SUPERSTABILITY FOR GENERALIZED MODULE LEFT DERIVATIONS AND GENERALIZED MODULE DERIVATIONS ON A BANACH MODULE (II)

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1 Volume 0 (2009), Issue 2, Article 85, 8 pp. SUPERSTABILITY FOR GENERALIZED MODULE LEFT DERIVATIONS AND GENERALIZED MODULE DERIVATIONS ON A BANACH MODULE (II) HUAI-XIN CAO, JI-RONG LV, AND J. M. RASSIAS COLLEGE OF MATHEMATICS AND INFORMATION SCIENCE SHAANXI NORMAL UNIVERSITY XI AN 70062, P. R. CHINA caohx@snnu.edu.cn r98@63.com PEDAGOGICAL DEPARTMENT SECTION OF MATHEMATICS AND INFORMATICS NATIONAL AND CAPODISTRIAN UNIVERSITY OF ATHENS ATHENS 5342, GREECE jrassias@primedu.uoa.gr Received 2 January, 2009; accepted 2 May, 2009 Communicated by S.S. Dragomir ABSTRACT. In this paper, we introduce and discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module. Key words and phrases: Superstability, Generalized module left derivation, Generalized module derivation, Module left derivation, Module derivation, Banach module Mathematics Subject Classification. Primary 39B52; Secondary 39B82.. INTRODUCTION The study of stability problems was formulated by Ulam in [28] during a tal in 940: Under what conditions does there exist a homomorphism near an approximate homomorphism? In the following year 94, Hyers in [2] answered the question of Ulam for Banach spaces, which states that if ε > 0 and f : X Y is a map with a normed space X and a Banach space Y such that (.) f(x + y) f(x) f(y) ε, for all x, y in X, then there exists a unique additive mapping T : X Y such that (.2) f(x) T (x) ε, for all x in X. In addition, if the mapping t f(tx) is continuous in t R for each fixed x in X, then the mapping T is real linear. This stability phenomenon is called the Hyers-Ulam stability of the additive functional equation f(x + y) = f(x) + f(y). A generalized version This subject is supported by the NNSFs of China (No. 0573, )

2 2 HUAI-XIN CAO, JI-RONG LV, AND J. M. RASSIAS of the theorem of Hyers for approximately additive mappings was given by Aoi in [] and for approximate linear mappings was presented by Th. M. Rassias in [26] by considering the case when the left hand side of the inequality (.) is controlled by a sum of powers of norms [25]. The stability of approximate ring homomorphisms and additive mappings were discussed in [6, 7, 8, 0,, 3, 4, 2]. The stability result concerning derivations between operator algebras was first obtained by P. Semrl in [27]. Badora [5] and Moslehian [7, 8] discussed the Hyers-Ulam stability and the superstability of derivations. C. Baa and M. S. Moslehian [4] discussed the stability of J -homomorphisms. Miura et al. proved the Hyers-Ulam-Rassias stability and Bourgin-type superstability of derivations on Banach algebras in [6]. Various stability results on derivations and left derivations can be found in [3, 9, 20, 2, 9]. More results on stability and superstability of homomorphisms, special functionals and equations can be found in J. M. Rassias papers [22, 23, 24]. Recently, S.-Y. Kang and I.-S. Chang in [5] discussed the superstability of generalized left derivations and generalized derivations. In the present paper, we will discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module. To give our results, let us give some notations. Let A be an algebra over the real or complex field F and X be an A -bimodule. Definition.. A mapping d : A A is said to be module-x additive if (.3) xd(a + b) = xd(a) + xd(b) (a, b A, x X). A module-x additive mapping d : A A is said to be a module-x left derivation (resp., module-x derivation) if the functional equation (.4) xd(ab) = axd(b) + bxd(a) (a, b A, x X) (resp., (.5) xd(ab) = axd(b) + d(a)xb (a, b A, x X)) holds. Definition.2. A mapping f : X X is said to be module-a additive if (.6) af(x + x 2 ) = af(x ) + af(x 2 ) (x, x 2 X, a A ). A module-a additive mapping f : X X is called a generalized module-a left derivation (resp., generalized module-a derivation) if there exists a module-x left derivation (resp., module-x derivation) δ : A A such that (.7) af(bx) = abf(x) + axδ(b) (x X, a, b A ) (resp., (.8) af(bx) = abf(x) + aδ(b)x (x X, a, b A )). In addition, if the mappings f and δ are all linear, then the mapping f is called a linear generalized module-a left derivation (resp., linear generalized module-a derivation). Remar. Let A = X and A be one of the following cases: (a) a unital algebra; (b) a Banach algebra with an approximate unit. Then module-a left derivations, module-a derivations, generalized module-a left derivations and generalized module-a derivations on A become left derivations, derivations, generalized left derivations and generalized derivations on A as discussed in [5]. J. Inequal. Pure and Appl. Math., 0(2) (2009), Art. 85, 8 pp.

