The Australian Journal of Mathematical Analysis and Applications

The Australian Journal of Mathematical Analysis and Applications http://ajmaa.org Volume 8, Issue, Article 3, pp. -8, 0 ULAM STABILITY OF RECIPROCAL DIFFERENCE AND ADJOINT FUNCTIONAL EQUATIONS K. RAVI, J. M. RASSIAS AND B. V. SENTHIL KUMAR 3 Received 9 November, 008; accepted 8 February, 00; published 8 November, 0. DEPARTMENT OF MATHEMATICS, SACRED HEART COLLEGE, TIRUPATTUR - 63560, INDIA. shckavi@yahoo.co.in PEDAGOGICAL DEPARTMENT E. E., SECTION OF MATHEMATICS AND INFORMATICS, NATIONAL AND CAPODISTRIAN UNIVERSITY OF ATHENS, 4, AGAMEMNONOS STR., AGHIA PARASKEVI, ATHENS, ATTIKIS 534, GREECE. jrassias@primedu.uoa.gr 3 DEPARTMENT OF MATHEMATICS, C.ABDUL HAKEEM COLLEGE OF ENGINEERING AND TECHNOLOGY, MELVISHARAM - 63 509, INDIA. bvssree@yahoo.co.in ABSTRACT. In this paper, the reciprocal difference functional equation (or RDF equation and the reciprocal adjoint functional equation (or RAF equation are introduced. Then the pertinent Ulam stability problem for these functional equations is solved, together with the extended Ulam (or Rassias stability problem and the generalized Ulam (or Ulam-Gavruta-Rassias stability problem for the same equations. Key words and phrases: Reciprocal Difference Functional equation;reciprocal Adjoint Functional equation; Ulam stability; Extended Ulam stability; Generalized Ulam stability. 000 Mathematics Subject Classification. 39B, 39B5, 39B7. ISSN (electronic: 449-590 c 0 Austral Internet Publishing. All rights reserved.

K. RAVI, J. M. RASSIAS AND B. V. SENTHIL KUMAR. INTRODUCTION In 940, Ulam [5] proposed the Ulam stability problem of additive mappings. In 94, D. H. Hyers [3] considered the case of approximately additive mappings f : E E where E and E are Banach spaces and f satisfies the inequality f( f(x f(y ɛ for all x, y E. It was shown that the limit L(x = lim n n f( n x exists for all x E and that L is the unique additive mapping satisfying f(x L(x ɛ. In 978, Th. M. Rassias [48] generalized the above result of an approximation involving a sum of powers of norms. In 98-994, a generalization of the result of D. H. Hyers was proved via theorems [9]-[3], [34], [35], using weaker conditions controlled by a product of different powers of norms. Theorem. (J. M. Rassias [9] Let f : E E be a mapping where E is a real-normed space and E is a Banach space. Assume that there exists θ > 0 such that f( f(x f(y θ x p y q for all x, y E where r = p + q. Then there exists a unique additive mapping L : E E such that θ f(x L(x r x r for all x E. However, the case r = in the above inequality is singular. A clever counter-example has been given by Gavruta []. A pertinent interesting paper about the stability of additive mappings was presented by Gavruta [0]. The above-mentioned stability involving a product of different powers of norms is called Ulam-Gavruta-Rassias stability by Bouikhalene and Elqorachi [3], Elqorachi and Sibaha [47] and Nakmahachalasint [4]. Besides, J. M. Rassias [33] introduced and investigated also the Euler-Lagrange quadratic mappings. Very recently, K. Ravi and B. V. Senthil Kumar [46] proved some interesting results on Ulam- Gavruta-Rassias stability of Rassias reciprocal functional equation (. r( = r(xr(y r(x + r(y. The reciprocal function r(x = is the solution of the functional equation (.. x In this paper, J. M. Rassias introduces the Reciprocal Difference Functional equation (or RDF equation ( (. r r( = r(xr(y r(x + r(y and the Reciprocal Adjoint Functional equation (or RAF equation ( (.3 r + r( = 3r(xr(y r(x + r(y and then we investigate these equations (. and (.3 controlled by the Product and the mixed product-sum of powers of norms" introduced by J. M. Rassias.. SOLUTION OF (. AND (.3 Theorem.. Let X and Y be sets of non-zero real numbers. A function r : X Y satisfies the functional equation (. if and only if r : X Y satisfies the functional equation (. if and only if r : X Y satisfies the functional equation (.3. Therefore, every solution of functional equations (. and (.3 is also a reciprocal function., Vol. 8, No., Art. 3, pp. -8, 0

