(0, x) LKϕ(0,x) for all x V, which implies that d(λg, Λh) Ld(g, h) (13) for any g, h S. That is, Λ is a strictly contractive self-mapping of S with the Lipschitz constant L. Moreover, by (8) we see that f(x) Λ f(x) = 1 5D 4 x) (14) 1 4 ϕ(0,x) for all x V. It means that d( f,λ f) 1 < by the definition of d. 4 Therefore, according to Theorem.1, the sequence {Λ n f} converges to the fixed point F : V Y of Λ satisfying F (x) = F (x) F ( x) + F (x)+ F ( x), 4 8 which is represented by [ f( F (x) = n Λ n f(x) n x) f( n x) = n + f( n x)+ f( n x) ] (15) n+1 4 n for all x V. In addition, it is unique in the set S 1 = {g S d( f,g) < }. Putting F (x) :=f(0)+ F (x), then one notes that the equality f(x) F (x) = f(x) F (x) for all x V and d( f, F ) 1 1 L d( f,λ f) 1 4(1 L), (16) which implies (6) and (7). By the definitions of F and F, together with (8) and (5), we have that DF(x, y) = D F (x, y) Df( n x, n y) f( n x, n y) = n n+1 + Df(n x, n y)+df( n x, n y) 4 n n +1( ϕ( n x, n y)+ϕ( n x, n y) ) n 4 n = 0

7 Approximate general quadratic mappings 81 for all x, y V. Thus, the mapping F satisfies the general quadratic functional equation (6). This completes the proof of this theorem. Theorem.3 Let f : V Y be a mapping for which there exists a mapping ϕ : V [0, ) such that Df(x, y) ϕ(x, y) (17) for all x, y V. If ϕ(x, y) =ϕ( x, y) for all x, y V and there exists a constant 0 <L<1 such that the mapping ϕ has the property Lϕ(x, y) 4ϕ(x, y) (18) for all x, y V, then there exists a unique mapping F : V Y such that DF(x, y) =0for all x, y V and f(x) F (x) Lϕ(0,x) 4(1 L) (19) for all x V In particular, F is represented by F (x) := f(0) (0) + n [ 4 n ( ( x ) ( x) ) ( f + f f(0) + n 1 f ( x ) ( x))] f n n n n for all x V. Proof. If we consider the mapping f = f f(0), then f : V Y satisfies f(0) = 0 and D f(x, y) = Df(x, y) ϕ(x, y) (1) for all x, y V. Let the space (S, d) be as in the proof of Theorem.. Now, we consider a mapping Λ : S S defined by Λg(x) :=g ( x ) g ( x ) + ( g ( x ) ( x )) + g for all g S and x V. Then we observe that Λ n g(x) = n 1( g ( x ) ( x )) 4 n ( ( x ) ( x )) g + g + g n n n n and Λ 0 g(x) =g(x) for all x V. Let g, h S and let K [0, ] bean arbitrary constant with d(g, h) K. From the definition of d, we have Λg(x) Λh(x) = 3g ( x ) ( x) g ( x) ( x) h + h ( 4Kϕ 0, x ) LKϕ(0,x)

8 8 K. Jun, Y. Lee and H. Kim for all x V.So d(λg, Λh) Ld(g, h) for any g, h S. That is, Λ is a strictly contractive self-mapping of S with the Lipschitz constant L. Also we see that f(x) Λ f(x) = 1 3 D f ( 0, x ϕ ( 0, x ) L 4 ϕ(0,x) ) x) D f( 0, for all x V, which implies that d( f,λ f) L <. Therefore according to 4 Theorem.1, the sequence {Λ n f} converges to a fixed point F of Λ in the set S 1 := {g S d(f,g) < }, that is, F (x) = F ( x which is represented by ) F ( x ) + ( F ( x ) ( x)) + F [ F (x) = ( n 1 f ( x ) x )) 4 f( n ( ( x ) x ))] + f + f( n n n n n for all x V. Putting F (x) := F (x) +f(0),x V, we then know that f(x) F (x) = f(x) F (x) for all x V and d( f, F ) 1 1 L d( f,λ f) L 4(1 L), which implies (19). From the definition of F and F, together with (18), and (1), we have DF(x, y) = D F (x, y) = ( n 1 Df ( x n, y ) ( x Df n n, y )) n n + 4n ( ( x Df, y ) ( x + Df n n, y )) n n ( n 1 + 4n )[ ( x ϕ n, y ) ( x + ϕ n n, y )] n n = 0 for all x, y V and so the mapping F satisfies the general quadratic functional equation (6). This completes the proof of this theorem.

