NON-ARCHIMEDEAN BANACH SPACE. ( ( x + y

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1 J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. ISSN(Print ISSN(Online Volume 5, Number 3 (August 08, Pages 9 7 ADDITIVE ρ-functional EQUATIONS IN NON-ARCHIMEDEAN BANACH SPACE Siriluk Paokanta a & Eon Hwa Shim b, Abstract. In this paper, we solve the additive ρ-unctional equations ( ( x + y (0. (x + y + (x y (x = ρ + (x y (x, where ρ is a ixed non-archimedean number with ρ <, and ( x + y (0. + (x y (x = ρ((x + y + (x y (x, where ρ is a ixed non-archimedean number with ρ <. Furthermore, we prove the Hyers-Ulam stability o the additive ρ-unctional equations (0. and (0. in non-archimedean Banach spaces.. Introduction and Preliminaries A valuation is a unction rom a ield K into [0, such that 0 is the unique element having the 0 valuation, rs = r s and the triangle inequality holds, i.e., r + s r + s, r, s K. A ield K is called a valued ield i K carries a valuation. The usual absolute values o R and C are examples o valuations. Let us consider a valuation which satisies a stronger condition than the triangle inequality. I the triangle inequality is replaced by r + s { r, s }, r, s K, then the unction is called a non-archimedean valuation, and the ield is called a non-archimedean ield. Clearly = = and n or all n N. A trivial Received by the editors June 08, 08. Accepted August 03, Mathematics Subject Classiication. Primary 46S0, 39B6, 39B5, 47S0, J5. Key words and phrases. Hyers-Ulam stability, non-archimedean normed space, additive ρ- unctional equation. Corresponding author. 9 c 08 Korean Soc. Math. Educ.

2 0 Siriluk Paokanta & Eon Hwa Shim example o a non-archimedean valuation is the unction taking everything except or 0 into and 0 = 0. Throughout this paper, we assume that the base ield is a non-archimedean ield, hence call it simply a ield. Deinition. ([7]. Let X be a vector space over a ield K with a non-archimedean valuation. A unction : X [0, is said to be a non-archimedean norm i it satisies the ollowing conditions: (i x = 0 i and only i x = 0; (ii rx = r x (r K, x X; (iii the strong triangle inequality x + y {x, y}, x, y X holds. Then (X, is called a non-archimedean normed space. Deinition.. (i Let {x n } be a sequence in a non-archimedean normed space X. Then the sequence {x n } is called Cauchy i or a given ε > 0 there is a positive integer N such that x n x m ε or all n, m N. (ii Let {x n } be a sequence in a non-archimedean normed space X. Then the sequence {x n } is called convergent i or a given ε > 0 there are a positive integer N and an x X such that x n x ε or all n N. Then we call x X a limit o the sequence {x n }, and denote by lim n x n = x. (iii I every Cauchy sequence in X converges, then the non-archimedean normed space X is called a non-archimedean Banach space. The stability problem o unctional equations originated rom a question o Ulam [0] concerning the stability o group homomorphisms. Hyers [6] gave a irst airmative partial answer to the question o Ulam or Banach spaces. Hyers Theorem was generalized by Aoki [] or additive mappings and by Rassias [8] or linear mappings by considering an unbounded Cauchy dierence. A generalization o the Rassias theorem was obtained by Găvruta [5] by replacing the unbounded Cauchy dierence by a general control unction in the spirit o Rassias approach. The unctional equation (x + y + (x y = (x is called the Jensen type additive equation.

