Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 195137, 8 pages doi:10.1155/2008/195137 ResearchArticleEuler-LagrangeTypeCubicOperatorsandTheirNormson X λ Space Abbas Najati 1 and Asghar Rahimi 2 1 Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, P.O. Box 56199-11367 Ardabili, Iran 2 Department of Mathematics, University of Maragheh, P.O. Box 55181-83111, Maragheh, East Azarbayjan, Iran Correspondence should be addressed to Abbas Najati, a.najati@uma.ac.ir Received 17 April 2008; Accepted 1 July 2008 Recommended by Jong Kim We will introduce linear operatorsand obtain their exact norms defined on the function spaces X λ and Z 5 λ . These operators are constructed from the Euler-Lagrangetypecubic functional equations andtheir Pexider versions. Copyright q 2008 A. Najati and A. Rahimi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let X and Y be complex normed spaces. For a fixed nonnegative real number λ, we denote by X λ the linear space of all functions f : X→Y with pointwise operations for which there exists a constant M f ≥ 0with fx≤M f e λx 1.1 for all x ∈ X. It is easy to show that the space X λ with the norm f : sup x∈X e −λx fx 1.2 is a normed space. Let us denote by X n λ the linear space of all functions φ : X ×···×X n times →Y with pointwise operations for which there exists a constant M φ ≥ 0with φx 1 , ,x n ≤M φ e λ n i1 x i 1.3 2 Journal of Inequalities and Applications for all x 1 , ,x n ∈ X. It is not difficult to show that the space X n λ with the norm φ : sup x 1 , ,x n ∈X φx 1 , ,x n e −λ n i1 x i 1.4 is a normed space. We denote by Z m λ the normed space m i1 X λ {f 1 , ,f m : f 1 , ,f m ∈ X λ } with pointwise operations together with the norm f 1 , ,f m : max{f 1 , ,f m }. 1.5 The norms of the Pexiderized Cauchy, quadratic, and Jensen operatorson the function space X λ have been investigated by Czerwik and Dlutek 1, 2.In3, Moslehian et al. have extended the results of 2 to the Pexiderized generalized Jensen and Pexiderized generalized quadratic operatorson the function space X λ and provided more general results regarding their norms. In 4, Jung investigated the norm of the cubic operator on the function space Z 5 λ . A function f : X→Y is called a cubic function if and only if f is a solution function of the cubic functional equation fx yfx − y2f 1 2 x y 2f 1 2 x − y 12f 1 2 x . 1.6 Jun and Kim 5 proved that when both X and Y are real vector spaces, a function f : X→Y satisfies 1.6 if and only if there exists a function B : X × X × X→Y such that fxBx, x, x for all x ∈ X, and B is symmetric for each fixed one variable and is additive for fixed two variables. In 6, the authors introduced the following Euler-Lagrange-type cubic functional equation, which is equivalent to 1.6, fx yfx − yaf 1 a x y af 1 a x − y 2aa 2 − 1f 1 a x 1.7 for fixed integers a with a / 0, ±1. Moreover, Jun and Kim 7 introduced the following Euler- Lagrange-type cubic functional