On the stability of the functional equation f(x+ y + xy) = f(x)+ f(y)+ xf (y) + yf (x)

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1 J. Math. Anal. Appl. 274 (2002) On the stability of the functional equation f(x+ y + xy) = f(x)+ f(y)+ xf (y) + yf (x) Yong-Soo Jung a, and Kyoo-Hong Park b a Department of Mathematics, Chungnam National University, Taejon , South Korea b Department of Mathematics Education, Seowon University, Chongju, Chungbuk , South Korea Received 30 April 2001 Submitted by T.M. Rassias Abstract In this paper we study the Hyers Ulam stability and the superstability of the functional equation f(x+ y + xy) = f(x)+ f(y)+ xf (y) + yf (x) Elsevier Science (USA). All rights reserved. Keywords: Functional equation; Stability 1. Introduction The problem of stability of functional equations was originally raised by S.M. Ulam [23] in 1940: Given a group G 1, a metric group G 2 with metric d(, ), andaε>0, does there exist a δ>0 such that if a mapping f : G 1 G 2 satisfies d(f(xy),f(x)f(y)) δ for all x,y G 1, then a homomorphism g : G 1 G 2 exists with d(f (x), g(x)) ε for all x G 1? * Corresponding author. addresses: ysjung@math.cnu.ac.kr (Y.S. Jung), parkkh@seowon.ac.kr (K.H. Park) X/02/$ see front matter 2002 Elsevier Science (USA). All rights reserved. PII: S X(02)

2 660 Y.S. Jung, K.H Park / J. Math. Anal. Appl. 274 (2002) For Banach spaces the Ulam problem was first solved by D.H. Hyers [3] in 1941, which states that if δ>0andf : X Y is a mapping with X, Y Banach spaces, such that f(x+ y) f(x) f(y) δ (1) for all x,y X, then there exists a unique additive mapping T : X Y such that f(x) T(x) δ for all x,y X. In such a case, the additive functional equation f(x+y) = f(x)+f(y)is said to have the Hyers Ulam stability property on (X, Y ). This terminology is applied to all kinds of functional equations which have been studied by many authors (see, for example, [4 6,14 19]). In 1978, Th.M. Rassias [13] succeeded in generalizing the Hyers result by weakening the condition for the bound of the left side of the inequality (1). Due to this fact, the additive functional equation f(x+ y) = f(x)+ f(y) is said to have the Hyers Ulam Rassias stability property on (X, Y ). A number of Rassias type results concerning the stability of different functional equations can be found in [2,4,7 10,17]. If each solution f : X Y of the inequality (1) is a solution of the additive functional equation f(x + y) = f(x) + f(y), then we say that the additive functional equation has the superstability property on (X, Y ). This property is also applied to the case of other functional equations (Refs. [1,9,16]). We now consider a functional equation which defines multiplicative derivations in algebras: f(xy)= xf (y) + yf (x). (2) It is immediate to observe that the real-valued function f(x)= x ln x is a solution of the functional equation (2) on the interval (0, ). During the 34th International Symposium on Functional Equations, Gy. Maksa [11] posed the problem concerning the Hyers Ulam stability on the interval (0, 1] of the functional equation (2), and J. Tabor gave an answer to the question of Maksa in [22]. On the other hand, Zs. Páles [12] remarked that the functional equation (2) on the interval [1, ) for real-valued functions is superstable. Here we introduce the following functional equation motivated by the functional equation (2): f(x+ y + xy) = f(x)+ f(y)+ xf (y) + yf (x). (3) In this paper, we will solve the functional equation (3) and then, by following the ideas of J. Tabor [22] and Zs. Páles [12], the Hyers Ulam stability on the interval ( 1, 0] and the superstability on the interval [0, ) of the functional equation (3) will be investigated, respectively.

