ABBAS NAJATI AND CHOONKIL PARK

Transcription

1 ON A CAUCH-JENSEN FUNCTIONAL INEQUALIT ABBAS NAJATI AND CHOONKIL PARK Abstract. In this paper, we investigate the following functional inequality f(x) + f(y) + f ( x + y + z ) f(x + y + z) in Banach modules over a C -algebra, and prove the generalized Hyers Ulam stability of additive mappings in Banach modules over a C -algebra to approximate homomorphisms. 1. Introduction and preliminaries The stability problem of functional equations originated from a question of Ulam [4] concerning the stability of group homomorphisms : Let (G 1, ) be a group and let (G,, d) be a metric group with the metric d(, ). Given ɛ > 0, does there exist a δ(ɛ) > 0 such that if a mapping h : G 1 G satisfies the inequality d(h(x y), h(x) h(y)) < δ for all x, y G 1, then there is a homomorphism H : G 1 G with d(h(x), H(x)) < ɛ for all x G 1? In other words, we are looking for situations when the homomorphisms are stable, i.e., if a mapping is almost a homomorphism, then there exists a true homomorphism near it. Hyers [17] gave a first affirmative answer to the question of Ulam for Banach spaces. Let and be Banach spaces. Assume that f : satisfies f(x + y) f(x) f(y) ε for all x, y and some ε 0. Then there exists a unique additive mapping T : such that f(x) T (x) ε for all x. Aoki [] and Th.M. Rassias [38] provided a generalization of Hyers theorem for additive mappings and linear mappings, respectively, which allows the Cauchy difference to be unbounded (see also [5] and [1]). 000 Mathematics Subject Classification. Primary 39B7; Secondary 46L05. Key words and phrases. Generalized Hyers Ulam stability, functional inequality, additive mapping in Banach modules over a C -algebra. The second author was supported by National Research Foundation of Korea (NRF ). 1

2 A. NAJATI, C. PARK Theorem 1.1. (Th.M. Rassias). Let f : E E be a mapping from a normed vector space E into a Banach space E subject to the inequality f(x + y) f(x) f(y) ɛ( x p + y p ) (1.1) for all x, y E, where ɛ and p are constants with ɛ > 0 and p < 1. Then the limit L(x) = lim n f( n x) n exists for all x E and L : E E is the unique additive mapping which satisfies f(x) L(x) ɛ p x p (1.) for all x E. If p < 0 then inequality (1.1) holds for x, y 0 and (1.) for x 0. Also, if for each x E the mapping t f(tx) is continuous in t R, then L is linear. For the case p = 1, a counter example has been given by Z. Gajda [11] (see also [40]). The generalized Hyers Ulam stability mentioned in Theorem 1.1 is known as Hyers Ulam Rassias stability (cf. the books of S. Czerwik [10], D.H. Hyers, G. Isac and Th.M. Rassias [18]). Theorem 1.. (J.M. Rassias [?] [37]). Let be a real normed linear space and a real Banach space. Assume that f : is a mapping for which there exist constants θ 0 and p, q R such that r = p + q 1 and f satisfies the functional inequality (Cauchy Gǎvruta Rassias inequality) f(x + y) f(x) f(y) θ x p y q for all x, y. Then there exists a unique additive mapping L : satisfying f(x) L(x) θ r x r for all x. If, in addition, f : is a mapping such that the transformation t f(tx) is continuous in t R for each fixed x, then L is linear. P. Gǎvruta [14] showed that Theorem 1. is not true when r = 1. The stability in Theorem 1. involving a product of different powers of norms is called Ulam Gǎvruta Rassias stability (see [4], [9] and [30]). During the last two decades, a number of papers and research monographs have been published on various generalizations and applications of Hyers Ulam Rassias stability and Ulam Gǎvruta Rassias stability to a number of functional equations and mappings (see [3], [6]-[9], [13]-[16], [0]-[], [4]-[34]). WWe also refer the readers to the books [1], [10], [18], [41] and [39]. Park, Cho and Han [33] investigated the functional inequality f(x) + f(y) + f(z) f(x + y + z) (1.3)