9 Approximate general quadratic mappings 83 3 Applications For a given mapping f : V Y, we use the following abbreviations Jf(x, y) := ( ) x + y f f(x) f(y), Qf(x, y) := f(x + y)+f(x y) f(x) f(y) for all x, y V. Using Theorem. and Theorem.3, we will obtain the stability results of the Jensen s functional equation Jf 0 and the quadratic functional equation Qf 0 in the following corollaries. Corollary 3.1 Let f i : V Y, i =1,, be mappings for which there exist functions φ i : V [0, ), i =1,, such that Jf i (x, y) φ i (x, y) () and φ i (x, y) =φ i ( x, y) for all x, y V, respectively. If there exists 0 < L<1 such that φ 1 (x, y) Lφ 1 (x, y), (3) Lφ (x, y) 4φ (x, y) (4) for all x, y V, then there exist unique Jensen mappings F i : V Y, i =1,, such that f 1 (x) F 1 (x) ϕ 1(0,x) 4(1 L), (5) f (x) F (x) Lϕ (0,x) (6) 4(1 L) for all x V, where ϕ i,i=1,, are defined by ϕ i (x, y) := φ i ( y, y)+φ i (y, 0) + φ i (x, y)+ 1 φ i(x, 4y) + 1 φ i(0, 4y)+φ i (y, y)+ 3 φ i(x, 0). In particular, the mappings F 1,F are represented by for all x V. f 1 ( n x) F 1 (x) = + f n n 1 (0), (7) F (x) = n( f ( x ) f (0) ) + f n n (0) (8)

10 84 K. Jun, Y. Lee and H. Kim Proof. First, we observe from () that Df i (x, y) = Jfi ( y, y)+jf i (y, 0) + Jf i (x, y)+ 1 Jf i(x, 4y) ϕ i (x, y), where we recall the following notation 1 Jf i(0, 4y) Jf i (y, y) 3 Jf i(x, 0) Df i (x, y) :=f i (x + y)+f i (x y)+3f i (y) 3f i (x) f i ( y) f i (y) for all x, y V and i =1,. Since ϕ 1 satisfies (5), according to Theorem., there exists a unique mapping F 1 : V Y satisfying the desired approximation (5), which is represented by (7). We observe from () and (3) that f 1( n x)+f 1 ( n x) n n+1 = f 1 ( n x)+f 1 ( n x) f 1 (0) n n+1 1 = n Jf 1( n x, n x) n+1 1 n φ 1( n x, n x) n+1 L n n φ 1(x, x) =0 as well as f 1 ( n x)+f 1 ( n x) n 4 n n L n n 4 φ 1(x, x) =0 n for all x V. From these properties and the representation (7), we get the mapping F 1 : V Y defined as (7). Moreover, we have Jf 1 ( n x, n y) n φ 1( n x, n y) L n φ n 1 (x, y) for all x, y V. Thus, taking the it as n in the above inequality, we get JF 1 (x, y) = 0 for all x, y V and so F 1 : V Y is a Jensen mapping. On the other hand, we know that ϕ satisfies (18). Therefore according to Theorem.3, there exists a unique mapping F : V Y satisfying (6) which is represented by (0). Observe that n 1 ( x ) ( x) f + f f (0) = n 1 ( x Jf n n n n, x ) n n n 1 φ ( x n, x ) n n n L n φ (x, x) =0