3 GENERALIZED MODULE DERIVATIONS 3 2. MAIN RESULTS Theorem 2.. Let A be a Banach algebra, X a Banach A -bimodule, and l be integers greater than, and ϕ : X X A X [0, ) satisfy the following conditions: (a) lim n [ϕ( n x, n y, 0, 0) + ϕ(0, 0, n z, w)] = 0 (x, y, w X, z A ). (b) lim 2n ϕ(0, 0, n z, n w) = 0 (z A, w X). (c) ϕ(x) := n=0 n+ ϕ( n x, 0, 0, 0) < (x X). Suppose that f : X X and g : A A are mappings such that f(0) = 0, δ(z) := g( n z) exists for all z A and n f,g (x, y, z, w) ϕ(x, y, z, w) lim (2.) for all x, y, w X and z A where ( x f,g(x, y, z, w) = f + y ) ( x l + zw + f y ) l + zw 2f(x) 2zf(w) 2wg(z). Then f is a generalized module-a left derivation and g is a module-x left derivation. Proof. By taing w = z = 0, we see from (2.) that ( (2.2) x f + y ) ( x + f l y ) 2f(x) l ϕ(x, y, 0, 0) for all x, y X. Letting y = 0 and replacing x by x in (2.2), we get (2.3) f(x) f(x) ϕ(x, 0, 0, 0) 2 for all x X. Hence, for all x X, we have from (2.3) that f (x) f(2 x) 2 f (x) f(x) + f(x) By induction, one can chec that (2.4) f(x) f(n x) n 2 f(2 x) 2 2 ϕ(x, 0, 0, 0) + 2 ϕ( 2 x, 0, 0, 0). n j+ ϕ( j x, 0, 0, 0) for all x in X and n =, 2,.... Let x X and n > m. Then by (2.4) and condition (c), we obtain that f( n x) f(m x) n m = f( n m m x) m f( m x) n m j= n m j+ ϕ( j m x, 0, 0, 0) m 2 2 j= s+ ϕ( s x, 0, 0, 0) s=m 0 (m ). J. Inequal. Pure and Appl. Math., 0(2) (2009), Art. 85, 8 pp.

4 4 HUAI-XIN CAO, JI-RONG LV, AND J. M. RASSIAS { } This shows that the sequence f( n x) is a Cauchy sequence in the Banach A -module X and n f( therefore converges for all x X. Put d(x) = lim n x) for every x X and f(0) = d(0) = n 0. By (2.4), we get (2.5) f(x) d(x) ϕ(x) (x X). 2 Next, we show that the mapping d is additive. To do this, let us replace x, y by n x, n y in (2.2), respectively. Then ( ) n f x n + n y + ( ) n l f x n n y l 2f(n x) n n ϕ( n x, n y, 0, 0) for all x, y X. If we let n in the above inequality, then the condition (a) yields that ( x (2.6) d + y ) ( x + d l y ) = 2 l d(x) for all x, y X. Since d(0) = 0, taing y = 0 and y = l x, respectively, we see that d ( x ) = d(x) and d(2x) = 2d(x) for all x X, and then we obtain that d(x + y) + d(x y) = 2d(x) for all x, y X. Now, for all u, v X, put x = (u + v), y = l (u v). Then by (2.6), we get that 2 2 ( x d(u) + d(v) = d + y ) ( x + d l y ) l = 2 d(x) = 2 d ( 2 (u + v) ) = d(u + v). This shows that d is additive. Now, we are going to prove that f is a generalized module-a left derivation. Letting x = y = 0 in (2.), we get that is f (zw) + f (zw) 2zf(w) 2wg(z) ϕ(0, 0, z, w), (2.7) f (zw) zf(w) wg(z) ϕ(0, 0, z, w) 2 for all z A and w X. By replacing z, w with n z, n w in (2.7) respectively, we deduce that (2.8) f ( 2n zw ) z 2n n f(n w) w n g(n z) 2 2n ϕ(0, 0, n z, n w) for all z A and w X. Letting n, condition (b) yields that (2.9) d(zw) = zd(w) + wδ(z) for all z A and w X. Since d is additive, δ is module-x additive. Put (z, w) = f (zw) zf(w) wg(z). Then by (2.7) we see from condition (a) that for all z A and w X. Hence n ( n z, w) 2 n ϕ(0, 0, n z, w) 0 (n ) f( n z w) d(zw) = lim ( n ) n zf(w) + wg( n z) + ( n z, w) = lim = zf(w) + wδ(z) n J. Inequal. Pure and Appl. Math., 0(2) (2009), Art. 85, 8 pp.