equation f 1 a x 1 b y f 1 b x 1 a y aba−b 2 f 1 ab x f 1 ab y ababf 1 ab x 1 ab y 1.8 for fixed integers a, b with a, b / 0,a± b / 0, and they proved the following theorem. Theorem 1.1 see 7, Theorem 2.1. Let X and Y be real vector spaces. If a mapping f : X→Y satisfies the functional equation 1.6,thenf satisfies the functional equation 1.8. We will introduce linear operators which are constructed from the Euler-Lagrange-type cubicand the Pexiderization of the Euler-Lagrange-type cubic functional equations 1.7 and 1.8. A. Najati and A. Rahimi 3 Definition 1.2. The operators C P 1 ,C P 2 : Z 5 λ →X 2 λ are defined by C P 1 f 1 , ,f 5 x, y : f 1 x yf 2 x − y − mf 3 1 m x y − mf 4 1 m x − y − 2m m 2 − 1 f 5 1 m x , C P 2 f 1 , ,f 5 x, y : f 1 1 a x 1 b y f 2 1 b x 1 a y − a ba − b 2 f 3 1 ab x f 4 1 ab y − aba bf 5 1 ab x 1 ab y , 1.9 where a, b,andm are fixed integers with a, b / 0, a ± b / 0, and m / 0, ±1. Definition 1.3. The operators C 1 ,C 2 : X λ →X 2 λ are defined by C 1 fx, y : fx yfx − y − mf 1 m x y − mf 1 m x − y − 2mm 2 − 1f 1 m x , C 2 fx, y : f 1 a x 1 b y f 1 b x 1 a y − a ba − b 2 f 1 ab x f 1 ab y − aba bf 1 ab x 1 ab y , 1.10 where a, b,andm are fixed integers with a, b / 0, a ± b / 0, and m / 0, ±1. In this paper, we will give the exact norms of the operators C P 1 ,C P 2 on the function space Z 5 λ , andnorms of the operators C 1 ,C 2 on the function space X λ . The results extend the results of 4. 2. Main results Throughout this section, a, b,andm are fixed integers with a, b / 0, a ± b / 0, and m / 0, ±1. The next theorems give us the exact norms of operators C P 1 , C P 2 , C 1 ,andC 2 . Theorem 2.1. The operator C P 1 : Z 5 λ →X 2 λ is a bounded linear operator with C P 1 2|m| 3 2. 2.1 Proof. First, we show that C P 1 ≤2|m| 3 2. Since max x y, x − y, 1 m x y , 1 m x − y , 1 m x ≤x y 2.2 4 Journal of Inequalities and Applications for all x, y ∈ X, we get C P 1 f 1 , ,f 5 sup x,y∈X e −λxy f 1 x yf 2 x − y − mf 3 1 m x y − mf 4 1 m x − y − 2mm 2 − 1f 5 1 m x ≤ sup x,y∈X e −λxy f 1 x y sup x,y∈X e −λx−y f 2 x − y |m| sup x,y∈X e −λ1/mxy f 3 1 m x y |m| sup x,y∈X e −λ1/mx−y f 4 1 m x − y 2|m|m 2 − 1sup x∈X e −λ1/mx f 5 1 m x f 1 f 2 |m|f 3 |m|f 4 2|m|m 2 − 1f 5 ≤ 2|m| 3 2 max{f 1 , f 2 , f 3 , f 4 , f 5 } 2|m| 3 2f 1 ,f 2 ,f 3 ,f 4 ,f 5 2.3 for each f 1 , ,f 5 ∈ Z 5 λ . This implies that C P 1 ≤2|m| 3 2. 2.4 Now, let ν ∈ Y be such that ν 1andlet{ξ n } n be a sequence of positive real numbers decreasing to 0. We define f n x ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ e 2λξ n ν, if x 2ξ n or x 0, − |m| m e 2λξ n ν, if mx |m 1|ξ n , mx |m − 1|ξ n or mx ξ n , 0, otherwise 2.5 for all x ∈ X. Hence we have e −λx f n x ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ e 2λξ n , if x 0, 1, if x 2ξ n , e 2−|m1/m|λξ n , if mx |m 1|ξ n , e 2−|m−1/m|λξ n , if mx |m − 1|ξ n , e 2−1/|m|λξ n , if mx ξ n , 0, otherwise 2.6 A. Najati and A. Rahimi 5 for all x ∈ X, so that f n ∈ X λ for all positive integers n, with f n e 2λξ n . 