ULAM STABILITY OF RECIPROCAL DIFFERENCE AND ADJOINT FUNCTIONAL EQUATIONS 3 Proof. Let r : X Y satisfy the functional equation (.. Letting y = x in (., we get (. r(x = r(x. Replacing x by x in (., we obtain (. r = r(x. Now, replacing (x, y by, x in (. and using (., we arrive ( (.3 r = r(xr(y r(x + r(y. Subtracting (. from (.3, we lead to (.. Next, let r : X Y satisfy the functional equation (.. Letting y = x in (., we arrive (.4 r(x = r(x. Replacing x by x in (.4, we obtain (.5 r = r(x. Applying the result (.5 in (., we get (.6 r( = r(xr(y r(x + r(y. Now, adding (.6 with (., we obtain (.3. Finally, let r : X Y satisfy the functional equation (.3. Putting y = x in (.3, we obtain (.7 r(x = r(x. Replacing x by x in (.7, we obtain (.8 r = r(x. Using (.8 in (.3, we obtain (.. This completes the proof of Theorem.. 3. HYERS-ULAM STABILITY OF RDF EQUATION (. Theorem 3.. Let X and Y be spaces of non-zero real numbers. Assume in addition that f:x Y is a mapping for which there exists a constant c (independent of x,y 0 such that the functional inequality ( (3. f f( f(xf(y f(x + f(y ɛ holds for all (x, y X. Then the limit (3. r(x = lim n n f( n x exists for all x X, n N and r:x Y is the unique mapping satisfying the functional equation (., such that (3.3 f(x r(x ɛ for all x X. Moreover, functional identity r(x = n r( n x holds for all x X and n N., Vol. 8, No., Art. 3, pp. -8, 0

4 K. RAVI, J. M. RASSIAS AND B. V. SENTHIL KUMAR Proof. Replacing (x, y by, x in (3., we obtain (3.4 f f(x ɛ. Replacing x by x in (3.4, dividing by and summing the resulting inequality with (3.4, we get f ( ( x f(x ɛ. Proceeding further and using induction on a positive integer n, we obtain (3.5 ( f f(x n n ɛ. n In order to prove the convergence of the sequence { n f( n x}, we have if n > p > 0, then n f( n x p f( p x = p (n p f( n x f( p x holds for all x X and n, p N. Setting p x = y in this relation and using (3.5, we obtain n f( n x p f( p x = p (n p f( n+p y f(y ( p ɛ (3.6 n p or or n f( n x p f( p x ɛ( p n < ɛ p lim p n f( n x p f( p x = 0. This shows that the sequence { n f( n x} is a Cauchy sequence and hence the limit (3. exists for all x X. To show that r satisfies (., replacing (x, y by ( n x, n y in (3. and dividing by n, we obtain ( ( (3.7 n f n f( n ( f( n xf( n y f( n x + f( n y ɛ n. Allowing n in (3.7, we see that r satisfies (. for all (x, y X. To prove r is a unique reciprocal function satisfying (. subject to (3.3, let us consider an s : X Y to be another reciprocal function which satisfies (. and the inequality (3.3. Clearly s( n x = n s(x, r( n x = n r(x and using (3.3, we arrive (3.8 s(x r(x = n s( n x r( n x n ( s( n x f( n x + f( n x r( n x n ɛ = n ɛ for all x X. Allowing n in (3.8, we find that r is unique. Applying (3. in (3.5, we arrive the result (3.3. This completes the proof of Theorem 3.. 4. HYERS-ULAM STABILITY OF RAF EQUATION (.3 Theorem 4.. Let X and Y be spaces of non-zero real numbers. Assume in addition that f:x Y is a mapping for which there exists a constant c (independent of x,y 0 such that the functional inequality ( (4. f + f( 3f(xf(y f(x + f(y ɛ, Vol. 8, No., Art. 3, pp. -8, 0

