On the Stability of the Quadratic Functional Equation of Pexider Type in Non- Archimedean Spaces

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1 I J C T A, 8(), 015, pp Iteratioal Sciece Press O the Stability of the Quadratic Fuctioal Equatio of Pexider Type i No- Archimedea Spaces M. Aohammady 1, Z. Bagheri ad C. Tuc 3 Abstract: I this article, we prove the geeralized Hyers-Ulam-Rassias stability of the pexiderial fuctioal equatio i o-archimedea ormed spaces. f(ax + ay) + f(ax ay) = ag(x) + 4ag(y) 8g(ay) Keywords: Hyers-Ulam-Rassias stability, No-Archimedea space, pexiderial fuctioal equatio. MSC: 39B8, 46S10, 1J5. 1. INTRODUCTION I 1940 S. Ulam origially raised the stability problem of fuctioal equatio. He had posed the followig questio. Let (G,.) be a group ad (B,.,d) be a metric group give >0 does there exist a > 0 such that if a fuctio f : G B satisfies the iequality d(f(xy), f(x) f(y)) for all x, y G there exists a homomorphism g : G B such that d(f(x), g(x)) for all x G. I 1941, D.H. Hyers[8] solved this questio i the cotext of Baach spaces. This was the first step towards more studies i this domai of research. I 1978, T.M. Rassias [19] has geeralized the Hyers theorem by cosiderig a ubouded Cauchy differece. Theorem 1.1. Let f : E F be a mappig from a orm vector space E ito a Baach space F subect to the iequality f(x + y) f(x) f(y) ( x p + y p ) For all x, y E, where ad p are costats with > 0 ad p < 1. The there exists a uique additive mappig T : E F such that p f ( x) T ( x) x, p For all x E. If p <0 the iequality f(xy) = f(x) + f(y) holds for all x, y6 = 0. Durig the last decade may mathematicias ivestigated several stability problems of fuctioal equatios. I 1897, Hesel [7] has itroduced a ormed space which does ot have the Archimedea property. It tured out that o-archimedea spaces have may ice applicatios (see [4], [8], [9], [15]). The Hyers- Ulam stability of the fuctioal equatio f(ax + ay) + f(ax ay) = af(x) + 4af(y) 8f(ay) was proved by G. ZamaiEsadai, H. Vaezi ad Y. N. Dehgha [] for mappigs f : X Y, where X is a o-archimedea Baach modules ad Y is a uital Baach algebra. I this paper, we cosider the followig Pexider fuctioal equatio f(ax + ay) + f(ax ay) = ag(x) + 4ag(y) 8g(ay). 1, Departmet of Mathematics, Faculty of Basic Scieces, Uiversity of Mazadara, Babolsar, Ira 3 Departmet of Mathematics, Faculty of Scieces, uiversity of YuzucuYil, Kampus Va, Turey amohse@umz.ac.ir, zohrehbagheri@yahoo.com, cemtuc@yahoo.com

