/ / Article

Research Article | Open Access

Volume 2010 |Article ID 849543 | https://doi.org/10.1155/2010/849543

Choonkil Park, "Fixed Points and the Stability of an AQCQ-Functional Equation in Non-Archimedean Normed Spaces", Abstract and Applied Analysis, vol. 2010, Article ID 849543, 15 pages, 2010. https://doi.org/10.1155/2010/849543

Fixed Points and the Stability of an AQCQ-Functional Equation in Non-Archimedean Normed Spaces

Accepted08 Apr 2010
Published02 Jun 2010

Abstract

Using fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation in non-Archimedean Banach spaces.

1. Introduction and Preliminaries

A valuation is a function from a field into such that 0 is the unique element having the 0 valuation, , and the triangle inequality holds, that is, A field is called a valued field if carries a valuation. The usual absolute values of and are examples of valuations.

Let us consider a valuation which satisfies a stronger condition than the triangle inequality. If the triangle inequality is replaced by then the function is called a non-Archimedean valuation, and the field is called a non-Archimedean field. Clearly and for all . A trivial example of a non-Archimedean valuation is the function taking everything except for 0 into 1 and .

Throughout this paper, we assume that the base field is a non-Archimedean field, hence call it simply a field.

Definition 1.1 (see [1]). Let be a vector space over a field with a non-Archimedean valuation . A function is said to be a non-Archimedean norm if it satisfies the following conditions:(i) if and only if ;(ii);(iii)the strong triangle inequality holds. Then is called a non-Archimedean normed space.

Definition 1.2. (i) Let be a sequence in a non-Archimedean normed space . Then the sequence is called Cauchy if for a given , there is a positive integer such that for all .
() Let be a sequence in a non-Archimedean normed space . Then the sequence is called convergent if for a given , there are a positive integer and an such that for all . Then we call a limit of the sequence , and denote it by.
() If every Cauchy sequence in converges, then the non-Archimedean normed space is called a non-Archimedean Banach space.

The stability problem of functional equations originated from a question of Ulam [2] concerning the stability of group homomorphisms. Hyers [3] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' Theorem was generalized by Aoki [4] for additive mappings and by Rassias [5] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias [5] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability or as Hyers-Ulam-Rassias stability of functional equations. A generalization of the Rassias theorem was obtained by Găvruţa [6] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach.

The functional equation is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [7] for mappings , where is a normed space and is a Banach space. Cholewa [8] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [9] proved the generalized Hyers-Ulam stability of the quadratic functional equation.

In [10], Jun and Kim considered the following cubic functional equation: which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping. In [11], Lee et al. considered the following quartic functional equation: which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping.

The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [1227]).

Let be a set. A function is called a generalized metric on if satisfies() if and only if ;() for all ;() for all .

We recall a fundamental result in fixed point theory.

Theorem 1.3 (see [28, 29]). Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either for all nonnegative integers or there exists a positive integer such that();()the sequence converges to a fixed point of ;() is the unique fixed point of in the set ;() for all .

In 1996, Isac and Rassias [30] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [3136]).

This paper is organized as follows: in Section 2, using the fixed point method, we prove the generalized Hyers-Ulam stability of the additive-quadratic-cubic-quartic functional equation in non-Archimedean Banach spaces for an odd case. In Section 3, using the fixed point method, we prove the generalized Hyers-Ulam stability of the additive-quadratic-cubic-quartic functional equation (1.10) in non-Archimedean Banach spaces for an even case.

Throughout this paper, assume that is a non-Archimedean normed vector space and that is a non-Archimedean Banach space.

2. Generalized Hyers-Ulam Stability of the Functional Equation (1.10): An Odd Case

One can easily show that an odd mapping satisfies (1.10) if and only if the odd mapping is an additive-cubic mapping, that is, It was shown in Lemma of [37] that and are cubic and additive, respectively, and that .

One can easily show that an even mapping satisfies (1.10) if and only if the even mapping is a quadratic-quartic mapping, that is, It was shown in Lemma of [38] that and are quartic and quadratic, respectively, and that .

For a given mapping , we define for all .

We prove the generalized Hyers-Ulam stability of the functional equation in non-Archimedean Banach spaces: an odd case.

Theorem 2.1. Let be a function such that there exists an with for all . Let be an odd mapping satisfying for all . Then there is a unique cubic mapping such that for all .