3 ON A CAUCH-JENSEN FUNCTIONAL INEQUALIT 3 in Banach spaces, and proved the generalized Hyers Ulam stability of the functional inequality (1.3) in Banach spaces. Throughout this paper, let A be a unital C -algebra with unit e, unitary group U(A) and norm. Assume that is a Banach A-module with norm and that is a Banach A-module with norm. For a A, let a = a, a or (a + a )/. An additive mapping T : is called A-additive if T (ax) = a T (x) for all a A and all x. In this paper, we investigate an A-additive mapping associated with the functional inequality (x + y f(x) + f(y) + f + z ) f(x + y + z) (1.4) and prove the generalized Hyers Ulam stability of A-additive mappings in Banach A-modules associated with the functional inequality (1.4). For convenience, we use the following abbreviation for a given a A and a mapping f : D a f(x, y, z) := f(ax) + f(ay) + a f ( x + y + z ) for all x, y, z.. Functional inequalities in Banach modules over a C -algebra Lemma.1. Let f : be a mapping such that D a f(x, y, z) f(ax + ay + az) (.1) for all x, y, z and all a U(A). Then f : is A-additive. Proof. Letting x = y = z = 0 and a = e U(A) in (.1), we get that f(0) = 0. Letting z = 0, y = x and a = e U(A) in (.1), we get f(x) + f( x) f(0) = 0 for all x. Hence f( x) = f(x) for all x. Letting z = x y and a = e U(A) in (.1) and using the oddness of f, we get f ( x + y ) f(x) f(y) f(0) = 0 for all x, y. So f ( x + y ) = f(x) + f(y) (.) for all x, y. Letting y = 0 in (.), we get f(x/) = f(x) for all x. Thus (.) implies that f(x + y) = f(x) + f(y) for all x, y. Hence f(rx) = rf(x) for all x and all r Q.

4 4 A. NAJATI, C. PARK Letting z = x and y = 0 in (.1) and using the oddness of f, we get for all x and all a U(A). Thus f(ax) a f(x) f(0) = 0 f(ax) = a f(x) (.3) for all a U(A) and all x. It is clear that (.3) holds for a = 0. Now let a A (a 0) and m be an integer greater than 4 a. Then a m < 1 4 < 1 3 = 1 3. By Theorem 1 of [3], there exist three elements u 1, u, u 3 U(A) such that 3 m a = u 1 + u + u 3. So 3 m a = ( 3 m a) = u 1 + u + u 3. Hence by (.3) we have f(ax) = m 3 f( 3 m ax) = m 3 f(u 1x + u x + u 3 x) = m 3 [f(u 1x) + f(u x) + f(u 3 x)] = m 3 (u 1 + u + u 3)f(x) = m 3 3 m a f(x) = a f(x) for all x. So f : is A-additive, as desired. Now we prove the generalized Hyers Ulam stability of A-additive mappings in Banach A-modules. Theorem.. Let r i > 1 and θ i be non-negative real numbers for all 1 i 3, and let f : be a mapping such that D a f(x, y, z) f(ax + ay + az) + θ 1 x r 1 + θ y r + θ 3 z r 3 (.4) for all x, y, z and all a U(A). Then there exists a unique A-additive mapping L : such that for all x. f(x) L(x) r 1 + r 1 θ 1 x r 1 + θ r x r + r 3 θ 3 r 3 x r 3 (.5) Proof. Letting x = y = z = 0 and a = e U(A) in (.4), we get that f(0) = 0. Letting a = e U(A), y = x and z = 0 in (.4), we get f(x) + f( x) θ 1 x r 1 + θ x r (.6) for all x. Letting a = e U(A), y = 0 and z = x in (.4), we get f(x) + f ( x) θ 1 x r 1 + θ 3 x r 3 (.7) for all x. It follows from (.6) and (.7) that ( x ) f f(x) + r 1 θ r 1 x r θ r x r + θ 3 x r 3