ULAM STABILITY OF RECIPROCAL DIFFERENCE AND ADJOINT FUNCTIONAL EQUATIONS 5 holds for all (x, y X. Then the limit (4. r(x = lim n n f( n x exists for all x X, n N and r:x Y is the unique mapping satisfying the functional equation (.3, such that (4.3 f(x r(x ɛ for all x X. Moreover, functional identity r(x = n r( n x holds for all x X and n N. Proof. Replacing (x, y by, x in (4., we obtain (4.4 f(x f ɛ. Now replacing x by x in (4.4, dividing by and summing the resulting inequality with (4.4, we get f(x f ( ( x ɛ. Proceeding further and using induction on a positive integer n, we obtain (4.5 f(x f ɛ (. n n n The proof of the rest of Theorem 4. is similar to that of Theorem 3.. This completes the proof of Theorem 4.. 5. GENERALIZED ULAM STABILITY OF RDF EQUATION (. The generalized Ulam (or Ulam-Gavruta-Rassias stability introduced by J. M. Rassias, concerns functional equations controlled by the product of powers of norms. Theorem 5.. Let f:x Y be a mapping on the spaces of non-zero real numbers. If there exist a, b : ρ = a + b > and c 0 such that ( (5. f f( f(xf(y f(x + f(y c x a y b for all x, y X, then there exists a unique reciprocal mapping r:x Y such that (5. r(x f(x c x ρ holds and r satisfies (., for all x, y X where c = c ρ+. Proof. Replacing (x, y by, x in (5., we obtain (5.3 f f(x c ρ x ρ. Now replacing x by x in (5.3, dividing by and summing the resulting inequality with (5.3, we get f ( x f(x c ρ x ρ. Proceeding further and using induction on a i(ρ+, Vol. 8, No., Art. 3, pp. -8, 0

6 K. RAVI, J. M. RASSIAS AND B. V. SENTHIL KUMAR positive integer n, we get f f(x n n c ρ (5.4 Setting c = c ρ+ (5.5 c ρ n i(ρ+ x ρ i(ρ+ x ρ c ρ+ x ρ. then the equation (5.4 reduces to f f(x n n c x ρ. In order to prove the convergence of the sequence { n f( n x}, we have if n > p > 0, then n f( n x p f( p x = p n+p f( n x f( p x holds for all x X and n, p N. Setting p x = y and using (5.5, we obtain n f( n x p f( p x = p n+p f( n+p y f(y (5.6 p(ρ+ c x ρ. As ρ >, the right-hand side of (5.6 tends to 0 as p. This shows that the sequence { n f( n x} is a Cauchy sequence. To show that r satisfies (., settting r(x = lim n n f( n x, replacing (x, y by ( n x, n y in (5. and dividing by n, we obtain (5.7 ( ( n f n f( n ( f( n xf( n y f( n x + f( n y c n(ρ+ x a y b. Allowing n in (5.7, we see that r satisfies (. for all (x, y X. To prove r is a unique reciprocal function satisfying (. subject to (5., let us consider an s : X Y to be another reciprocal function which satisfies (. and the inequality (5.. Clearly s( n x = n s(x, r( n x = n r(x and using (5., we arrive s(x r(x = n s( n x r( n x n ( s( n x f( n x + f( n x r( n x (5.8 n(ρ++ c x ρ for all x X. Allowing n in (5.8, we find that r is unique. This completes the proof of the Theorem 5.. Theorem 5.. Let f:x Y be a mapping on the spaces of non-zero real numbers. If there exist a, b : ρ = a + b < and c 0 such that ( (5.9 f f( f(xf(y f(x + f(y c x a y b for all x, y X, then there exists a unique reciprocal mapping r:x Y such that (5.0 f(x r(x c x ρ and r satisfies (., for all x, y X where c = c ρ+., Vol. 8, No., Art. 3, pp. -8, 0