5 GENERALIZED MODULE DERIVATIONS 5 for all z A and w X. It follows from (2.9) that zf(w) = zd(w) for all z A and w X, and then d(w) = f(w) for all w X. Since d is additive, f is module-a additive. So, for all a, b A and x X by (2.9), and af(bx) = ad(bx) = abf(x) + axδ(b) xδ(ab) = d(abx) abf(x) = af(bx) + bxδ(a) abf(x) = a(d(bx) bf(x)) + bxδ(a) = axδ(b) + bxδ(a). This shows that if δ is a module-x left derivation on A, then f is a generalized module-a left derivation on X. Lastly, we prove that g is a module-x left derivation on A. To do this, we compute from (2.7) that f( n zw) z f(n w) wg(z) n n 2 n ϕ(0, 0, z, n w) for all z A and all w X. By letting n, we get from condition (a) that d(zw) = zd(w) + wg(z) for all z A and all w X. Now, (2.9) implies that wg(z) = wδ(z) for all z A and all w X. Hence, g is a module-x left derivation on A. This completes the proof. Corollary 2.2. Let A be a Banach algebra, X a Banach A -bimodule, ε 0, p, q, s, t [0, ) and and l be integers greater than. Suppose that f : X X and g : A A are mappings such that f(0) = 0, δ(z) := lim g( n z) exists for all z A and n (2.0) f,g(x, y, z, w) ε( x p + y q + z s w t ) for all x, y, w X and all z A (0 0 := ). Then f is a generalized module-a left derivation and g is a module-x left derivation. Proof. It is easy to chec that the function ϕ(x, y, z, w) = ε( x p + y q + z s w t ) satisfies conditions (a), (b) and (c) of Theorem 2.. Corollary 2.3. Let A be a Banach algebra with unit e, ε 0, and and l be integers greater than. Suppose that f, g : A A are mappings with f(0) = 0 such that f,g (x, y, z, w) ε for all x, y, w, z A. Then f is a generalized left derivation and g is a left derivation. Proof. By taing w = e in (2.8), we see that the limit δ(z) := lim g( n z) exists for all n z A. It follows from Corollary 2.2 and Remar that f is a generalized left derivation and g is a left derivation. This completes the proof. Lemma 2.4. Let X, Y be complex vector spaces. Then a mapping f : X Y is linear if and only if f(αx + βy) = αf(x) + βf(y) for all x, y X and all α, β T := {z C : z = }. J. Inequal. Pure and Appl. Math., 0(2) (2009), Art. 85, 8 pp.

6 6 HUAI-XIN CAO, JI-RONG LV, AND J. M. RASSIAS Proof. It suffices to prove the sufficiency. Suppose that f(αx + βy) = αf(x) + βf(y) for all x, y X and all α, β T := {z C : z = }. Then f is additive and f(αx) = αf(x) for all x X and all α T. Let α be any nonzero complex number. Tae a positive integer n such that α/n < 2. Tae a real number θ such that 0 a := e iθ α/n < 2. Put β = arccos a 2. Then α = n(e i(β+θ) + e i(β θ) ) and therefore f(αx) = nf(e i(β+θ) x) + nf(e i(β θ) x) = ne i(β+θ) f(x) + ne i(β θ) f(x) = αf(x) for all x X. This shows that f is linear. The proof is completed. Theorem 2.5. Let A be a Banach algebra, X a Banach A -bimodule, and l be integers greater than, and ϕ : X X A X [0, ) satisfy the following conditions: (a) lim n [ϕ( n x, n y, 0, 0) + ϕ(0, 0, n z, w)] = 0 (x, y, w X, z A ). (b) lim 2n ϕ(0, 0, n z, n w) = 0 (z A, w X). (c) ϕ(x) := n=0 n+ ϕ( n x, 0, 0, 0) < (x X). Suppose that f : X X and g : A A are mappings such that f(0) = 0, δ(z) := g( n z) exists for all z A and n 3 f,g (x, y, z, w, α, β) ϕ(x, y, z, w) lim (2.) for all x, y, w X, z A and all α, β T := {z C : z = }, where 3 f,g (x, y, z, w, α, β) stands for ( αx f + βy ) ( αx + zw + f l βy ) + zw 2αf(x) 2zf(w) 2wg(z). l Then f is a linear generalized module-a left derivation and g is a linear module-x left derivation. Proof. Clearly, the inequality (2.) is satisfied. Hence, Theorem 2. and its proof show that f is a generalized left derivation and g is a left derivation on A with f( n x) (2.2) f(x) = lim, g(x) = f(x) xf(e) n for every x X. Taing z = w = 0 in (2.) yields that ( (2.3) αx f + βy ) ( αx + f l βy ) 2αf(x) l ϕ(x, y, 0, 0) for all x, y X and all α, β T. If we replace x and y with n x and n y in (2.3) respectively, then we see that ( ) α n f x + βn y + ( ) α n n l f x βn y 2αf( n x) n l n n ϕ( n x, n y, 0, 0) 0 as n for all x, y X and all α, β T. Hence, ( αx (2.4) f + βy ) ( αx + f l βy ) l = 2αf(x) for all x, y X and all α, β T. Since f is additive, taing y = 0 in (2.4) implies that (2.5) f(αx) = αf(x) J. Inequal. Pure and Appl. Math., 0(2) (2009), Art. 85, 8 pp.