5 ON A CAUCH-JENSEN FUNCTIONAL INEQUALIT 5 for all x. Hence n f ( x ) m f ( x ) n m n 1 j+1 f ( x ) j f ( x ) j+1 j + r 1 n 1 ( θ r 1 x r 1 1 r 1 ) j + θ x r n 1 ( r (.8) ) j+1 n 1 ( + θ3 x r ) 3 j for all non-negative integers m and n with n > m and all x. It follows from (.8) that the sequence { n f( x )} is Cauchy for all x. Since is complete, the n sequence { n f( x )} converges. So one can define the mapping L : by n L(x) := lim n f ( x ) n n for all x. Moreover, letting m = 0 and passing the limit n in (.8), we get (.5). It follows from (.4) that D a L(x, y, z) = lim n n Da f ( x n, y n, z n ) lim n n+1 f ( ax n + ay n + az n ) r 3 + lim n[ θ 1 n nr x r θ nr y r + θ 3 nr z r 3 3 = L(ax + ay + az) for all x, y, z and all a U(A). So by Lemma.1, the mapping L : is A-additive. Now, let T : be another A-additive mapping satisfying (.5). Then we have L(x) T (x) = lim n n f ( x n ) T ( x n ) [ ( lim n r1 + )θ 1 n nr 1 ( r 1 x r 1 ) + θ nr ( r x r ) + r 3 θ 3 nr 3 ( r 3 x r 3 ) ] ] = 0 for all x. So we can conclude that L(x) = T (x) for all x. This proves the uniqueness of L. Thus the mapping L : is a unique A-additive mapping satisfying (.5). Theorem.3. Let 0 < r i < 1 and θ i, δ be non-negative real numbers for all 1 i 3, and let f : be a mapping satisfying f(0) = 0 and the inequality D a f(x, y, z) f(ax + ay + az) + δ + θ 1 x r 1 + θ y r + θ 3 z r 3 (.9)

6 6 A. NAJATI, C. PARK for all x, y, z and all a U(A). Then there exists a unique A-additive mapping L : such that for all x. f(x) L(x) 3δ + + r 1 θ 1 x r 1 r 1 + θ r x r + r 3 θ 3 r x r 3 3 Proof. Similarly to the proof of Theorem., it follows from (.9) that ( x ) f f(x) 3δ + + r 1 θ r 1 x r θ r x r + θ 3 x r 3 for all x. Hence 1 n f(n x) 1 m f(m x) n 1 1 j+1 f(j+1 x) 1 j f(j x) 3δ +1 (1 + θ r x r ) j + + r 1 +1 r 1 ( r θ 1 x r 1 ) j + θ3 x r 3 +1 ( r 1 +1 for all non-negative integers m and n with n > m and all x. The rest of the proof is similar to the proof of Theorem. and we omit the details. Theorem.4. Let {r i } 3 i=1 and θ be non-negative real numbers such that λ := r 1 + r + r 3 (0, 1) (1, + ), r 1 + r > 0, r 3 > 0, and let f : be a mapping such that ) j ( r 3 D a f(x, y, z) f(ax + ay + az) + θ x r 1 y r z r 3 (.10) for all x, y, z and all a U(A) (by letting 0 A-additive. ) j = 1). Then f : is Proof. Since r 1 + r > 0, r j > 0 for some 1 j. Without loss of generality, we may assume that r > 0. Letting x = y = z = 0 and a = e U(A) in (.10), we get that f(0) = 0. Letting a = e U(A), y = x and z = 0 in (.10), we get f(x) + f( x) f(0) = 0 for all x. So the mapping f is odd. Letting a = e U(A), y = 0 and z = x in (.10) and using the oddness of f, we get (x) f f(x) f(0) = 0 for all x. Hence f(x/) = f(x) and so n f ( x ) = f(x) (.11) n