ULAM STABILITY OF RECIPROCAL DIFFERENCE AND ADJOINT FUNCTIONAL EQUATIONS 7 Proof. Replacing (x, y by (x, x in (5.9 and multiplying by, we obtain (5. f(x f(x c x ρ. Now replacing x by x in (5., multiplying by and summing the resulting inequality with (5., we get f(x f( x c i(ρ+ x ρ. Proceeding further and using induction on a positive integer n, we get (5. Setting c = c ρ+ n f(x n f( n x c i(ρ+ x ρ c then the equation (5. reduces to i(ρ+ x ρ c ρ+ x ρ. (5.3 f(x n f( n x c x ρ. In order to prove the convergence of the sequence { n f( n x}, we have if n > p > 0, then n f( n x p f( p x = p n p f( n x f( p x holds for all x X and n, p N. Setting p x = y in this relation and using (5.3, we obtain (5.4 n f( n x p f( p x = p n p f( n p y f(y p(ρ+ c x ρ. As ρ <, the right-hand side of (5.4 tends to 0 as p. This shows that the sequence { n f( n x} is a Cauchy sequence. To prove r satisfies (. and it is unique, the proof is similar to that of Theorem 5.. This completes the proof of the Theorem 5.. Theorem 5.3. Let f:x Y be a mapping for which there exists a constant θ > 0 and f satisfies ( (5.5 f f( f(xf(y f(x + f(y θh(x, y where H : X Y be a function such that (5.6 φ(x = H i, x i+ i+ with the condition lim (5.7 n H n, x = 0 n+ n+ holds. Then there exists a unique reciprocal mapping A : X Y which satisfies (. and the inequality (5.8 f(x A(x θφ(x for all x X. Proof. Replacing (x, y by, x in (5.5, we obtain (5.9 f ( f(x x θh, x., Vol. 8, No., Art. 3, pp. -8, 0

8 K. RAVI, J. M. RASSIAS AND B. V. SENTHIL KUMAR Again replacing x by x in (5.9, dividing by and summing the resulting inequality with (5.9, we get f ( x f(x θ H ( x x, i i+. Proceeding further and using i+ induction on a positive integer n, we get n f f(x n n θ H i, x i+ i+ (5.0 θ i H i+, x i+ for all x X. In order to prove the convergence of the sequence { n f( n x}, replace x by p x in (5.0 and divide by p,we find that for n > p > 0 p f( p x n p f( n p x = p f( p x n f( n p x (5. θ p+i H p+i+, x p+i+. Allow p and using (5.7, the right-hand side of the inequality (5. tends to 0. Thus the sequence { n f( n x} is a Cauchy sequence. Allowing n in (5.0, we arrive (5.8. To show that A satisfies (., setting A(x = lim n n f( n x, replacing (x, y by ( n x, n y in (5.5 and dividing by n, we obtain (5. ( ( n f n f( n ( f( n xf( n y f( n x + f( n y n θh( n x, n y. Allowing n in (5., we see that A satisfies (. for all (x, y X. To prove A is a unique reciprocal function satisfying (.. Let B : X Y be another reciprocal function which satisfies (. and the inequality (5.8. Clearly B( n x = n B(x, A( n x = n A(x and using (5.8, we arrive (5.3 B(x A(x = n B( n x A( n x ( n B( n x f( n x + f( n x A( n x θ H n+i, x n+i+ n+i+ for all x X. Allowing n in (5.3 and using (5.7, we find that A is unique. This completes the proof of the Theorem 5.3. Theorem 5.4. Let f:x Y be a mapping for which there exists a constant θ > 0 and f satisfies ( (5.4 f f( f(xf(y f(x + f(y θh(x, y where H : X Y be a function such that (5.5 φ(x = i H( i+ x, i+ x with the condition (5.6 lim n n H( n+ x, n+ x = 0, Vol. 8, No., Art. 3, pp. -8, 0

ULAM STABILITY OF RECIPROCAL DIFFERENCE AND ADJOINT FUNCTIONAL EQUATIONS 9 holds. Then there exists a unique reciprocal mapping A : X Y which satisfies (. and the inequality (5.7 f(x A(x θφ(x for all x X. Proof. Replacing (x, y by (x, x in (5.4 and multiplying by, we obtain (5.8 f(x f(x θh(x, x. Again replacing x by x in (5.8, multiplying by and summing the resulting inequality with (5.8, we get f(x f( x θ i H( i+ x, i+ x. Proceeding further and using induction on a positive integer n, we get (5.9 n f(x n f( n x θ i H( i+ x, i+ x θ i H( i+ x, i+ x for all x X. In order to prove the convergence of the sequence { n f( n x}, replace x by p x in (5.9 and multiply by p, we find that for n > p > 0 p f( p x n+p f( n+p x = p f( p x n f( n+p x (5.30 θ p+i H( p+i+ x, p+i+ x. Allowing p and using (5.6, the right-hand side of the inequality (5.30 tends to 0. Thus the sequence { n f( n x} is a Cauchy sequence. Allowing n in (5.9, we arrive (5.7. To prove A satisfies (. and it is unique, the proof is similar to that of Theorem 5.3. This completes the proof of the Theorem 5.4. 6. EXTENDED ULAM STABILITY OF RDF EQUATION (. The extended Ulam(or Rassias stability introduced by J. M. Rassias, concerns functional equations controlled by the mixed product-sum of powers of norms. Theorem 6.. Let f:x Y be a mapping on the spaces of non-zero real numbers. If there exist k and α with k > 0 and α > such that ( (6. f f( f(xf(y f(x + f(y k ( x α y α + ( x α + y α for all x, y X, then there exists a unique reciprocal mapping r:x Y such that (6. r(x f(x c x α and r satisfies (., for all x, y X where c = 6k. α+ Proof. Replacing (x, y by, x in (6., we obtain (6.3 f f(x 3k α x α., Vol. 8, No., Art. 3, pp. -8, 0

