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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 401762, 9 pages
Stability of the -Dimensional Mixed-Type Additive and Quadratic Functional Equation in Non-Archimedean Normed Spaces
1Department of Mathematics Education, Gongju National University of Education, Gongju 314-711, Republic of Korea
2Mathematics Section, College of Science and Technology, Hongik University, 339-701 Jochiwon, Republic of Korea
3Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece
Received 19 November 2011; Accepted 23 January 2012
Academic Editor: Muhammad Aslam Noor
Copyright © 2012 Yang-Hi Lee 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.
We will prove the stability of the functional equation in non-Archimedean normed spaces.
A classical question in the theory of functional equations is “when is it true that a function, which approximately satisfies a functional equation, must be somehow close to an exact solution of the equation?’’ Such a problem, called a stability problem of the functional equation, was formulated by Ulam in 1940 (see ). In the following year, Hyers  gave a partial solution of Ulam’s problem for the case of approximate additive functions. Subsequently, his result was generalized by Aoki  for additive functions and by Rassias  for linear functions. Indeed, they considered the stability problem for unbounded Cauchy differences. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians (see [5–23]).
A non-Archimedean field is a field equipped with a function (valuation) such that() if and only if ;();() for all .
Clearly, it holds that and for all .
Let be a vector space over a scalar field with a non-Archimedean and nontrivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions:() if and only if ;() for all and ;() for all .
Then is called a non-Archimedean space. Due to the fact that a sequence is Cauchy if and only if converges to zero in a non-Archimedean space. A complete non-Archimedean space is a non-Archimedean space in which every Cauchy sequence is convergent.
Recently, Moslehian and Rassias  proved the Hyers-Ulam stability of the Cauchy functional equation and the quadratic functional equation in non-Archimedean normed spaces.
We now consider the -dimensional mixed-type quadratic and additive functional equation whose solution is called a quadratic-additive function.
In 2009, Towanlong and Nakmahachalasint  obtained a stability result for the functional equation (1.4), in which they constructed a quadratic-additive function by composing an additive function and a quadratic function , where and approximate the odd part and the even part of the given function , respectively.
In this paper, we investigate a general stability problem for the -dimensional mixed-type quadratic and additive functional equation (1.4) in non-Archimedean normed spaces.
2. Solutions of (1.4)
In this section, we prove the generalized Hyers-Ulam stability of the -dimensional mixed-type quadratic and additive functional equation (1.4). Assume that is an additive group and is a complete non-Archimedean space.
For a given function , we use the abbreviations for all and for an arbitrarily fixed .
Theorem 2.1. Assume that is an integer. Let and be an additive group and a complete non-Archimedean space, respectively. A function is a solution of (1.4) if and only if is quadratic, is additive, and .
Proof. If a function is a solution of (1.4), then we have ,
for all , that is, is quadratic and is additive.
Conversely, assume that is quadratic, is additive, and . We apply an induction on to prove for all . For , we have If and for some integer () and for all , then a routine calculation yields for all . Hence, we conclude that for all .
Since is additive, a long calculation yields Hence, it follows from (2.5) and (2.6) that for all ; that is, is a solution of (1.4).
3. Generalized Hyers-Ulam Stability of (1.4)
In the following theorem, we will investigate the stability problem of the functional equation (1.4).
Theorem 3.1. Assume that is an integer. Let and be an additive group and a complete non-Archimedean space, respectively. Assume that is a function such that for all . Moreover, assume that the limit exists for each . If a function satisfies the inequality for any , then there exists a unique quadratic-additive function such that for each . In particular, is given by for all .
Proof. If we replace in (3.1) with 0 for each , then we have
Since , it holds that and
Hence, we conclude that .
Let be a function defined by for all and . A tedious calculation, together with (), (), and (3.3), yields for all and . It follows from (3.1) and (3.9) that the sequence is Cauchy. Since is complete, we conclude that is convergent.
Let us define for any . It follows from and (3.9) that for all and . In view of (3.2), if we let in (3.11), then we obtain the inequality (3.4).
Replacing in (3.3) with for and considering () and (), we get for all and . If we let in the last inequality, then it follows from the condition (3.1) that for all ; that is, is a quadratic-additive function.
Assume that is another quadratic-additive function satisfying (3.4). By the definition of , a routine calculation yields for each and . Hence, it follows from (3.8) that for any and . Since is a solution of (1.4), it follows from the last equality that for any and . Obviously, this equality also holds for .
Consequently, by considering that , it follows from (), (3.1), (3.4), and (3.8) that for all . Therefore, , which proves the uniqueness of .
Corollary 3.2. Let and be non-Archimedean normed spaces over with . If is complete and satisfies the inequality for all and for some , then there exists a unique quadratic-additive function such that for all .
Proof. Let . Since and , we get for all . Therefore, the conditions of Theorem 3.1 are satisfied. Indeed, it is easy to see that . By Theorem 3.1, there exists a unique quadratic-additive function such that (3.18) holds.
The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (no. 2011-0004919).
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