3 ADDITIVE ρ-functional EQUATIONS The unctional equation (x + y + (x y = (x + (y is called the quadratic unctional equation. In particular, every solution o the quadratic unctional equation is said to be a quadratic mapping. The stability o quadratic unctional equation was proved by Sko [9] or mappings : E E, where E is a normed space and E is a Banach space. Cholewa [4] noticed that the theorem o Sko is still true i the relevant domain E is replaced by an Abelian group. The stability problems o various unctional equations have been extensively investigated by a number o authors (see [, 3]. In this paper, we solve the additive ρ-unctional equations (0. and (0. and prove the Hyers-Ulam stability o the additive ρ-unctional equations (0. and (0. in non-archimedean Banach spaces. Throughout this paper, assume that X is a non-archimedean normed space and that Y is a non-archimedean Banach space. Let =.. Additive ρ-unctional Equation (0. in Non-Archimedean Normed Spaces Throughout this section, assume that ρ is a ixed non-archimedean number with ρ <. In this section, we solve the additive ρ-unctional equation (0. in non-archimedean normed spaces. Lemma.. I a mapping : X Y satisies (0 = 0 and ( ( x + y (. (x + y + (x y (x = ρ + (x y (x or all x, y X, then : X Y is additive. Proo. Assume that : X Y satisies (.. Letting y = x in (., we get (x (x = 0 and so (x = (x or all x X. Thus (. ( x = (x It ollows rom (. and (. that ( ( x + y (x + y + (x y (x = ρ + (x y (x = ρ((x + y + (x y (x

4 Siriluk Paokanta & Eon Hwa Shim and so (x + y + (x y = (x or all x, y X. It is easy to show that is additive. We prove the Hyers-Ulam stability o the additive ρ-unctional equation (. in non-archimedean Banach spaces. Theorem.. Let r < and θ be nonnegative real numbers and let : X Y be a mapping satisying (0 = 0 and ( x + y ( (x + y + (x y (x ρ + (x y (x (.3 θ(x r + y r or all x, y X. Then there exists a unique additive mapping A : X Y such that (.4 Proo. Letting y = x in (.3, we get (x A(x θ r xr (.5 (x (x θx r So ( (x x θx r Hence r ( x ( x l l m (.6 { ( m x ( x ( l l l+ x ( x },, m l+ m m ( m = max { l x ( x ( l,, m x ( x } l+ m { m l rl+r,, m } r(m +r θx r θ = xr (r l+r or all nonnegative integers m and l with m > l and all x X. It ollows rom (.6 that the sequence { n ( x n } is a Cauchy sequence Since Y is complete, the sequence { n ( x n } converges. So one can deine the mapping A : X Y by A(x := lim n n ( x n Moreover, letting l = 0 and passing the limit m in (.6, we get (.4.

5 ADDITIVE ρ-functional EQUATIONS 3 It ollows rom (.3 that ( x + y (A A(x + y + A(x y A(x ρ ( ( = lim n n x + y x y ( x n + n ( ( ( n x + y x y ρ n+ + n n θ lim n nr (xr + y r = 0 or all x, y X. So A(x + y + A(x y A(x = ρ ( A ( x + y + A (x y A(x ( x n + A (x y A(x or all x, y X. By Lemma., the mapping A : X Y is additive. Now, let T : X Y be another additive mapping satisying (.4. Then we have ( x ( x A(x T (x = q A q q T { ( q x ( x ( q A q q x ( x }, q q T q q q θ (r q+r xr, which tends to zero as q So we can conclude that A(x = T (x This proves the uniqueness o h. Thus the mapping A : X Y is a unique additive mapping satisying (.4. Theorem.3. Let r > and θ be nonnegative real numbers and let : X Y be a mapping satisying (0 = 0 and (.3. Then there exists a unique additive mapping A : X Y such that (x A(x θ xr Proo. It ollows rom (.5 that (x (x θxr

6 4 Siriluk Paokanta & Eon Hwa Shim Hence ( l l x m (m x { ( l l x ( l+ x l+,, m ( m x } m (m x { = max l ( l x ( x l+,, m ( m x } (m x { } lr l+,, r(m (m + θx r θ = xr ( rl+ or all nonnegative integers m and l with m > l and all x X. The rest o the proo is similar to the proo o Theorem.. 3. Additive ρ-unctional Equation (0. Throughout this section, assume that ρ is a ixed non-archimedean number with ρ <. In this section, we solve the additive ρ-unctional equation (0. in non-archimedean normed spaces. Lemma 3.. I a mapping : X Y satisies ( x + y (3. + (x y (x = ρ((x + y + (x y (x or all x, y X, then : X Y is additive. Proo. Assume that : X Y satisies (3.. (3. Letting x = y = 0 in (3., we get (0 = 0. Letting y = 0 in (3., we get ( x = (x It ollows rom (3. and (3. that ( x + y (x + y + (x y (x = + (x y (x = ρ((x + y + (x y (x and so (x + y + (x y = (x or all x, y X. It is easy to show that is additive.