Proof. Letting in (2.5), we get for all .
Replacing by in (2.5), we get for all .
By (2.7) and (2.8), for all .
Letting and for all , we get for all .
Consider the set and introduce the generalized metric on where, as usual, . It is easy to show that is complete. (See the proof of Lemma of [39].)
Now we consider the linear mapping such that for all .
Let be given such that . Then for all . Hence for all . So implies that . This means that for all .
It follows from (2.10) that for all . So .
By Theorem 1.3, there exists a mapping satisfying the following.
() is a fixed point of , that is, for all . The mapping is a unique fixed point of in the set This implies that is a unique mapping satisfying (2.18) such that there exists a satisfying for all ; since is odd, is an odd mapping.
() as . This implies the equality for all .
() , which implies the inequality This implies that the inequality (2.6) holds.
By (2.5), for all and all . So for all and all . So for all . Thus the mapping is cubic, as desired.

Corollary 2.2. Let and be positive real numbers with . Let be an odd mapping satisfying for all . Then there exists a unique cubic mapping such that for all .

Proof. The proof follows from Theorem 2.1 by taking for all . Then we can choose and we get the desired result.

Theorem 2.3. Let be a function such that there exists an with for all . Let be an odd mapping satisfying (2.5). Then there is a unique cubic mapping such that for all .

Proof. It follows from (2.10) that for all .
The rest of the proof is similar to the proof of Theorem 2.1.

Theorem 2.4. Let be a function such that there exists an with for all . Let be an odd mapping satisfying (2.5). Then there is a unique additive mapping such that for all .

Proof. Letting and for all in (2.9), we get for all .
The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.5. Let and be positive real numbers with . Let be an odd mapping satisfying (2.26). Then there exists a unique additive mapping such that for all .

Proof. The proof follows from Theorem 2.4 by taking for all . Then we can choose and we get the desired result.

Theorem 2.6. Let be a function such that there exists an with for all . Let be an odd mapping satisfying (2.5). Then there is a unique additive mapping such that for all .

Proof. It follows from (2.34) that for all .
The rest of the proof is similar to the proof of Theorem 2.1.

3. Generalized Hyers-Ulam Stability of the Functional Equation (1.10): An Even Case

Now we prove the generalized Hyers-Ulam stability of the functional equation in non-Archimedean Banach spaces: an even case.

Theorem 3.1. Let be a function such that there exists an with for all . Let be an even mapping satisfying (2.5) and . Then there is a unique quartic mapping such that for all .

Proof. Letting in (2.5), we get for all .
Replacing by in (2.5), we get for all .
By (3.3) and (3.4), for all .
Letting and for all , we get for all .
The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 3.2. Let and be positive real numbers with . Let be an even mapping satisfying (2.26) and . Then there exists a unique quartic mapping such that for all .

Proof. The proof follows from Theorem 3.1 by taking for all . Then we can choose and we get the desired result.

Theorem 3.3. Let be a function such that there exists an with for all . Let be an even mapping satisfying (2.5) and . Then there is a unique quartic mapping such that for all .

Proof. It follows from (3.6) that for all .
The rest of the proof is similar to the proof of Theorem 2.1.

Theorem 3.4. Let be a function such that there exists an with for all . Let be an even mapping satisfying (2.5) and . Then there is a unique quadratic mapping such that for all .

Proof. Letting and for all in (3.5), we get for all .
The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 3.5. Let and be positive real numbers with . Let be an even mapping satisfying (2.26) and . Then there exists a unique quadratic mapping such that for all .

Proof. The proof follows from Theorem 3.4 by taking for all . Then we can choose and we get the desired result.

Theorem 3.6. Let be a function such that there exists an with for all . Let be an even mapping satisfying (2.5) and . Then there is a unique quadratic mapping such that for all .

Proof. It follows from (3.14) that for all .
The rest of the proof is similar to the proof of Theorem 2.1.

For a given , let and . Then is odd and is even. Let and . Then . Let and . Then . Thus

Theorem 3.7. Let be a function such that there exists an with for all . Let be a mapping satisfying and (2.5). Then there exist an additive mapping , a quadratic mapping , a cubic mapping , and a quartic mapping such that for all .

Corollary 3.8. Let and be positive real numbers with . Let be a mapping satisfying and (2.5). Then there exist an additive mapping , a quadratic mapping , a cubic mapping and a quartic mapping , such that for all .

