Research Article Approximate Riesz Algebra-Valued Derivations

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1 Abstract ad Applied Aalysis Volume 2012, Article ID , 5 pages doi: /2012/ Research Article Approximate Riesz Algebra-Valued Derivatios Faruk Polat Departmet of Mathematics, Faculty of Sciece, Çakırı Karateki Uiversity, Çakırı, Turkey Correspodece should be addressed to Faruk Polat, faruk.polat@gmail.com Received 7 May 2012; Accepted 26 August 2012 Academic Editor: Jausz Brzdek Copyright q 2012 Faruk Polat. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Let F be a Riesz algebra with a exteded orm u such that F, u is complete. Also, let v be aother exteded orm i F weaker tha u such that wheever a x x ad x y z i v,thez x y; b y y ad x y z i v,thez x y. Letε ad δ>be two oegative real umbers. Assume that a map f : F F satisfies f x y f x f y u ε ad f x y x f y f x y v δ for all x, y F. I this paper, we prove that there exists a uique derivatio d : F F such that f x d x u ε, x F.Moreover,x f y d y 0 for all x, y F. 1. Itroductio Let E ad E be Baach spaces ad let δ > 0. A fuctio f : E E is called δ-additive if f x y f x f y < δ for all x, y E. The well-kow problem of stability of fuctioal equatio f x y f x f y started with the followig questio of Ulam 1. Does there exist for each ε>0, a δ>0 such that, to each δ-additive fuctio f of E ito E there correspods a additive fuctio l of E ito E satisfyig the iequality f x l x ε for each x E? I 1941, Hyers 2 aswered this questio i the affirmative way ad showed that δ may be take equal to ε. The aswer of Hyers is preseted i a great umber of articles ad books. For the theory of the stability of fuctioal equatios see Hyers et al 3. Let F be a algebra. A mappig d : F F is called a derivatio if ad oly if it satisfies the followig fuctioal equatios: d a b d a d b, d ab ad b d a b, for all a, b F.

2 2 Abstract ad Applied Aalysis The stability of derivatios was first studied by Ju ad Park 4. Further, approximate derivatios were ivestigated by a umber of mathematicias see, e.g., 5 7. The aim of the preset paper is to examie the stability problem of derivatios for Riesz algebras with exteded orms. 2. Prelimiaries A vector space F with a partial order satisfyig the followig two coditios: 1 x y αx z αy z for all z F ad 0 α R, 2 for all x, y F, the supremum x y ad ifimum x y exist i F hece, the modulus x : x x exists for each x F, is called a Riesz space or vector lattice. Typical examples of Riesz spaces are provided by the fuctio spaces. C K the spaces of real valued cotiuous fuctios o a topological space K, l p real valued absolutely summable sequeces, c the spaces of real valued coverget sequeces, ad c 0 the spaces of real valued sequeces covergig to zero are atural examples of Riesz spaces uder the poitwise orderig. A Riesz space F is called Archimedea if 0 u, v F ad u v for each N imply u 0. A subset S i a Riesz space F is said to be solid if it follows from u v i F ad v S that u S. A solid liear subspace of a Riesz space F is called a ideal. Every subset D of a Riesz space F is icluded i a smallest ideal F D, called ideal geerated by D. A pricipal ideal of a Riesz space F is ay ideal geerated by a sigleto {u}. This ideal will be deoted by I u.itiseasytoseethat I u {v F : λ 0 such that v λ u }. 2.1 Let F be a Riesz space ad 0 u F. Firstly, we give the followig defiitio. Defiitio The sequece x i F is said to be u-uiformly coverget to the elemet x F wheever, for every ε>0, there exists 0 such that x 0 k x εu holds for each k. 2 The sequece x i F is said to be relatively uiformly coverget to x wheever x coverges u-uiformly to x F for some 0 u F. Whe dealig with relative uiform covergece i a Archimedea Riesz space F, it is atural to associate with every positive elemet u F a exteded orm u i F by the followig formula: x u if{λ 0: x λu} x F. 2.2 Note that x u < if ad oly if x I u.also x δu if ad oly if x u δ. A Baach lattice is a vector lattice F that is simultaeously a Baach space whose orm is mootoe i the followig sese. For all x, y F, x y implies x y. Hece, x x for all x F. The sequece x i F, u is called a exteded u-ormed Cauchy sequece, if for every ε>0 there exists k such that x k x m k u <εfor all m,. If every exteded u-ormed Cauchy sequece is coverget i F, the F is called a exteded u-ormed Baach lattice.