2.7 Let u ∈ X be such that u 1 and take x 0 ,y 0 ∈ X as x 0 y 0 ξ n u. Then it follows from the definition of f n that C P 1 f n , ,f n sup x,y∈X e −λxy f n x yf n x − y − mf n 1 m x y − mf n 1 m x − y − 2mm 2 − 1f n 1 m x ≥ e −2λξ n e 2λξ n ν e 2λξ n ν |m|e 2λξ n ν |m|e 2λξ n ν 2|m|m 2 − 1e 2λξ n ν 2|m| 3 2. 2.8 If on the contrary C P 1 < 2|m| 3 2, then there exists a δ>0 such that C P 1 f n , ,f n ≤2|m| 3 2 − δf n , ,f n 2.9 for all positive integers n. So it follows from 2.7, 2.8,and2.9 that 2|m| 3 2 ≤C P 1 f n , ,f n ≤2|m| 3 2 − δe 2λξ n 2.10 for all positive integers n. Since lim n→∞ e 2λξ n 1, the right-hand side of 2.10 tends to 2|m| 3 2 − δ as n→∞, whence 2|m| 3 2 ≤ 2|m| 3 2 − δ, which is a contradiction. Hence we have C P 1 2|m| 3 2. Theorem 2.1 of 4 is a result of Theorem 2.1 for m 2. Corollary 2.2. The operator C 1 : X λ →X 2 λ is a bounded linear operator with C 1 2|m| 3 2. 2.11 Proof. The result follows from the proof of Theorem 2.1. Theorem 2.3. The operator C P 2 : Z 5 λ →X 2 λ is a bounded linear operator with C P 2 2|a b|a − b 2 |aba b| 2. 2.12 Proof. Since max 1 a x 1 b y , 1 b x 1 a y , 1 ab x , 1 ab y , 1 ab x 1 ab y ≤x y 2.13 6 Journal of Inequalities and Applications for all x, y ∈ X, we get C P 2 f 1 , ,f 5 sup x,y∈X e −λxy f 1 1 a x 1 b y f 2 1 b x 1 a y − a ba − b 2 f 3 1 ab x f 4 1 ab y − aba bf 5 1 ab x 1 ab y ≤ sup x,y∈X e −λ1/ax1/by f 1 1 a x 1 b y sup x,y∈X e −λ1/bx1/ay f 2 1 b x 1 a y |a b|a − b 2 sup x∈X e −λ1/abx f 3 1 ab x |a b|a − b 2 sup y∈X e −λ1/aby f 4 1 ab y |aba b| sup x,y∈X e −λ1/abx1/aby f 5 1 ab x 1 ab y ≤f 1 f 2 |a b|a − b 2 f 3 f 4 |aba b|f 5 ≤ 2|a b|a − b 2 |aba b| 2 max{f 1 , f 2 , f 3 , f 4 , f 5 } 2|a b|a − b 2 |aba b| 2f 1 ,f 2 ,f 3 ,f 4 ,f 5 2.14 for each f 1 , ,f 5 ∈ Z 5 λ . This implies that C P 2 ≤2|a b|a − b 2 |aba b| 2. 2.15 Let η be a real number such that η / ∈ 0, 1, 1 − a b , 1 − b a , a − 1 1 − b , b − 1 1 − a , a 1 − b , b 1 − a . 2.16 Now, let u ∈ X, ν ∈ Y be such that u ν 1andlet{ξ n } n be a sequence of positive real numbers decreasing to 0. We define f n x ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ e λ1|η|ξ n ν, if x 1 a η b ξ n u, or x 1 b η a ξ n u, − |a b| a b e λ1|η|ξ n ν, if x 1 ab ξ n u, or x η ab ξ n u, − |aba b| aba b e λ1|η|ξ n ν, if x 1 η ab ξ n u, 0, otherwise 2.17 A. Najati and A. Rahimi 7 for all x ∈ X. Hence we have e −λx f n x ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ e 1|η|−|1/aη/b|λξ n , if x 1 a η b ξ n u, e 1|η|−|1/bη/a|λξ n , if x 1 b η a ξ n u, e 1|η|−|1/ab|λξ n , if x 1 ab ξ n u, e 1|η|−|η/ab|λξ n , if x η ab ξ n u, e 1|η|−|1η/ab|λξ n , if x 1 η ab ξ n u, 0, otherwise 2.18 for all x ∈ X, so that f n ∈ X λ for all positive integers n, with f n max{e 1|η|−|1/aη/b|λξ n ,e 1|η|−|1/bη/a|λξ n , e 1|η|−|1/ab|λξ n ,e 1|η|−|η/ab|λξ n ,e 1|η|−|1η/ab|λξ n }. 