7 ADDITIVE ρ-functional EQUATIONS 5 We prove the Hyers-Ulam stability o the additive ρ-unctional equation (3. in non-archimedean Banach spaces. Theorem 3.. Let r < and θ be nonnegative real numbers, and let : X Y be a mapping such that ( x + y (3.3 + (x y (x ρ((x + y + (x y (x θ(x r + y r or all x, y X. Then there exists a unique additive mapping A : X Y such that (3.4 (x A(x θx r Proo. Letting y = 0 in (3.3, we get ( x (3.5 (x θx r So (3.6 ( x ( x l l m { ( m x ( x ( l l l+ x ( x },, m l+ m m ( m = max { l x ( x ( l,, m x ( x } l+ m { m l rl,, m } r(m θx r θ = xr (r l or all nonnegative integers m and l with m > l and all x X. It ollows rom (3.6 that the sequence { n ( x n } is a Cauchy sequence Since Y is complete, the sequence { n ( x n } converges. So one can deine the mapping A : X Y by A(x := lim n n ( x n Moreover, letting l = 0 and passing the limit m in (3.6, we get (3.4. The rest o the proo is similar to the proo o Theorem.. Theorem 3.3. Let r > and θ be positive real numbers, and let : X Y be a mapping satisying (3.3. Then there exists a unique additive mapping A : X Y

8 6 Siriluk Paokanta & Eon Hwa Shim such that (3.7 (x A(x r θ x r Proo. It ollows rom (3.5 that (x (x r θ x r Hence (3.8 l (l x m (m x { ( l l x ( l+ x l+,, m ( m x } m (m x { = max l ( l x ( x l+,, m ( m x } (m x { } rl l+,, r(m (m + r θx r = r θ xr ( rl+ or all nonnegative integers m and l with m > l and all x X. It ollows rom (3.8 that the sequence { n ( n x} is a Cauchy sequence Since Y is complete, the sequence { n ( n x} converges. So one can deine the mapping A : X Y by A(x := lim n n (n x Moreover, letting l = 0 and passing the limit m in (3.8, we get (3.7. The rest o the proo is similar to the proos o Theorems. and 3.. Acknowledgments S. Paokanta was supported by Basic Science Research Program through the National Research Foundation o Korea unded by the Ministry o Education, Science and Technology (NRF-07RDAB

9 ADDITIVE ρ-functional EQUATIONS 7 Reerences. T. Aoki: On the stability o the linear transormation in Banach spaces. J. Math. Soc. Japan (950, L. Cădariu, L. Găvruta & P. Găvruta: On the stability o an aine unctional equation. J. Nonlinear Sci. Appl. 6 (03, A. Chahbi & N. Bounader: On the generalized stability o d Alembert unctional equation. J. Nonlinear Sci. Appl. 6 (03, P.W. Cholewa: Remarks on the stability o unctional equations. Aequationes Math. 7 (984, P. Gǎvruta: A generalization o the Hyers-Ulam-Rassias stability o approximately additive mappings. J. Math. Anal. Appl. 84 (994, D.H. Hyers: On the stability o the linear unctional equation. Proc. Natl. Acad. Sci. U.S.A. 7 (94, M.S. Moslehian & Gh. Sadeghi: A Mazur-Ulam theorem in non-archimedean normed spaces. Nonlinear Anal. TMA 69 (008, Th.M. Rassias: On the stability o the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 7 (978, F. Sko: Propriet locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano 53 (983, S.M. Ulam: A Collection o the Mathematical Problems. Interscience Publ. New York, 960. a Department o Mathematics, Research Institute or Natural Sciences, Hanyang University, Seoul 04763, Korea address: siriluk@hanyang.ac.kr b Department o Mathematics, Daejin University, Kyunggi 59, Korea address: stareun0@nate.com