7 GENERALIZED MODULE DERIVATIONS 7 for all x X and all α T. Lemma 2.4 yields that f is linear and so is g. Next, similar to the proof of Theorem 2.3 in [5], one can show that g(a ) Z(A ) rad(a ). This completes the proof. Corollary 2.6. Let A be a complex semi-prime Banach algebra with unit e, ε 0, p, q, s, t [0, ) and and l be integers greater than. Suppose that f, g : A A are mappings with f(0) = 0 and satisfy following inequality: (2.6) 3 f,g (x, y, z, w, α, β) ε( x p + y q + z s w t ) for all x, y, z, w A and all α, β T (0 0 := ). Then f is a linear generalized left derivation and g is a linear left derivation which maps A into the intersection of the center Z(A ) and the Jacobson radical rad(a ) of A. Proof. Since A has a unit e, letting w = e in (2.8) shows that the limit δ(z) := lim g( n z) n exists for all z A. Thus, using Theorem 2.5 for ϕ(x, y, z, w) = ε( x p + y q + z s w t ) yields that f is a linear generalized left derivation and g is a linear left derivation since A has a unit. Similar to the proof of Theorem 2.3 in [5], one can chec that the mapping g maps A into the intersection of the center Z(A ) and the Jacobson radical rad(a ) of A. This completes the proof. Corollary 2.7. Let A be a complex semiprime Banach algebra with unit e, ε 0, and l be integers greater than. Suppose that f, g : A A are mappings with f(0) = 0 and satisfy the following inequality: 3 f,g (x, y, z, w, α, β) ε for all x, y, z, w A and all α, β T. Then f is a linear generalized left derivation and g is a linear left derivation which maps A into the intersection of the center Z(A ) and the Jacobson radical rad(a ) of A. Remar 2. Inequalities (2.0) and (2.6) are controlled by their right-hand sides by the mixed sum-product of powers of norms", introduced by J. M. Rassias (in 2007) and applied afterwards by K. Ravi et al. ( ). Moreover, it is easy to chec that the function ϕ(x, y, z, w) = P x p + Q y q + S z s + T w t satisfies conditions (a), (b) and (c) of Theorem 2. and Theorem 2.5, where P, Q, T, S [0, ) and p, q, s, t [0, ) are all constants. REFERENCES [] T. AOKI, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (950), [2] M. AMYARI, F. RAHBARNIA, AND Gh. SADEGHI, Some results on stability of extended derivations, J. Math. Anal. Appl., 329 (2007), [3] M. AMYARI, C. BAAK, AND M.S. MOSLEHIAN, Nearly ternary derivations, Taiwanese J. Math., (2007), [4] C. BAAK, AND M.S. MOSLEHIAN, On the stability of J -homomorphisms, Nonlinear Anal., 63 (2005), [5] R. BADORA, On approximate derivations, Math. Inequal. & Appl., 9 (2006), [6] R. BADORA, On approximate ring homomorphisms, J. Math. Anal. Appl., 276 (2002), [7] J.A. BAKER, The stability of the cosine equation, Proc. Amer. Soc., 80 (980), J. Inequal. Pure and Appl. Math., 0(2) (2009), Art. 85, 8 pp.

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