3 Y.S. Jung, K.H Park / J. Math. Anal. Appl. 274 (2002) Throughout this paper, we will denote by R and N the sets of real numbers and of positive integers, respectively. 2. Solutions of Eq. (3) It is easy to see that the real-valued function f(x)= (x + 1) ln(x + 1) is a solution of the functional equation (3) on the interval ( 1, ). In the following theorem, we will find out the general solution of the functional equation (3) on the interval ( 1, ). Theorem 2.1. Let X be a real (or complex) linear space. A function f : ( 1, ) X satisfies the functional equation (3) for all x ( 1, ) if and only if there exists a solution D : (0, ) X of the functional equation (2) such that f(x)= D(x + 1) for all x ( 1, ). Proof. Necessity. Let us define the mapping D : (0, ) X by D(x) := f(x 1). We claim that D is a solution of the functional equation (2). Indeed, for all x,y (0, ),wehave D(xy) = f(xy 1) = f ( (x 1) + (y 1) + (x 1)(y 1) ) = f(x 1) + f(y 1) + (x 1)f (y 1) + (y 1)f (x 1) = xd(y) + yd(x). Therefore D is a solution of the functional equation (2), as claimed, and f(x)= D(x + 1) for all x ( 1, ). Sufficiency. This is obvious. 3. Hyers Ulam stability of Eq. (3) We first state a theorem of F. Skof [20] concerning the stability of the additive functional equation f(x+ y) = f(x)+ f(y)on a restricted domain: Theorem 3.1. Let X be a real (or complex) Banach space. Given c>0, leta mappingf : [0,c) X satisfy the inequality f(x+ y) f(x) f(y) δ

4 662 Y.S. Jung, K.H Park / J. Math. Anal. Appl. 274 (2002) for some δ 0 and for all x,y [0,c) with x + y [0,c). Then there exists an additive mapping A : R X such that f(x) A(x) 3δ for all x [0,c). Our main theorem in this section is the Hyers Ulam stability on the interval ( 1, 0] of the functional equation (3) and the proof is similar to the one given in [22]. Theorem 3.2. Let X be a real (or complex) Banach space, and let f : ( 1, 0] X be a mapping satisfying the inequality f(x+ y + xy) f(x) f(y) xf (y) yf (x) δ (4) for some δ>0 and for all x, y ( 1, 0]. Then there exists a solution H : ( 1, 0] X of the functional equation (3) such that f(x) H(x) (4e)δ (5) for all x ( 1, 0]. Proof. Let g : ( 1, 0] X be a mapping defined by g(x) = f(x) x + 1 forall x ( 1, 0]. Then, by (4), we see that g satisfy the inequality g(x + y + xy) g(y) g(x) δ (x + 1)(y + 1) for all x,y ( 1, 0]. If we define the mapping F : [0, ) X by F ( ln(x + 1) ) = g(x) for all x ( 1, 0], then, by setting u = ln(x + 1) and v = ln(y + 1), we obtain F(u+ v) F(u) F(v) δe u+v (6) for all u, v [0, ).Thismeansthat F(u+ v) F(u) F(v) δe c for all u, v [0,c)with u + v<c,wherec>1 is an arbitrary given constant. According to Theorem 3.1, there exists an additive mapping A : R X such that F(u) A(u) 3δe c for all u [0,c).Ifweletc 1 in the last inequality, we then get

5 Y.S. Jung, K.H Park / J. Math. Anal. Appl. 274 (2002) F(u) A(u) 3eδ (7) for all u [0, 1]. Moreover, it follows from (6) that F(u+ 1) F(u) F(1) δe u+1, F(u+ 2) F(u+ 1) F(1) δe u+2,. F(u+ k) F(u+ k 1) F(1) δe u+k for all u [0, 1] and k N. Summing up these inequalities we obtain F(u+ k) F(u) kf(1) δe e u+k ( 1 + e 1 + +e k+1) δe e u+k (8) for all u [0, 1] and k N. We assert that F(v) A(v) 4δe e v (9) for all v [0, ). In fact, let v 0andletk N {0} be given with v k [0, 1]. Then, by (7) and (8), we have F(v) A(v) F(v) F(v k) kf(1) + F(v k) A(v k) + A(k) kf(1) δe e v + 3δe + A(k) kf(1) δe e v + 3δe + k A(1) F(1) δe e v + 3δe + 3δev δe ( e v + 3(1 + v) ) 4δe e v. Hence, from (9) and the definition of F, it follows that ( ) g(x) A ln(x + 1) 4δe e ln(x+1) = 4δe x + 1 for all x ( 1, 0], i.e., f(x) x + 1 A( ln(x + 1) ) 4δe (10) x + 1 for all x ( 1, 0]. If we put H(x)= (x + 1)A( ln(x + 1)) for all x ( 1, 0], we can easily check that H is a solution of the functional equation (3) by Theorem 2.1. This and (10) yield that f(x) H(x) (4e)δ for all x ( 1, 0] which proves (5). The proof of the theorem is complete.