7 ON A CAUCH-JENSEN FUNCTIONAL INEQUALIT 7 for all n Z and all x. Let λ > 1 (we have a similar proof when 0 < λ < 1). It follows from (.10) and (.11) that D a f(x, y, z) = lim n Da f ( x n, y n, z ) n n lim n+1 (ax f n + ay n + az ) + lim n θ n n n nλ x r 1 y r z r 3 = f(ax + ay + az) for all x, y, z. By Lemma.1, the mapping f : is A-additive. Theorem.5. Let {r i } 3 i=1 and θ, δ be non-negative real numbers such that λ := r 1 + r +r 3 (0, 1), r 1 +r > 0, r 3 > 0, and let f : be a mapping satisfying f(0) = 0 and the inequality D a f(x, y, z) f(ax + ay + az) + δ + θ x r 1 y r z r 3 (.1) for all x, y, z and all a U(A) (by letting 0 = 1). Then there exists a unique A-additive mapping L : such that for all x. f(x) L(x) 3δ Proof. Without loss of generality, we may assume that r > 0. Similarly to the proof of Theorem., letting a = e U(A), y = x and z = 0 in (.1), we get f(x) + f( x) δ (.13) for all x. Letting a = e U(A), y = 0 and z = x in (.1), we get ( x) f(x) + f δ (.14) for all x. It follows from (.13) and (.14) that ( x ) f f(x) 3δ for all x. Hence 1 n f(n x) 1 m f(m x) n 1 1 j+1 f(j+1 x) 1 j f(j x) 3δ +1 (1) j for all non-negative integers m and n with n > m and all x. The rest of the proof is similar to the proof of Theorem. and we omit the details.

8 8 A. NAJATI, C. PARK Theorem.6. Let {r i } i=1 and θ, δ be non-negative real numbers and let f : be a mapping satisfying f(0) = 0 and the inequality f(ax + ay + az) + δ + θ x r 1 y r if 0 < λ < 1, D a f(x, y, z) (.15) f(ax + ay + az) + θ x r 1 y r if λ > 1 for all x, y, z and all a U(A) (by letting 0 = 1), where λ := r 1 + r. Then there exists a unique A-additive mapping L : such that 3δ + θ x λ λ if 0 < λ < 1, f(x) L(x) (.16) θ λ x λ if λ > 1 for all x. Proof. Since r 1 +r > 0, without loss of generality, we may assume that r > 0. Letting a = e U(A), y = x and z = 0 in (.15), we get { δ + θ x λ f(x) + f( x) if 0 < λ < 1, θ x λ (.17) if λ > 1 for all x. Letting a = e U(A), y = 0 and z = x in (.15), we get { ( x) f(x) + f δ if 0 < λ < 1, 0 if λ > 1 for all x. It follows from (.17) and (.18) that 3δ + θ x λ λ ( x ) f f(x) if 0 < λ < 1, θ x λ λ if λ > 1 for all x. Hence we have the following cases: Case I. Let 0 < λ < 1. In this case, we get 1 n f(n x) 1 m f(m x) n 1 1 j+1 f(j+1 x) 1 j f(j x) 3δ +1 for all non-negative integers m and n with n > m and all x. Case II. Let λ > 1. In this case, we get (.18) (.19) (1) j n 1 + θ x λ ( λ ) j n f ( x ) m f ( x ) n 1 j+1 f ( x ) j f ( x ) n m j+1 j θ x λ n 1 ( λ ) j+1 (.0)