0 K. RAVI, J. M. RASSIAS AND B. V. SENTHIL KUMAR Now replacing x by x in (6.3, dividing by and summing the resulting inequality with (6.3, we get f ( x f(x 3k α x α. Using induction on a positive integer n, we get i(α+ f f(x n n 3k n α x α i(α+ 3k α x α i(α+ (6.4 Setting c = (6.5 6k α+ 6k α+ x α. then the equation (6.4 reduces to f f(x n n c x α. In order to prove the convergence of the sequence { n f( n x}, we have if n > p > 0, then n f( n x p f( p x = p n+p f( n x f( p x holds for all x X and n, p N. Setting p x = y in this relation and using (6.5, we obtain n f( n x p f( p x = p n+p f( n+p y f(y (6.6 p(α+ c x α. As α >, the right-hand side of (6.6 tends to 0 as p. This shows that the sequence { n f( n x} is a Cauchy sequence. To show that r satisfies (., setting r(x = lim n n f( n x, replacing (x, y by ( n x, n y in (6. and dividing by n, we obtain ( ( n f n f( n ( f( n xf( n y f( n x + f( n y (6.7 n(α+ k ( x α y α + ( x α + y α. Allowing n in (6.7, we see that r satisfies (. for all (x, y X. To prove r is a unique reciprocal function satisfying (. subject to (6.. Let s : X Y be another reciprocal function which satisfies (. and the inequality (6.. Clearly s( n x = n s(x, r( n x = n r(x and using (6., we arrive s(x r(x = n s( n x r( n x n ( s( n x f( n x + f( n x r( n x (6.8 n(α++ c x α for all x X. Allowing n in (6.8, we find that r is unique. This completes the proof of the Theorem 6.. Theorem 6.. Let f:x Y be a mapping on the spaces of non-zero real numbers. If there exist k and α with k > 0 and α < such that ( (6.9 f f( f(xf(y f(x + f(y k ( x α y α + ( x α + y α for all x, y X, then there exists a unique reciprocal mapping r:x Y such that (6.0 f(x r(x c x α, Vol. 8, No., Art. 3, pp. -8, 0

ULAM STABILITY OF RECIPROCAL DIFFERENCE AND ADJOINT FUNCTIONAL EQUATIONS and r satisfies (. for all x, y X where c = 6k. α+ Proof. Replacing (x, y by (x, x in (6.9 and multiplying by, we obtain (6. f(x f(x 6k x α. Now replacing x by x in (6., multiplying by and summing the resulting inequality with (6.,we get f(x f( x 6k i(α+ x α. Using induction on a positive integer n, we get (6. Setting c = 6k α+ n f(x n f( n x 6k i(α+ x α 6k i(α+ x α then the equation (6. reduces to 6k α+ x α. (6.3 f(x n f( n x c x α. In order to prove the convergence of the sequence { n f( n x}, we have if n > p > 0, then n f( n x p f( p x = p n p f( n x f( p x holds for all x X and n, p N. Setting p x = y in this relation and using (6.3, we obtain (6.4 n f( n x p f( p x = p n p f( n p y f(y p(α+ c x α. As α <, the right-hand side of (6.4 tends to 0 as p. This shows that the sequence { n f( n x} is a Cauchy sequence. To prove r satisfies (. and it is unique, the proof is similar to that of Theorem 6.. This completes the proof of the Theorem 6.. 7. GENERALIZED ULAM STABILITY OF RAF EQUATION (.3 Theorem 7.. Let f:x Y be a mapping on the spaces of non-zero real numbers. If there exist a, b : ρ = a + b > and c 0 such that ( (7. f + f( 3f(xf(y f(x + f(y c x a y b for all x, y X, then there exists a unique reciprocal mapping r:x Y such that (7. f(x r(x c x ρ holds and r satisfies (.3, for all x, y X where c = c ρ+. Proof. Replacing (x, y by, x in (7., we obtain (7.3 f(x f c ρ x ρ., Vol. 8, No., Art. 3, pp. -8, 0