11 Approximate general quadratic mappings 85 as well as n f ( x ) ( x) + f f (0) =0 n n n for all x V. From these and (0), we get (8). Moreover, we have n Jf ( x, y ) n φ ( x n n, y ) L n n n φ (x, y) n for all x, y V. Taking the it as n in the above inequality, we get JF (x, y) = 0 for all x, y V and so F : V Y is a Jensen mapping. This completes the corollary. Now we obtain a stability result of quadratic functional equations. Corollary 3. Let φ i : V [0, ), (i =1, ) be given function. Suppose that each f i : V Y,(i =1, ) satisfies Qf i (x, y) φ i (x, y) for all x, y V. If φ i, (i =1, ) satisfies the symmetrical property φ i (x, y) = φ i ( x, y) and there exists a constant 0 <L<1 such that the mapping φ 1 satisfies the property (3) and φ satisfies (4) for all x, y V, then we have unique quadratic mappings F 1,F : V Y such that f 1 (x) f 1 (0) F 1 (x) φ 1(0, x)+φ 1 (x, x)+φ 1 (0,x)+φ 1 (0, 0), (9) 4(1 L) f (x) F (x) L[φ (0, x)+φ (x, x)+φ (0,x)+φ (0, 0)] (30) 4(1 L) for all x V, where F 1 and F are represented by f F 1 (x) = 1 ( n x) n, (31) 4 n ( ) F (x) = n 4 n f x (3) n for all x V. Moreover, if 0 <L< 1 and φ 1 is continuous, then f 1 f 1 (0) is itself a quadratic mapping. Proof. We note that Df i (x, y) = Qf i (x, y) Qf i (x + y, y) Qf i (0,y)+Qf i (0, 0) φ i (x, y)+φ i (x + y, y)+φ i (0,y)+φ i (0, 0) =: ϕ i (x, y) for all x, y V and i =1,. Then ϕ 1 satisfies (5) and ϕ satisfies (18). According to Theorem., there exists a unique mapping F 1 : V Y satisfying (9) which is represented by F 1 (x) = n ( f1 ( n x)+f 1 ( n x) + f 1( n x) f 1 ( n x) 4 n n+1 ).

12 86 K. Jun, Y. Lee and H. Kim Observe that n f 1 ( n x) f 1 ( n x) n+1 = n 1 Qf n+1 1 (0, n x) n 1 n+1 φ 1 (0, n x) n L n φ 1(0,x)=0 as well as n f 1 ( n x) f 1 ( n x) 4 n n L n n+1 φ 1(0,x)=0 for all x V. From these, we lead to (31) for all x V. Moreover, we have Qf 1 ( n x, n y) 4 n φ 1( n x, n y) Ln 4 n φ 1(x, y) n for all x, y V. Taking the it as n in the above inequality, we get QF 1 (x, y) =0 for all x, y V and so F 1 : V Y is a quadratic mapping. On the other hand, since Lφ (0, 0) 4φ (0, 0) and f (0) = Qf (0, 0) φ (0, 0) we can show that φ (0, 0) = 0 and f (0) = 0. According to Theorem.3, there exists a unique mapping F : V Y satisfying (30), which is represented by (0). We have as well as n 4 n ( ) x f + f n ( x ) 4 n = Qf n (0, x ) n n 4 n n φ (0, x ) n ( ) ( n 1 x f f n n x ) =0 n L n n φ (0,x)=0 for all x V. From these and (11), we get (3). Notice that 4 n Qf ( x n, y n ) 4 n φ ( x n, y n ) L n φ (x, y)

