Research Article Nearly Quadratic Mappings over p-adic Fields

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1 Abstract and Applied Analysis Volume 202, Article ID , 2 pages doi:0.55/202/ Research Article Nearly Quadratic Mappings over p-adic Fields M. Eshaghi Gordji, H. Khodaei, and Gwang Hui Kim 2 Department of Mathematics, Semnan University, P.O. Box , Semnan, Iran 2 Department of Mathematics, Kangnam University, Yongin, Gyeonggi , Republic of Korea Correspondence should be addressed to Gwang Hui Kim, ghkim@kangnam.ac.kr Received 30 October 20; Revised 20 November 20; Accepted 2 November 20 Academic Editor: John Rassias Copyright q 202 M. Eshaghi Gordji et al. 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. e establish some stability resultsover p-adic fields for the generalized quadratic functional equation n k k n i 2 i 2 i i f n n k i n k,i / i,...,i n k x i n k r x ir f n x i 2 n n f x i, where n N and n 2.. Introduction and Preliminaries In 899, Hensel discovered the p-adic numbers as a number of theoretical analogue of power series in complex analysis. Fix a prime number p. For any nonzero rational number x, there exists a unique integer n x such that x a/b p n x, where a and b are integers not divisible by p. Then, p-adic absolute value x p : p n x defines a non-archimedean norm on Q. The completion of Q with respect to the metric d x, y x y p is denoted by Q p, and it is called the p-adic number field. In fact, Q p is the set of all formal series x k n x a k p k, where a k p are integers see, e.g., 2, 3. Notethatifp>2, then 2 n p for each integer n. During the last three decades, p-adic numbers have gained the interest of physicists for their research, in particular, in problems coming from quantum physics, p-adic strings, and superstrings 4, 5. A key property of p-adic numbers is that they do not satisfy the Archimedean axiom: For x, y > 0, there exists n N such that x<ny. Let K denote a field and function valuation absolute from K into 0,. A non- Archimedean valuation is a function that satisfies the strong triangle inequality; namely, x y max{ x, y } x y for all x, y K. The associated field K is referred to as a non-archimedean field. Clearly, and n for all n. A trivial example of a non-archimedean valuation is the function taking everything except 0 into and 0 0. e always assume in addition that is nontrivial, that is, there is a z K such that z / 0,.

2 2 Abstract and Applied Analysis Let X be a linear space over a field K with a non-archimedean nontrivial valuation. A function : X 0, is said to be a non-archimedean norm if it is a norm over K with the strong triangle inequality ultrametric ; namely, x y max{ x, y } for all x, y X. Then, X, is called a non-archimedean space. In any such a space, a sequence {x n } n N is Cauchy if and only if {x n x n } n N converges to zero. By a complete non-archimedean space, we mean one in which every Cauchy sequence is convergent. The study of stability problems for functional equations is related to a question of Ulam 6 concerning the stability of group homomorphisms, which was affirmatively answered for Banach spaces by Hyers 7. Subsequently, the result of Hyers was generalized by Aoki 8 for additive mappings and by Rassias 9 for linear mappings by considering an unbounded Cauchy difference. The paper by Rassias has provided a lot of influences in the development of what we now call the generalized Hyers-Ulam stability or Hyers- Ulam-Rassias stability of functional equations. Rassias 0 considered the Cauchy difference controlled by a product of different powers of norm. The above results have been generalized by Forti and Găvruţa 2 who permitted the Cauchy difference to become arbitrary unbounded see also Arriola and Beyer 23 investigated stability of approximate additive functions f : Q p R. They showed that if f : Q p R is a continuous function for which there exists a fixed ε such that f x y f x f y ε for all x, y Q p, then there exists a unique additive function T : Q p R such that f x T x ε for all x Q p. For more details about the results concerning such problems, the reader is referred to Recently, Khodaei and Rassias 46 introduced the generalized additive functional equation k i n k i n k n k a i x i a ir x ir f a i x i 2 n a f x,i/ i,...,i n k r. and proved the generalized Hyers-Ulam stability of the above functional equation. The functional equation f x x 2 f x x 2 2f x 2f x 2.2 is related to symmetric biadditive function and is called a quadratic functional equation 47, 48. Every solution of the quadratic equation.2 is said to be a quadratic function. Now, we introduce the generalized quadratic functional equation in n-variables as follows: k i n k i n k n k x i x ir f x i 2 n f x i,,i/ i,...,i n k r.3 where n 2. Moreover, we investigate the generalized Hyers-Ulam stability of functional equation.3 over the p-adic field Q p.