2.19 Let x 0 ,y 0 ∈ X be such that x 0 ξ n u and y 0 ηξ n u. Then it follows from the definition of f n that C P 2 f n , ,f n sup x,y∈X e −λxy f n 1 a x 1 b y f n 1 b x 1 a y − a ba − b 2 f n 1 ab x f n 1 ab y − aba bf n 1 ab x 1 ab y ≥ e −λ1|η|ξ n e λ1|η|ξ n e λ1|η|ξ n 2|ab|a − b 2 e λ1|η|ξ n |abab|e λ1|η|ξ n 2|a b|a − b 2 |aba b| 2, 2.20 so that C P 2 f n , ,f n ≥2|a b|a − b 2 |aba b| 2. 2.21 If on the contrary C P 2 < 2|a b|a − b 2 |aba b| 2, then there exists a δ>0 such that C P 2 f n , ,f n ≤2|a b|a − b 2 |aba b| 2 − δf n , ,f n 2.22 8 Journal of Inequalities and Applications for all positive integers n. So it follows from 2.21 and 2.22 that 2|a b|a − b 2 |aba b| 2 ≤C P 2 f n , ,f n ≤2|a b|a − b 2 |aba b| 2 − δf n 2.23 for all positive integers n. Since lim n→∞ ξ n 0, it follows from 2.19 that lim n→∞ f n 1, so the right-hand side of 2.23 tends to 2|a b|a − b 2 |aba b| 2 − δ as n→∞, whence 2|a b|a − b 2 |aba b| 2 ≤ 2|a b|a − b 2 |aba b| 2 − δ, 2.24 which is a contradiction. Hence we have C P 2 2|a b|a − b 2 |aba b| 2. Corollary 2.4. The operator C 2 : X λ →X 2 λ is a bounded linear operator with C 2 2|a b|a − b 2 |aba b| 2. 2.25 Proof. The result follows from the proof of Theorem 2.3. Acknowledgment The authors would like to thank the referee for his/her useful comments. References 1 S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002. 2 S. Czerwik and K. Dlutek, “Cauchy and Pexider operators in some function spaces,” in Functional Equations, Inequalities and Applications, pp. 11–19, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. 3 M. S. Moslehian, T. Riedel, and A. Saadatpour, “Norms of operators in X λ spaces,” Applied Mathematics Letters, vol. 20, no. 10, pp. 1082–1087, 2007. 4 S M. Jung, “Cubic operator norm on X λ space,” Bulletin of the Korean Mathematical Society, vol. 44, no. 2, pp. 309–313, 2007. 5 K W. Jun and H M. Kim, “The generalized Hyers-Ulam-Rassias stability of a cubic functional equation,” Journal of Mathematical Analysis and Applications, vol. 274, no. 2, pp. 867–878, 2002. 6 K W. Jun, H M. Kim, and I S. Chang, “On the Hyers-Ulam stability of an Euler-Lagrangetypecubic functional equation,” Journal of Computational Analysis and Applications, vol. 7, no. 1, pp. 21–33, 2005. 7 K W. Jun and H M. Kim, “On the stability of Euler-Lagrangetypecubic mappings in quasi-Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1335–1350, 2007. . Corporation Journal of Inequalities and Applications Volume 2008, Article ID 195137, 8 pages doi:10.1155/2008/195137 Research Article Euler-Lagrange Type Cubic Operators and Their Norms on X λ Space Abbas. obtain their exact norms defined on the function spaces X λ and Z 5 λ . These operators are constructed from the Euler-Lagrange type cubic functional equations and their Pexider versions. Copyright. functional equation 1.8. We will introduce linear operators which are constructed from the Euler-Lagrange- type cubic and the Pexiderization of the Euler-Lagrange- type cubic functional equations