9 ON A CAUCH-JENSEN FUNCTIONAL INEQUALIT 9 for all non-negative integers m and n with n > m and all x. It follows from (.19) (respectively, (.0)) that the sequence { 1 f( n x)} (respectively, { n f( x )}) is n n Cauchy for all x. Since is complete, the sequence { 1 f( n x)} (respectively, n { n f( x )}) converges. So one can define the mapping L : by n 1 lim n f( n x) if 0 < λ < 1, n L(x) := lim n n f( x ) if λ > 1 n for all x. Moreover, letting m = 0 and passing the limit n in (.19) and (.0), we get (.16). The rest of the proof is similar to the proof of Theorem.. Theorem.7. Let r, δ and θ be non-negative real numbers and let f : be a mapping satisfying f(0) = 0 and the inequality f(ax + ay + az) + δ + θ z r if 0 < r < 1, D a f(x, y, z) f(ax + ay + az) + θ z r if r > 1 (.1) for all x, y, z and all a U(A). Then there exists a unique A-additive mapping L : such that 3δ + r θ x r r if 0 < r < 1, f(x) L(x) r θ r x r if r > 1 for all x. Proof. Letting a = e U(A), y = x and z = 0 in (.1), we get f(x) + f( x) { δ if 0 < r < 1, 0 if r > 1 (.) for all x. Letting a = e U(A), y = 0 and z = x in (.1), we get { ( x) f(x) + f δ + θ x r if 0 < r < 1, θ x r if r > 1 (.3) for all x. It follows from (.) and (.3) that ( x ) f f(x) 3δ + θ x r if 0 < r < 1, θ x r if r > 1

10 10 A. NAJATI, C. PARK for all x. Hence we have the following cases: Case I. Let 0 < r < 1. In this case, we get 1 n f(n x) 1 m f(m x) n 1 1 j+1 f(j+1 x) 1 j f(j x) 3δ +1 (1) j + θ x r ( r +1 for all non-negative integers m and n with n > m and all x. Case II. Let r > 1. In this case, we get n f ( x ) m f ( x ) n 1 j+1 f ( x ) j f ( x ) n m j+1 j θ x r n 1 ( r ) j for all non-negative integers m and n with n > m and all x. The rest of the proof is similar to the proof of Theorem. and Theorem.6 and we omit the details. Acknowledgment The authors would like to thank the referee for a number of valuable suggestions regarding a previous version of this paper. References [1] J. Aczél, J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, [] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan (1950), [3] C. Baak, Cauchy Rassias stability of Cauchy-Jensen additive mappings in Banach spaces, Acta Math. Sinica (006), [4] B. Bouikhalene and E. Elqorachi, Ulam Gǎvruta Rassias stability of the Pexider functional equation, International Journal of Applied Mathematics and Statistics, 7 (007), [5] D.G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), [6] L. Cǎdariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory Appl. 008, Art. ID 74939, 15 pages. [7] L. Cǎdariu and V. Radu, Remarks on the stability of monomial functional equations, Fixed Point Theory 8 (007), [8] L. Cǎdariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber. 346 (004), [9] L. Cǎdariu and V. Radu, Fixed points and the stability of Jensen s functional equation, J. Inequalities in Pure and Appl. Math. 4 (003) article 4, 7 pages. [10] P. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 00. [11] Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), ) j

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12 1 A. NAJATI, C. PARK [35] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (198), [36] J.M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. 108 (1984), [37] J.M. Rassias, Solution of a problem of Ulam, J. Approximation Theory, 57 (1989), [38] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 7 (1978), [39] Th.M. Rassias (ed.), Functional Equations and Inequalities, Kluwer Academic Publishers, Dordrecht, Boston, London, 000. [40] Th.M. Rassias and P. Šemrl, On the behaviour of mappings which do not satisfy Hyers Ulam stability, Proc. Amer. Math. Soc. 114 (199), [41] Th.M. Rassias and J. Tabor (eds.), Stability of Mappings of Hyers Ulam Type, Hadronic Press Inc., Florida, [4] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New ork, Abbas Najati Department of Mathematics Faculty of Sciences University of Mohaghegh Ardabili Ardabil Iran address: a.nejati@yahoo.com Choonkil Park Department of Mathematics Hanyang University Seoul, Republic of Korea address: baak@hanyang.ac.kr