3 Abstract and Applied Analysis 3 As a special case, if n 2in.3, then we have the functional equation.2. Also, if n 3in.3, weobtain x i,i/ i,i 2 2 x ir r x i x i f x i 2 2 f x i,,i/ i i 2.4 that is, f x x 2 x 3 f x x 2 x 3 f x x 2 x 3 f x x 2 x 3 4f x 4f x 2 4f x Stability of Quadratic Functional Equation.3 over p-adic Fields e will use the following lemma. Lemma 2.. Let X and Y be real vector spaces. A function f : X Y satisfies the functional equation.3 if and only if the function f is quadratic. Proof. Let f satisfy the functional equation.3. Setting x i 0 i,...,n in.3, we have k i n k i n k f 0 f 0 2 n f 0, 2. that is, f 0 4 f 0 f 0 f 0 2 n f 0, i n i n 2 i n 2 i n 3 i or n n n f 0 2 n f 0, n n but n j n j j j n j n j j 0 j 2 n j,andalson>j so2 n n f 0 0. Putting x i 0 i 2,...,n in.3 and then using f 0 0, we get n 2 n 2 f x x n f x x n n 2 f x x n n 2 n 2 n 3 f x x n 2 f x x n

4 4 Abstract and Applied Analysis n 2 n 2 f x x n f x x n f x x n n 2 2 n f x 2 n f x n, 2.4 that is, n 2 n 2 f x x n f x x n 2 n f x 2 n f x n, 2.5 j j for all x,x n X, this shows that f satisfies the functional equation.2. So the function f is quadratic. Conversely, suppose that f is quadratic, thus f satisfies the functional equation.2. Hence, we have f 0 0andf is even. e are going to prove our assumption by induction on n 2. It holds on n 2. Assume that it holds on the case where n t; that is, we have k i t k i t k t k x i x ir f x i 2 t f x i,i/ i,...,i t k r 2.6 for all x,...,x t X. It follows from.2 that f x i x t f x i x t 2f x i 2f x t 2.7 for all x,...,x t X. Replacing x t by x t in 2.7,weobtain f t t t x i x t x t f x i x t x t 2f x i x t 2f x t 2.8 for all x,...,x t X. Adding 2.7 to 2.8, we have f t t t t x i x t x t f x i x t x t f x i x t x t f x i x t x t [ ] t t 2 f x i x t f x i x t 4f x t 2.9

5 Abstract and Applied Analysis 5 for all x,...,x t X. Replacing x t by x t in 2.9,weget f x i x t x t x t f x i x t x t x t f x i x t x t x t [ ] f x i x t x t x t 2 f x i x t x t f x i x t x t 4f x t 2.0 for all x,...,x t X. Adding 2.9 to 2.0, onegets f x i x t x t x t f x i x t x t x t f x i x t x t x t f x i x t x t x t f x i x t x t x t f x i x t x t x t t f x i x t x t x t f x i [ 2 f x i x t x t f x i x t x t f x i x t x t f ] x i x t x t 8f x t 2. for all x,...,x t X. By using the above method, for x t 2 until x 2, we infer that t k 2 t k t i t i t k t t t x i x ir f x i,i/ i,...,i t r t k x i x ir f x i 2 t f x t i t k i t k,i/ i,...,i t k r 2.2 for all x,...,x t X. Now, by the case n t, weleadto t k t i t i t k [ ] 2 2 t f x i 2 t f x t t t t x i x ir f x i,i/ i,...,i t r 2.3 for all x,...,x t X,so.3 holds for n t. This completes the proof of the lemma.

