Int. J. Nonlinear Anal. Appl. 3 (0) No., 75-8 ISSN: 008-68 (electronic) http://www.ijnaa.semnan.ac.ir Approximate additive and quadratic mappings in -Banach spaces and related topics Y. J. Cho a, C. Park b, M. Eshaghi Gordji c, a Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-70, Korea. b Research Institute for Natural Sciences, Hanyang University, Seoul 33-79, Korea. c Department of Mathematics, Semnan University, P. O. Box 3595-363, Semnan, Iran. (Communicated by A. Ebadian) Abstract Won Gil Park [Won Gil Park, J. Math. Anal. Appl., 376 () (0) 93 0] proved the Hyers Ulam stability of the Cauchy functional equation, the Jensen functional equation and the quadratic functional equation in Banach spaces. One can easily see that all results of this paper are incorrect. Hence the control functions in all theorems of this paper are not correct. In this paper, we correct these results. Keywords: Hyers Ulam Stability, Cauchy Functional Equation, Jensen Functional Equation, Quadratic Functional Equation. 00 MSC: 39B8,39B5, 46A70, 47H99.. Introduction One of the most interesting questions in the theory of functional analysis concerning the Ulam stability problem of functional equations is as follows: When is it true that a mapping satisfying a functional equation approximately must be close to an exact solution of the given functional equation? The first stability problem concerning group homomorphisms was raised by Ulam [] in 940 and affirmatively solved by Hyers [3]. The result of Hyers was generalized by Aoki [4] for approximate additive mappings and by Th.M. Rassias [5] for approximate linear mappings by allowing the difference Cauchy equation f(x + y) f(x) f(y) to be controlled by ε( x p + y p ). Corresponding author Email addresses: yjcho@gnu.ac.kr (Y. J. Cho), baak@hanyang.ac.kr (C. Park), madjid.eshaghi@gmail.com (M. Eshaghi Gordji ) Received: September 0 Revised: October 0
76 Cho, Park and Eshaghi Gordji The functional equation f(x + y) + f(x y) = f(x) + f(y) (.) is related to symmetric bi-additive function and is called a quadratic functional equation and every solution of the quadratic equation (.) is said to be a quadratic mapping. Skof [6] proved the Hyers-Ulam stability of the quadratic functional equation (.). The theory of linear -normed spaces was first developed by S. Gähler [7] in the mid 960 s, while that of -Banach spaces was studied later by S. Gähler [8] and A. White [9]. For more details, the readers refer to the papers [0] - []. Definition.. ([7]) Let X be a real linear space over R with dim X > and, : X X R be a function. Then (X,, ) is called a linear -normed space if (N ) x, y > 0 and x, y = 0 if and only if x and y are linearly dependent; (N ) x, y = y, x ; (N 3 ) αx, y = α x, y ; (N 4 ) x, y + z x, y + x, z for all x, y, z X and α R. If x X and x, y = 0 for all y X, then x = 0. Moreover, the functions x x, y are continuous functions of X into R for each fixed y X [7, 0]. The basic definitions of a -Banach space are given in [] (see also [8, 9]): () A sequence {x n } in a linear -normed space X is called a convergent sequence if there is an x X such that lim n x n x, y = 0 for all y X. In this case, we write lim n x n = x. () A sequence {x n } in a linear -normed space X is called a Cauchy sequence if there are y, z X such that y and z are linearly independent, lim n,m x n x m, y = 0 and lim n,m x n x m, z = 0. (3) A linear -normed space in which every Cauchy sequence is a convergent sequence is called a -Banach space. From now on, let X be a normed linear space and Y a Banach space. Recently, W. G. Park [] proved the Hyers Ulam stability of the Cauchy functional equation, the Jensen functional equation and the quadratic functional equation in Banach spaces. One can easily see that all results of the paper [] are incorrect. We remained that for z, y Y, we can not define z r and y r in relations (.), (.), (.3), (.4), (3.), (3.), (3.3), (3.4), (3.5), (3.8), (4.), (4.), (4.3) and (4.4) in the paper []. So all relations and all results in paper [] are incorrect. In the present paper, we correct the results of [].. Approximate Additive Mappings In this section, we investigate the Hyers Ulam stability of additive mappings from X into Y. Theorem.. Let θ [0, ), p, q (0, ) and p+q < and let f : X Y be a mapping satisfying f(x + y) f(x) f(y), z θ x p y q (.) for all x, y X and all z Y. Then there is a unique additive mapping A : X Y such that f(x) A(x), z p+q (.)