K. RAVI, J. M. RASSIAS AND B. V. SENTHIL KUMAR Now replacing x by x in (7.3, dividing by and summing the resulting inequality with (7.3,we get f(x f ( x c ρ x ρ. Proceeding further and using induction on a positive integer n, we get i(ρ+ f(x f c n n n ρ x ρ i(ρ+ c ρ x ρ i(ρ+ (7.4 Setting c = c ρ+ (7.5 c ρ+ x ρ. then the equation (7.4 reduces to f(x f c x ρ. n n The proof of the rest of Theorem 7. goes through the same way as in Theorem 5.. This completes the proof of the Theorem 7.. Theorem 7.. Let f:x Y be a mapping on the spaces of non-zero real numbers. If there exist a, b : ρ = a + b < and c 0 such that ( (7.6 f + f( 3f(xf(y f(x + f(y c x a y b for all x, y X, then there exists a unique reciprocal mapping r:x Y such that (7.7 r(x f(x c x ρ and r satisfies (.3, for all x, y X where c = c ρ+. Proof. Replacing (x, y by (x, x in (7.6 and multiplying by, we obtain (7.8 f(x f(x c x ρ. Now replacing x by x in (7.8, multiplying by and summing the resulting inequality with (7.8, we get f( x f(x c i(ρ+ x ρ. Proceeding further and using induction on a positive integer n, we get (7.9 Setting c = c ρ+ n n f( n x f(x c i(ρ+ x ρ c then the equation (7.9 reduces to i(ρ+ x ρ c ρ+ x ρ. (7.0 n f( n x f(x c x ρ. The proof of the rest of Theorem 7. can be done by similar arguments as in Theorem 5.. This completes the proof of the Theorem 7.., Vol. 8, No., Art. 3, pp. -8, 0

ULAM STABILITY OF RECIPROCAL DIFFERENCE AND ADJOINT FUNCTIONAL EQUATIONS 3 Theorem 7.3. Let f:x Y be a mapping for which there exists a constant θ > 0 and f satisfies ( (7. f + f( 3f(xf(y f(x + f(y θh(x, y where H : X Y be a function such that (7. φ(x = H i, x i+ i+ with the condition lim (7.3 n H n, x = 0 n+ n+ holds. Then there exists a unique reciprocal mapping A : X Y which satisfies (.3 and the inequality (7.4 f(x A(x θφ(x for all x X. Proof. Replacing (x, y by, x in (7., we obtain (7.5 f(x f ( x θh, x. Again replacing x by x in (7.5, dividing by and summing the resulting inequality with (7.5, we get f(x f ( x θ H ( x x, i i+. Proceeding further and using i+ induction on a positive integer n, we get f(x f n θ n n H i, x i+ i+ (7.6 θ i H i+, x i+ for all x X. The proof of the rest of Theorem 7.3 goes through the same way as in Theorem 5.3. This completes the proof of the Theorem 7.3. Theorem 7.4. Let f:x Y be a mapping for which there exists a constant θ > 0 and f satisfies ( (7.7 f + f( 3f(xf(y f(x + f(y θh(x, y where H : X Y be a function such that (7.8 φ(x = i H( i+ x, i+ x with the condition (7.9 i=o lim n n H( n+ x, n+ x = 0 holds. Then there exists a unique reciprocal mapping A : X Y which satisfies (. and the inequality (7.0 A(x f(x θφ(x for all x X., Vol. 8, No., Art. 3, pp. -8, 0