2 750 M. Aohammady, Z. Bagheri ad C. Tuc. PRELIMINARIES I this sectio, we give some defiitios ad related lemmas for our mai result. Defiitio.1. A triagular orm (shorter t-orm) is a biary operatio o the uit iterval [0,1], i.e., a fuctio T : [0,1] [0,1] [0,1] satisfyig the followig four axioms: For all a,b,c [0,1] (i) T(a,b) = T(b,a) (commutativity), (ii) T(a,T(b,c)) = T(T(a,b),c) (associativity), (iii)t(a,1) = a(boudary coditio), (iv)t(a,b) T(a,c) wheever b c (mootoicity). Basic examples are the L uasiewicz t-orm T L ad the t-orms T P ad T M where T L (a, b) := max{a + b 1,0}, T P (a,b) := abad T M (a,b) := mi{a,b}. Defiitio.: Let K be a field. A o-archimedea absolute value o K is a fuctio : K [0, + ) such that, for ay a, b K, (i) a 0 ad the equality holds if ad oly if a = 0, (ii) ab = a b, (iii) a + b max{ a, b } (the strict triagle iequality). Note that 1 for each iteger. We always assume, i additio, that is o-trivial, i.e., there exists a a 0 K such that a 0 6 = 0,1. Defiitio.3: Let X be a vector space over a scalar field K with a o-archimedea o-trivial valuatio.. A fuctio. : X R is a o- Archimedea orm (valuatio) if it satisfies the followig coditios: (i) x = 0 if ad oly if x = 0, (ii) rx = r x(r K; x X), (iii)the strog triagle iequality (ultrametric); amely x + y max{x; y : x; y X}. The (X;. ) is called a o-archimedea space. Due to the fact that x x m max{x +1 x m 1} ( > m). Defiitio.4: A sequece {x } is Cauchy if ad oly if {x +1 x } coverges to zero i a o- Archimedea space. By a complete o-archimedea space we mea oe i which every Cauchy sequece is coverget. I this paper, we solve the stability problem for the pexiderial fuctioal equatios f(ax + ay) + f(ax ay) = ag(x) + 4ag(y) 8g(ay) whe the uow fuctios are with values i a o-archimedea space. 3. MAIN RESULTS Throughout this sectio, we assume that H is a additive semigroup ad X is a complete o-archimedea space ad a N. Theorem 3.1: Let : H H [0, ] be a fuctio such that

3 O the Stability of the Quadratic Fuctioal Equatio of Pexider type i No-Archimedea Spaces 751 ( x, y) 0 for all x, y H ad let for each x H the it (3.1) ( x,0) ( x) max 0 (3.) exists. Suppose that f, g: H X are mappigs with f(0) = g(0) = 0 ad satisfyig the followig iequality f(ax + ay) + f(ax ay) + 8g(ay) ag(x) 4ag(y) (x,y) (3.3) for all x X. The there exists a mappig T : H X such that ad 1 f ( x) T ( x) ( x ) (3.4) for all x X. Moreover, if ( x,0) 1 g( x) T ( x) max, ( x) (3.5) ( x,0) max 0 the T is the uique mappig satisfyig (3.4) ad (3.5). Proof. Puttig y = 0 ad a = 1 i (3.3), we get f ( x) g( x) ( x,0) Substitutig y = 0 ad a = i (3.3), we have f ( x) ( x,0) g( x) so f ( x) ( x,0) f ( x) for all x H. Replacig x by 1 x i (3.8) ad dividig both sides by 1, the f x f x x 1 1 ( ) ( ) (,0) 1 (3.6) (3.7) (3.8) (3.9) It follows from (3.1) ad (3.9) that the sequece f ( x) is a Cauchy sequece. Sice X is complete, so f ( x) f ( x) is coverget. Set T ( x ) :. Usig iductio we see

4 75 M. Aohammady, Z. Bagheri ad C. Tuc that f ( x) 1 ( x,0) f ( x) max 0 (3.10) It s clear that (3.10) holds for = 1 by (3.9). Now, if (3.10) holds for every 0 <, we obtai f ( x) f ( x ) = f x f x f x ( ) ( ) ( ) f ( x) max 1 1 f ( x) f ( x) f ( x), 1 1 f ( x) max 1 ( x,0) 1 ( x,0), max ( x,0) max 0. So for all N ad all x H, (3.10) holds. By taig to approach ifiity i (3.10) ad usig (3.) oe obtais (3.4). O the other had, by (3.7), we obtai g( x) T ( x) max{ g( x) f ( x), f ( x) T( x) } ( x,0) 1 max, ( x). If S be aother mappig satisfies (3.4) ad (3.5), the for all x H, we get f ( x) f ( x) T ( x) S( x) max T( x), S( x) Therefore T = S. This completes the proof. 1 ( x,0) max 0 Corollary 3.. Let : [0; ) [0; ) be a fuctio satisfyig ( t) ( ) (t)(t 0); ( ) < 4 Let >0, H be a ormed space ad let f; g: H X are mappigs with f(0) = g(0) = 0 ad satisfyig f(ax + ay) + f(ax ay) ag(x) 4ag(y) + 8g(ay) ( x + y) for all x; y H. The there exists a uique mappig T : H X such that ad 1 f ( x) T( x) ( x )