Approximate additive and quadratic mappings in...3 (0) No.,75-8 77 Proof. Setting y = x in (.), we get f(x) f(x), z θ x p+q It follows that f(x) f(x), z θ x p+q Replacing x by j x and dividing j, we obtain j f(j x) j+ f(j+ x), z (p+q )(j ) θ x p+q l f(l x) m f(m x), z (p+q )l (p+q )m p+q θ x p+q (.3) lim l,m l f(l x) m f(m x), z = 0 for all x X and all z Y. This means that { j f( j x)} is a Cauchy sequence in Y. Since Y is a Banach space, the sequence { j f( j x)} converges for all x X. So one can define the mapping A : X Y by A(x) := lim j j f( j x) for all x X. It is easy to show that A(x + y) A(x) A(y), z θ x p y q lim j (p+q )j = 0 for all x, y X and all z Y. It follows that A : X Y is additive. To prove the uniqueness property of A, let B : X Y be an additive mapping satisfying (.). Then one can show that A(x) B(x), z (p+q )j p+q for all x X, all z Y and all j N. The right hand side of the last inequality tends to zero as j. It follows that A(x) = B(x) for all x X. This completes the proof. Theorem.. Let θ [0, ), p, q (0, ) and p+q > and let f : X Y be a mapping satisfying f(x + y) f(x) f(y), z θ x p y q (.4) for all x, y X and all z Y. Then there is a unique additive mapping A : X Y such that f(x) A(x), z p+q Proof. Setting y = x in (.4), we get f(x) f(x), z θ x p+q It follows that f(x) f( x ), z θ x p+q p+q Replacing x by j x and multiplying j, we obtain j f( x ) j j+ f( x ), z j θ j+ (p+q)(j+) x p+q
78 Cho, Park and Eshaghi Gordji l f( x ) l m f( x ), z m [ p+q (p+q )l lim l,m l f( x ) l m f( x ), z = 0 m (p+q )m ] for all x X and all z Y. This means that { j f( x j )} is a Cauchy sequence in Y. Since Y is a Banach space, the sequence { j f( x j )} converges for all x X. So one can define the mapping A : X Y by A(x) := lim j j f( x j ) for all x X. It is easy to show that A : X Y is additive. The rest of proof is similar to the proof of Theorem.. 3. Approximate Jensen Mappings In this section, we investigate the Hyers Ulam stability of Jensen mappings from X into Y. Theorem 3.. Let θ [0, ), p, q (0, ) and p+q < and let f : X Y be a mapping satisfying f( x + y ) f(x) f(y), z θ x p y q (3.) for all x, y X and all z Y. Then there is a unique Jensen mapping J : X Y such that f(x) J(x), z Proof. Define g : X Y by g(x) := f(x) f(0) 3 3 p+q (3.) for all x X. Then it is easy to show that g(0) = 0 and g satisfies (3.). That is g( x + y ) g(x) g(y), z θ x p y q (3.3) for all x, y X and all z Y. Putting y = x in (3.3) to get g(x) + g( x), z θ x p+q One can show that g(x) 3 g(3x), z θ x p+q 3 Replacing x by 3 j x and dividing 3 j, we obtain 3 j g(3j x) 3 j+ g(3j+ x), z ( + 3 q )3 (p+q )(j ) θ x p+q 3 l g(3l x) 3 m g(3m x), z 3 3 p+q [3(p+q )l 3 (p+q )m ]θ x p+q (3.4)
Approximate additive and quadratic mappings in...3 (0) No.,75-8 79 lim l,m 3 l g(3l x) 3 m g(3m x), z = 0 for all x X and all z Y. This means that { g(3 j x)} is a Cauchy sequence in Y. Since Y is a 3 j Banach space, the sequence { g(3 j x)} converges for all x X. So one can define the mapping 3 j A : X Y by A(x) := lim j g(3 j x) = lim 3 j j f(3 j x) for all x X. It is easy to see that 3 j g(x) A(x), z 3 3p+q Define a mapping J : X Y by J(x) := A(x) + f(0) for all x X. Then we have f(x) J(x), z = g(x) A(x), z 3 3p+q By using the same technique of proving Theorem 3. of [], one can show that J is a unique Jensen mapping from X into Y such that (3.). Theorem 3.. Let θ [0, ), p, q (0, ) and p + q > and let f : X Y be a mapping such that f(0) = 0 and that f( x + y ) f(x) f(y), z θ x p y q (3.5) for all x, y X and all z Y. Then there is a unique additive mapping A : X Y such that f(x) A(x), z 3 p+q 3 Proof. By the same argument as in the proof of Theorem 3., we get f(x) 3f( x 3 ), z 3p+q Replacing x by 3 j x and multiplying 3 j, we obtain 3 j f( x 3 j ) 3j+ f( x 3 j+ ), z 3 (p+q)(j+) 3 j θ x p+q l f( x ) l m f( x ), z m 3 [ p+q 3 3 (p+q )l 3 lim l,m 3l f( x 3 ) l 3m f( x ), z = 0 3m (p+q )m ] for all x X and all z Y. This means that {3 j f( x 3 j )} is a Cauchy sequence in Y. Since Y is a Banach space, the sequence {3 j f( x 3 j )} converges for all x X. So one can define the mapping A : X Y by A(x) := lim j 3 j f( x 3 j ) for all x X. It is easy to show that A : X Y is additive. The rest of proof is similar to the proof of Theorem 3..