13 Approximate general quadratic mappings 87 for all x, y V. Taking the it as n, then we have shown that QF (x, y) =0 for all x, y V and so F : V Y is a quadratic mapping. This completes the corollary. Now, we obtain generalized Hyers-Ulam stability results in the framework of normed spaces using Theorem. and Theorem.3. Corollary 3.3 Let X be a normed space, θ 0, and p (0, 1) (, ). Suppose that a mapping f : X Y satisfies the inequality Df(x, y) θ( x p + y p ) for all x, y X. Then there exists a unique general quadratic mapping F : X Y satisfying DF(x, y) =0such that f(x) F (x) { θ ( p ) x p, if 0 <p<1; θ p 4 x p, if p >, for all x X. Proof. By putting ϕ(x, y) :=θ( x p + y p ) together with L := p 1 < 1 if 0 <p<1 and L := p < 1ifp>, then we get the desired results by applying Theorem. and Theorem.3. Corollary 3.4 Let X be a normd space and p, q > 0 with p + q (0, 1) (, ). If a mapping f : X Y satisfies Df(x, y) θ x p y q for all x, y X, then f is itself a general quadratic mapping. Proof. It follows from Theorem. and Theorem.3, by putting L := p+q 1 < 1ifp + q<1 and L := p q < 1ifp + q>. ACKNOWLEDGEMENTS. This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (No. 01R1A1A008139) and (No. 01R1A1A4A ). A corresponding author is Hark-Mahn Kim: hmkim@cnu.ac.kr.

14 88 K. Jun, Y. Lee and H. Kim References [1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, (1950), [] Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1, Colloq. Publ. vol. 48, Amer. Math. Soc., Providence, RI, 000. [3] D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), [4] L. Cădariu, V. Radu, Fixed points and the stability of Jensen s functional equation, J. Inequal. Pure Appl. Math. 4(1), (003), Art.4. [5] L. Cădariu, V. Radu, Fixed points and the stability of quadratic functional equations, An. Univ. Timisoara Ser. Mat.-Inform. 41 (003), [6] L. Cădariu, V. Radu, Fixed Point Methods for the Generalized Stability of Functional Equations in a Single Variable, Fixed Point Theory and Applications, Volume 008 (008), Article ID 74939, 15 pages. [7] L. Cădariu, V. Radu, Fixed Points and Stability for Functional Equations in Probabilistic Metric and Random Normed Spaces, Fixed Point Theory and Applications, Volume 009 (009), Article ID , 18 pages. [8] Z. Gajda, On the stability of additive mappings, Intern. J. Math. Math. Sci. 14 (1991), [9] P. Gǎvruta, A generalization of the Hyers Ulam Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), [10] M.E. Gordji, C. Park, M.B. Savadkouhi, The stability of a quartic type functional equation with the fixed point alternative, Fixed Point Theory, 11 (010) No., [11] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 7 (1941), 4. [1] S. Jin, Y. Lee, A fixed point approach to the stability of the Cauchy additive and quadratic type functional equation, Journal of Applied Mathematics, Volume 011 (011), Article ID , 16 pages. [13] S. Jin, Y. Lee, Fuzzy stability of a general quadratic functional equation, Advances in Fuzzy Systems, Volume 011 (011), Article ID , 9 pages.

15 Approximate general quadratic mappings 89 [14] S. Jung, On the Hyers Ulam Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 04 (1996), 1 6. [15] B. Margolis and J. B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), [16] A.K. Mirmostafaee, Non-Archimedean stability of quadratic equations, Fixed Point Theory, 11 (010) No. 1, [17] M.S. Moslehian, A. Najati, An application of a fixed point theorem to a functional inequality, Fixed Point Theory, 10 (009) No. 1, [18] A. Najati, M.B. Moghimi, Stability of a functional equation deriving from quadratic and additive function in quasi-banach spaces, J. Math. Anal. Appl. 337 (008), [19] C. Park, A fixed point approach to the stability of additive functional inequalities in RN-spaces, Fixed Point Theory, 1 (011) No., [0] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (198), [1] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984), [] T.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 7 (1978), [3] T.M. Rassias, The stability of mappings and related topics, In Report on the 7th ISFE, Aequationes. Math. 39 (1990), [4] T.M. Rassias, P. Šemrl, On the behaviour of mappings which do not satisfy Hyers Ulam Rassias stability, Proc. Amer. Math. Soc. 114 (199), [5] S. Rolewicz, Metric Linear Spaces, PWN-Polish Sci. Publ./Reidel, Warszawa/Dordrecht, [6] I.A. Rus, Principles and applications of fixed point theory, Ed. Dacia, Cluj-Napoca, 1979 (in Romanian). [7] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, Received: September, 01