3 Abstract ad Applied Aalysis 3 A Riesz space F is called a Riesz algebra or a lattice ordered algebra if there exists a associative multiplicatio i F with the usual algebra properties such that 0 u v for all 0 u, v F. For more detailed iformatio about Riesz spaces, the reader ca cosult the book Riesz Spaces by Luxemburg ad Zaae 8. I the sequel, all the Riesz spaces are assumed to be Archimedea. 3. Mai Result Recetly, Polat 9 geeralized the Hyers result 2 to Riesz spaces with exteded orms ad proved the followig. Theorem 3.1. Let E be a liear space ad F a Riesz space equipped with a exteded orm u such that the space F, u is complete. If, for some δ>0, amapf : E F, u is δ-additive, the limit l x lim f 2 x /2 exists for each x E. l x is the uique additive fuctio satisfyig the iequality f x l x u δ for all x E. By usig Theorem 3.1, we give the mai result of the paper as follows. Theorem 3.2. Let F be a Riesz algebra with a exteded orm u such that F, u is complete. Also, let v be aother exteded orm i F weaker tha u such that wheever a x x ad x y z i v,thez x y; b y y ad x y z i v,thez x y. Let ε ad δ be two oegative real umbers. Assume that a map f : F F satisfies ( ) ( ) f x y f x f y u ε, 3.1 ( ) ( ) f x y x f y f x y v δ, 3.2 for all x, y F. The, there exists a uique derivatio d : F F such that f x d x u ε, (x F). Moreover, x f y d y 0 for all x, y F. Proof. By Coditio 3.1, Theorem 3.1 shows that there exists a uique additive fuctio d : F F such that f x d x u ε, 3.3 for each x F. It is eough to show that d satisfies Coditio 1.2. The iequality 3.3 implies that f x d x u ε x F, N. 3.4 By the additivity of d, we the have 1 u f x d x 1 ε x F, N, 3.5

4 4 Abstract ad Applied Aalysis which meas that d x lim f x, x F, with respect to u orm ad so is with respect to v orm. Coditio 3.2 implies that the fuctio r : F F F defied by r x, y f x y x f y f x y is bouded. Hece 1 lim r( x, y ) 0, ), 3.7 with respect to v orm. Applyig 3.6 ad 3.7, we have d ( x y ) x f ( y ) d x y, ). 3.8 Ideed, we have the followig with respect to v orm, d ( x y ) 1 lim f( ( x y )) 1 lim f( x y ) 1 ( ( ) ( )) lim x f y f x y r x, y ( lim x f ( y ) f x y r( x, y ) ) 3.9 x f ( y ) d x y, ). Let x, y F ad N be fixed. The usig 3.8 ad additivity of d, we have x f ( y ) d x y x f ( y ) d x y d ( x y ) d ( x y ) x f ( y ) d x y 3.10 x f ( y ) d x y. Therefore, x f ( y ) x f( y ),, N ) Sedig to ifiity, by 3.6, weseethat x f ( y ) x d ( y ), ) Combiig this formula with 3.8, we have that d satisfies 1.2 which is the desired result. Moreover, the last formula yields x f y d y 0 for all x, y F.

5 Abstract ad Applied Aalysis 5 Refereces 1 S. M. Ulam, A Collectio of Mathematical Problems, Itersciece Publishers, Lodo, UK, D. H. Hyers, O the stability of the liear fuctioal equatio, Proceedigs of the Natioal Academy of Scieces of the Uited States of America, vol. 27, pp , D. H. Hyers, G. Isac, ad Th. M. Rassias, Stability of Fuctioal Equatios i Several Variables,Birkhauser Bosto, Bosto, Mass, USA, K.-W. Ju ad D.-W. Park, Almost derivatios o the Baach algebra C 0, 1, Bulleti of the Korea Mathematical Society, vol. 33, o. 3, pp , M. S. Moslehia, Terary derivatios, stability ad physical aspects, Acta Applicadae Mathematicae, vol. 100, o. 2, pp , M. E. Gordji ad M. S. Moslehia, A trick for ivestigatio of approximate derivatios, Mathematical Commuicatios, vol. 15, o. 1, pp , A. Fošer, O the geeralized Hyers-Ulam stability of module left m, derivatios, Aequatioes Mathematicae, vol. 84, o. 1-2, pp , W. A. J. Luxemburg ad A. C. Zaae, Riesz Spaces, vol. 1, North-Hollad Publishig Compay, Amsterdam, The Netherlads, F. Polat, Some geeralizatios of Ulam-Hyers stability fuctioal equatios to Riesz algebras, Abstract ad Applied Aalysis, vol. 2012, Article ID , 9 pages, 2012.

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