6 6 Abstract and Applied Analysis Corollary 2.2. A function f : X Y satisfies the functional equation.3 if and only if there exists a symmetric biadditive function B : X X Y such that f x B x, x for all x X. Now, we investigate the stability of the functional equation.3 from a Banach space B into p-adic field Q p. For convenience, we define the difference operator D f for a given function f: k n k D f x,...,x n : x i x ir i n k i n k,i/ i,...,i n k r f x i 2 n f x i. 2.4 Theorem 2.3. Let B be a Banach space and let ε>0, λ be real numbers. Suppose that a function f : Q p B with f 0 0 satisfies the inequality Df x,...,x n ε x i λ p 2.5 for all x,...,x n Q p. Then there exists a unique quadratic function Q : Q p B such that for all nonzero x Q p. f x Q x ε 2 n 2 n λ 3 x λ p, p 2, λ> 2; 2.6 ε 3.2 n 3 x λ p, p > 2; Proof. Letting x x 2 x / 0andx i 0 i 3,...,n in 2.5, weobtain for all x Q p. Hence, f x 4 f 2x ε 2 n x λ p f 2 l x 2l 2 2m f 2m x ε 2 n m j l 2 λj p 2 2j x λ p 2.8 for all nonnegative integers m and l with m>land for all x Q p. It follows from 2.8 that the sequence { /2 2m f 2 m x } is a Cauchy sequence for all x Q p. Since B is complete, the sequence { /2 2m f 2 m x } converges. Therefore, one can define the function Q : Q p B by Q x : lim m 2 2m f 2m x 2.9

7 Abstract and Applied Analysis 7 for all x Q p. It follows from 2.5 and 2.9 that D Q x,...,x n lim m D 2 2m f 2 m x,...,2 m x n 2 λm p lim m 2 2m ε x i λ p for all x,...,x n Q p.sod Q x,...,x n 0. By Lemma 2., the function Q : Q p B is quadratic. Taking the limit m in 2.8 with l 0, we find that the function Q is quadratic function satisfying the inequality 2.6 near the approximate function f : Q p B of.3. To prove the aforementioned uniqueness, we assume now that there is another additive function Q : Q p B which satisfies.3 and the inequality 2.6. So Q x Q x 2 2m Q 2 m x Q 2 m x Q 2 m x f 2 m x f 2 m x Q 2 m x 2 2m ε 2 2m λm 2 n 2 2 n λ 4 x λ p, p 2, λ > 2; ε 3.2 2m n 4 x λ p, p > 2; 2.2 which tends to zero as m for all nonzero x Q p. This proves the uniqueness of Q, completing the proof of uniqueness. The following example shows that the above result is not valid over p-adic fields. Example 2.4. Let p>2beaprime number and define f : Q p 2 n p, Q p by f x x 2 2x. Since Df x,...,x n p 2n i 2 x i x i p p i 2 x i p 2.22 for all x,...,x n Q p. Hence, the conditions of Theorem 2.3 for ε andλ hold. However for each n N, we have 2 2 m f 2m x 2 2m f 2m x x p p 2 m x p 2.23 p for all x Q p. Hence { /2 2m f 2 m x } is not convergent for all nonzero x Q p. In the next result, which can be compared with Theorem 2.3, we will show that the stability of the functional equation.3 in non-archimedean spaces over p-adic fields.