80 Cho, Park and Eshaghi Gordji 4. Approximate Quadratic Mappings In 995, C. Borelli and G.L. Forti [3] obtained the result on the stability theorem for a class of functional equations including the quadratic functional equation. In this section, we investigate the Hyers Ulam stability of quadratic mappings from normed X into Banach space Y. Theorem 4.. Let θ [0, ), p, q (0, ) and p+q < and let f : X Y be a mapping satisfying f(x + y) + f(x y) f(x) f(y), z θ x p y q (4.) for all x, y X and all z Y. Then there is a unique quadratic mapping Q : X Y such that f(x) Q(x), z 4 p+q + f(0), z (4.) 3 Proof. Putting y = x in (4.), we get f(x) 4f(x) + f(0), z θ x p+q for all x X and all z Y. So, we get f(x) 4 f(x) 4 f(0), z θ 4 x p+q Replacing x by j x and dividing 4 j, we obtain 4 j f(j x) 4 j+ f(j+ x), z (p+q )(j ) θ x p+q 4 l f(l x) 4 m f(m x) 3 (4 l 4 m )f(0), z [ (p+q )l (p+q )m 4 p+q ]θ x p+q (4.3) lim l,m 4 l f(l x) 4 m f(m x), z = 0 for all x X and all z Y. This means that { 4 j f( j x)} is a Cauchy sequence in Y. Since Y is a Banach space, the sequence { 4 j f( j x)} converges for all x X. So one can define the mapping Q : X Y by Q(x) := lim j 4 j f( j x) for all x X. It is easy to see that Q(x + y) + Q(x y) Q(x) Q(y), z θ x p y q lim j (p+q )j = 0 for all x, y X and all z Y. It follows that Q(x + y) + Q(x y) = Q(x) + Q(y) for all x, y X. Hence Q is a quadratic mapping. On the other hand, by (4.3), it follows that f(x) Q(x) 3 f(0), z 4 p+q It follows inequality (4.). Now, let Q : X Y be another quadratic mapping satisfying (4.). Then we have Q(x) Q (x), z (p+q )j+ θ x p+q + f(0), z 4 p+q 3.4j which tends to zero as j for all x X. It follows that Q = Q.
Approximate additive and quadratic mappings in...3 (0) No.,75-8 8 Theorem 4.. Let θ [0, ), p, q (0, ) and p + q > and let f : X Y be a mapping such that f(0) = 0 and that f(x + y) + f(x y) f(x) f(y), z θ x p y q (4.4) for all x, y X and all z Y. Then there is a unique quadratic mapping Q : X Y such that f(x) Q(x), z p+q 4 Proof.Putting y = x in (4.4), we get f(x) 4f(x), z θ x p+q So we get f(x) 4f( x ), z θ x p+q p+q Replacing x by j x and multiplying 4 j, we obtain 4 j f( x ) j 4j+ f( x ), z 4 j j+ (p+q)(j+) 4 l f( x ) l 4m f( x ), z m [ p+q 4 (p+q )l ] (p+q )m lim l,m 4l f( x ) l 4m f( x ), z = 0 m for all x X and all z Y. This means that {4 j f( x j )} is a Cauchy sequence in Y. Since Y is a Banach space, the sequence {4 j f( x j )} converges for all x X. So one can define the mapping Q : X Y by Q(x) := lim j 4 j f( x j ) for all x X. It is easy to show that Q : X Y is a quadratic mapping. The rest of proof is similar to the proof of Theorem 4.. Acknowledgements The authors would like to extend their thanks to referee for his (her) valuable comments and suggestions which helped simplify and improve the results of paper. References [] W. G. Park, Approximate additive mappings in Banach spaces and related topics, J. Math. Anal. Appl., 376 () (0) 93 0. [] S. M. Ulam, Problems in modern mathematics, Chapter VI, science ed., Wiley, New York, 940. [3] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 7 (94) 4. [4] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, (950) 64 66. [5] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 7 (978) 97 300. [6] F. Skof, Propriet locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53 (983) 3 9. [7] S. Gähler, Lineare -normierte Räume, Math. Nachr., 8 (964) 43 (German). [8] S. Gähler, Über -Banach-Räume, Math. Nachr., 4 (969) 335 347 (German). [9] A. White, -Banach spaces, Math. Nachr., 4 (969) 43 60. [0] Y. J. Cho, P. C. S. Lin, S. S. Kim and A. Misiak, Theory of -Inner Product Spaces, Nova Science Publishers, New York, 00. [] R. W. Freese and Y. J. Cho, Geometry of Linear -Normed Spaces, Nova Science Publishers, New York, 00. [] S. Elumalai, Y. J. Cho and S. S. Kim, Best approximation sets in linear -normed spaces, Commun. Korean Math. Soc., (997) 69 69. [3] C. Borelli and G. L. Forti, On a general Hyers-Ulam stability result, Int. J. Math. Math. Sci., 8 (995) 9 36.