6 664 Y.S. Jung, K.H Park / J. Math. Anal. Appl. 274 (2002) Superstability of Eq. (3) For the purpose, we will introduce the next result [21] which is essential to prove the main theorem: Theorem 4.1. Let X be a real (or complex) Banach space, and let c>0 be a given constant. Suppose that a mapping f : R X satisfies the inequality f(x+ y) f(x) f(y) δ for some δ 0 and for all x,y R with x + y >c. Then there exists a unique additive mapping A : R X such that f(x) A(x) 9δ for all x R. Now let us prove the main theorem of the section which is the superstability of the functional equation (3) on the interval [0, ). Theorem 4.2. Let X be a real (or complex) Banach space, and let f : [0, ) X be a mapping satisfying the inequality f(x+ y + xy) f(x) f(y) xf (y) yf (x) δ (11) for some δ>0 and for all x,y [0, ).Thenf satisfies the functional equation (3) for all x,y [0, ). Proof. Defining the mapping g : [0, ) X by g(x) = f(x) x + 1 for all x [0, ) as in the proof of Theorem 3.2, and defining the mapping F : [0, ) X by F ( ln(x + 1) ) = g(x) for all x [0, ), we see, by putting u = ln(x + 1) and v = ln(y + 1),that F(u+ v) F(u) F(v) δe (u+v) (12) for all u, v [0, ). We claim that F is additive. From (12) with δ n = δe n (n N), we obtain F(u+ v) F(u) F(v) δn for all u, v [0, ) with u + v>n. We now define a mapping T : R X by

7 Y.S. Jung, K.H Park / J. Math. Anal. Appl. 274 (2002) { F(u) for u 0, T(u)= F( u) for u<0. It is not difficult to see that T(u+ v) T(u) T(v) δn for all u, v R with u + v >n. Therefore, by Theorem 4.1, there exists a unique additive mapping A n : R X satisfying T(u) A n (u) 9δ n (13) for all u R. Letm, n N with n>m. Then the additive mapping A n : R X satisfies T(u) A n (u) 9δ m for all u R. The uniqueness argument now implies A n = A m for all n N with n>m>0, and thus A 1 = A 2 = = A n =. Taking the limit in (13) as n, we obtain T = A 1 and we deduce that F is additive. Now, according to the definitions of F and g, wehave f(x) x + 1 = F ( ln(x + 1) ) for all x [0, ), i.e., f(x)= (x + 1)F ( ln(x + 1) ) for all x [0, ), and hence we see that f satisfies the functional equation (3) for all x,y [0, ) by Theorem 2.1 since F is additive and D(x) = xf(ln(x)) (x [1, )) is a solution of the functional equation (2). This completes the proof of the theorem. Acknowledgments We thank Professor Themistocles M. Rassias and the referees for their valuable comments and help. References [1] J.A. Baker, J. Lawrence, F. Zorzitto, The stability of the equation f(x+ y) = f(x)f(y), Proc. Amer. Math. Soc. 74 (1979) [2] S. Czerwik, On the stability of the quadratic mappings in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992) [3] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27 (1941) [4] D.H. Hyers, G. Isac, Th.M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston, [5] D.H. Hyers, G. Isac, Th.M. Rassias, On the asymptoticity aspect of Hyers Ulam stability of mappings, Proc. Amer. Math. Soc. 126 (1998)

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