5 O the Stability of the Quadratic Fuctioal Equatio of Pexider type i No-Archimedea Spaces 753 ( x ) g( x) T ( x) for all x H. Proof. Defiig :H [0; ) by (x; y) : = ( x + y), the we have for all x, y H. O the other had ( x,0) ( ) ( x, y) 0 exists for all x H. Also ( x,0) ( x,0) ( x) max 0 ( x,0) max 0.. Applyig Theorem (3.1), we coclude desired result. Refereces [1] T. Aoi, O the stability of the liear trasformatio i Baach spaces. J. Math. Soc, Japa, (1950), [] Joh A. Baer, A geeral fuctioal equatio ad its stability. Proc. Amer. Math. Soc, 133(005), o. 6, (electroic). [3] Joh A. Baer, J. Lawrece, ad F. Zorzitto, the stability of the equatio f(x+y)=f(x).f(y), Proc.Amer. Math. Soc, 74(1979), o., [4] Joh A. Baer, The stability of the cosie equatio. Proc. Amer. Math. Soc, 80(1980), o.3, [5] R. Ger ad P. Semrl, The stability of the expoetial equatio, Proc. Amer. Math. Soc, 14 (1996), o. 3, [6] J. A. Gogue, L-Fuzzy sets, J. Math. Aal. Appl. 18(1967), [7] K. Hesel, Ober Eieeue Theorie der algebraishe Zahle. (Germa) Math. Z. (1918), o. 3-4, [8] D. H. Hyers, O the stability of the liear fuctioal equatio. Proc. Nat. Acad. Sci. U. S. A. 7, (1941), 4. [9] K. Jarosz, Almost multiplicative fuctioals. Studia Math. 14(1997), o. 1, [10] B. E. Johso, Approximately multiplicative fuctioals. J. Lodo Math. Soc. 34 (1998), o. 3, [11] A. K. Katsaras, Fuzzy topological vector spaces. II. Fuzzy Sets ad Systems, 1(1984), o [1] Peter J.Nyios, O some o-archimedea spaces of Alexadroff ad Urysoh, Topology Appl. 91(1999), o.1, 1 3. [13] Ji Ha Par, Ituitioistic fuzzy metric spaces. chaos Solitos Fratals (004), o. 5, [14] C. Par, M. Eshaghi Gordi, A. Naati, Geeralized Hyers-Ulam stability of a AQCQ fuctioal equatio i o- Archimedea Baach spaces. J. Noliear Sci. Appl. 3(010), o. 4, [15] J. M. Rassias, Solutio of a quadratic stability Hyers-Ulam type problem, Ricerche Mat. 50(001), o. 1, [16] J. M. Rassias,Solutio of the Ulam stability problem for Euler-Lag rage quadratic mappigs. J. Math. Aal. Appl. 0(1998), o., [17] Themistocles M. Rassias, O the stability of the quadratic fuctioal equatio ad its applicatios. Studia Uiv. Babe s- Bolyai Math. 43 (1998), o. 3, [18] Themistocles M. Rassias, The problem of S. M. Ulam for approximately multiplicative mappigs. J. Math. Aal. Appl. 46 (000), o., [19] Themistocles M. Rassias, O the stability of the liear mappig i Baach spaces. Proc. Amer. Math. Soc. 7 (1978), o.,

6 754 M. Aohammady, Z. Bagheri ad C. Tuc [0] S. M. Ulam, problems i moder mathematics. Sciece Editios, Jho Wiley sos, Ic., New yor (1964). [1] L. A. Zadeh, Fuzzy sets. Iformatio ad Cotrol 8 (1965), [] G. Zamai Esadai, Hamid Vaezi ad Y. N. Dehgha Stability of a Mixed Additive ad Quadratic Fuctioal Equatio i Noarchimedea Baach Modules. Taiwaese J. Math. 14 (010), o. 4,