8 8 Abstract and Applied Analysis Theorem 2.5. Let l {, } be fixed. Let V be a non-archimedean space and be a complete non-archimedean space over Q p,wherep>2 is a prime number. Suppose that a function f : V satisfies the inequality ε x i λ V, λl > 2l; D f x,...,x n ε x λ V x i λ i V, λ λ i l>2l; i 2 } ε max { x i λ V ; i n, λl > 2l; 2.24 for all x,...,x n V,whereε, λ,...,λ n and λ are nonnegative real numbers. Then, the limit Q x : lim m p f p lm x 2lm 2.25 exists for all x Vand Q : V is a unique quadratic function satisfying f x Q x 2p l l λ/2 ε x λ V, p l l λ λ 2 /2 ε x λ λ 2 p l l λ/2 ε x λ V, V, 2.26 for all x V. Proof. By 2.24, Df x,...,x n ε x i λ V 2.27 for all x,...,x n V, where λl > 2l. Putting x i 0 i,...,n in 2.27 to obtain f 0 0, setting x i 0 i 3,...,n in 2.27, weobtain 2 n 2 f x x 2 2 n 2 f x x 2 2 n f x 2 n f x 2 ε x λ V x 2 λ V 2.28 for all x,x 2 V.So f x x 2 f x x 2 2f x 2f x 2 ε x λ V x 2 λ V 2.29 for all x,x 2 V. Letting x x 2 x in 2.29, we have f 2x 4f x 2ε x λ V 2.30

9 Abstract and Applied Analysis 9 for all x V. By induction on j, we will show that for each j 2, f jx j 2 f x 2ε x λ V 2.3 for all x V. It holds on j 2; see Let 2.3 hold for j 2,...,k. Replacing x and x 2 by kx and x in 2.29, respectively, we get f k x f k x 2f kx 2f x ε k λ p x λ V 2.32 for all x V. It follows from 2.32 and our induction hypothesis that f k x k 2 f x f k x f k x 2f kx 2f x f k x k 2 f x 2 f kx k 2 f x } max {2ε x λ V,ε k λ p x λ V 2ε x λ V 2.33 for all x V. This proves 2.3 for each j 2. In particular, f px p 2 f x 2ε x λ V 2.34 for all x V.So f x p f px 2p 2 ε x λ 2 V, x f x p2 f 2p λ ε x λ V p 2.35 for all x V. Hence, p f p lj x 2lj p f p l j x 2l j 2p2lj l λ/2 l p λlj ε x λ V 2.36 for all x V. Since the right side of the above inequality tends to zero as j, { /p 2lm f p lm x } is a Cauchy sequence in complete non-archimedean space, thusit

10 0 Abstract and Applied Analysis converges to some function Q x lim m /p 2lm f p lm x for all x V.Using 2.35 and induction, one can show that for any m N, we have f x m p 2lm f plm x p 2lj f plj x p 2l j f pl j x j 0 max{ } p 2lj f plj x p f p l j x ;0 j<m 2l j } max {2p l l λ/2 lj 2 λ ε x λ V ;0 j<m 2.37 for all x V. Letting m in this inequality, we see that f x Q x 2p l l λ/2 ε x λ V 2.38 for all x V. Moreover, DQ x,...,x n lim m p D f p lm x 2lm,...,p lm x n lim p 2lm m p λlm ε x i λ V for all x,...,x n V.SoD Q x,...,x n 0. By Lemma 2., the function Q : V is quadratic. Now, let Q : V be another quadratic function satisfying.3 and 2.38.So Q x Q x { p 2lm Q max p lm x f p x lm, f p lm x Q } p lm x 2p2lm l λ/2 l p λlm ε x λ V, which tends to zero as m for all x V. This proves the uniqueness of Q. The rest of the proof is similar to the above proof, hence it is omitted Acknowledgments The third author of this work was partially supported by Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education, Science and Technology Grant number: References K. Hensel, Uber eine neue Begrundung der theorie der algebraischen Zahlen, Jahresbericht der Deutschen Mathematiker Vereinigung, vol. 6, pp , 897.

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