4 K. RAVI, J. M. RASSIAS AND B. V. SENTHIL KUMAR Proof. Replacing (x, y by (x, x in (7.7 and multiplying by, we obtain (7. f(x f(x θh(x, x. Again replacing x by x in (7., multiplying by and summing the resulting inequality with (7., we get f( x f(x θ i H( i+ x, i+ x. Proceeding further and using induction on a positive integer n, we get (7. n n f( n x f(x θ i H( i+ x, i+ x θ i H( i+ x, i+ x for all x X. The proof of the rest of Theorem 7.4 is similar to that of Theorem 5.4. This completes the proof of the Theorem 7.4. 8. EXTENDED ULAM STABILITY OF RAF EQUATION (.3 Theorem 8.. Let f:x Y be a mapping on the spaces of non-zero real numbers. If there exist k and α with k > 0 and α > such that ( (8. f + f( 3f(xf(y f(x + f(y k ( x α y α + ( x α + y α for all x, y X, then there exists a unique reciprocal mapping r:x Y such that (8. f(x r(x c x α and r satisfies (.3, for all x, y X where c = 6k. α+ Proof. Replacing (x, y by, x in (8., we obtain (8.3 f(x f 3k α x α. Now replacing x by x in (8.3, dividing by and summing the resulting inequality with (8.3, we get f(x f ( x 3k α x α. Using induction on a positive integer n, we get i(α+ f(x f 3k n n n α x α i(α+ 3k α x α i(α+ (8.4 Setting c = (8.5 6k α+ 6k α+ x α. then the equation (8.4 reduces to f(x f c x α. n n The proof of the rest of Theorem 8. is similar to that of Theorem 6.. This completes the proof of the Theorem 8.., Vol. 8, No., Art. 3, pp. -8, 0

ULAM STABILITY OF RECIPROCAL DIFFERENCE AND ADJOINT FUNCTIONAL EQUATIONS 5 Theorem 8.. Let f:x Y be a mapping on the spaces of non-zero real numbers. If there exist k and α with k > 0 and α < such that ( (8.6 f + f( 3f(xf(y f(x + f(y k ( x α y α + ( x α + y α for all x, y X, then there exists a unique reciprocal mapping r:x Y such that (8.7 r(x f(x c x α and r satisfies (.3, for all x, y X where c = 6k. α+ Proof. Replacing (x, y by (x, x in (8.6 and multiplying by, we obtain (8.8 f(x f(x 6k x α. Now replacing x by x in (8.8, multiplying by and summing the resulting inequality with (8.8,we get f( x f(x 6k i(α+ x α. Using induction on a positive integer n, we get (8.9 Setting c = 6k α+ n n f( n x f(x 6k i(α+ x α 6k i(α+ x α 6k α+ x α. then the equation (8.9 reduces to (8.0 n f( n x f(x c x α. The proof of the rest of Theorem 8. is similar to that of Theorem 6.. This completes the proof of the Theorem 8.. REFERENCES [] J. ACZEL and J. DHOMBRES, Functional Equations in Several Variables, Cambridge Univ.Press, 989. [] J. H. BAE and K. W. JUN, On the generalized Hyers-Ulam-Rassias stability of quadratic functional equation, Bull. Koeran. Math. Soc., 38 (00, pp. 35-336. [3] B. BOUIKHALENE and E. ELQORACHI, Ulam-Gavruta-Rassias stability of the Pexider functional equation, Internat. J. Appl. Math. Stat., 7 (007, pp. 7-39. [4] I. S. CHANG and H. M. KIM, On the Hyers-Ulam stability of quadratic functional equations, J. Ineq. Appl. Math., 3 (00, pp. -. [5] I. S. CHANG and Y. S. JUNG, Stability of functional equations deriving from cubic and quadratic functions, J. Math. Anal. Appl., 83 (003, pp. 49-500. [6] P. W. CHOLEWA, Remarks on the stability of functional equations, Aequationes Math., 7 (984, pp. 76-86. [7] CHOONKIL PARK, JONG SU AN and JIANLIAN CUI, Isomorphisms and Derivations in Lie C*-Algebras, Abstract and Applied Analysis, Article ID 85737, 4 pages(007. [8] CH. PARK, Hyers-Ulam-Rassias stability of homomorphisms in quasi-banach algebras, Bull. Sci. Math., 3 (008, pp. 87-96., Vol. 8, No., Art. 3, pp. -8, 0

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8 K. RAVI, J. M. RASSIAS AND B. V. SENTHIL KUMAR [5] S. M. ULAM, Problems in Modern Mathematics, Chapter VI, Wiley-Interscience, New York, 964., Vol. 8, No., Art. 3, pp. -8, 0