Proof. The proof follows from Theorem 3.7 by taking for all . Then we can choose and we get the desired result.

Theorem 3.9. Let be a function such that there exists an with for all . Let be a mapping satisfying and (2.5). Then there exist an additive mapping , a quadratic mapping , a cubic mapping and a quartic mapping such that for all .

Acknowledgment

This work was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788).

References

1. M. S. Moslehian and G. Sadeghi, “A Mazur-Ulam theorem in non-Archimedean normed spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 10, pp. 3405–3408, 2008.
2. S. M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience, New York, NY, USA, 1960. View at: MathSciNet
3. D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222–224, 1941.
4. T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950.
5. Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.
6. P. Găvruţa, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994.
7. F. Skof, “Proprietà locali e approssimazione di operatori,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113–129, 1983. View at: Publisher Site | Google Scholar
8. P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76–86, 1984.
9. S. Czerwik, “On the stability of the quadratic mapping in normed spaces,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 62, pp. 59–64, 1992.
10. K.-W. Jun and H.-M. Kim, “The generalized Hyers-Ulam-Rassias stability of a cubic functional equation,” Journal of Mathematical Analysis and Applications, vol. 274, no. 2, pp. 267–278, 2002.
11. S. H. Lee, S. M. Im, and I. S. Hwang, “Quartic functional equations,” Journal of Mathematical Analysis and Applications, vol. 307, no. 2, pp. 387–394, 2005.
12. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002. View at: MathSciNet
13. Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431–434, 1991.
14. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, vol. 34 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, 1998. View at: MathSciNet
15. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001. View at: MathSciNet
16. C. Park, “Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras,” Bulletin des Sciences Mathématiques, vol. 132, no. 2, pp. 87–96, 2008.
17. C. Park, “Hyers-Ulam-Rassias stability of a generalized Apollonius-Jensen type additive mapping and isomorphisms between ${C}^{\ast }$-algebras,” Mathematische Nachrichten, vol. 281, no. 3, pp. 402–411, 2008. View at: Publisher Site | Google Scholar | MathSciNet
18. C. Park and J. Cui, “Generalized stability of ${C}^{\ast }$-ternary quadratic mappings,” Abstract and Applied Analysis, vol. 2007, Article ID 23282, 6 pages, 2007. View at: Google Scholar | MathSciNet
19. C. Park and A. Najati, “Homomorphisms and derivations in ${C}^{\ast }$-algebras,” Abstract and Applied Analysis, vol. 2007, Article ID 80630, 12 pages, 2007. View at: Google Scholar | MathSciNet
20. Th. M. Rassias, “Problem 16; 2, Report of the 27th International Symposium on Functional Equations,” Aequationes Mathematicae, vol. 39, pp. 292–293; 309, 1990. View at: Google Scholar
21. Th. M. Rassias, “On the stability of the quadratic functional equation and its applications,” Studia Universitatis Babeş-Bolyai. Mathematica, vol. 43, no. 3, pp. 89–124, 1998.
22. Th. M. Rassias, “The problem of S. M. Ulam for approximately multiplicative mappings,” Journal of Mathematical Analysis and Applications, vol. 246, no. 2, pp. 352–378, 2000.
23. Th. M. Rassias, “On the stability of functional equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 264–284, 2000.
24. Th. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23–130, 2000.
25. Th. M. Rassias and P. Šemrl, “On the behavior of mappings which do not satisfy Hyers-Ulam stability,” Proceedings of the American Mathematical Society, vol. 114, no. 4, pp. 989–993, 1992.
26. Th. M. Rassias and P. Šemrl, “On the Hyers-Ulam stability of linear mappings,” Journal of Mathematical Analysis and Applications, vol. 173, no. 2, pp. 325–338, 1993.
27. Th. M. Rassias and K. Shibata, “Variational problem of some quadratic functionals in complex analysis,” Journal of Mathematical Analysis and Applications, vol. 228, no. 1, pp. 234–253, 1998.
28. L. Cădariu and V. Radu, “Fixed points and the stability of Jensen's functional equation,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 4, 2003. View at: Google Scholar | Zentralblatt MATH
29. J. B. Diaz and B. Margolis, “A fixed point theorem of the alternative, for contractions on a generalized complete metric space,” Bulletin of the American Mathematical Society, vol. 74, pp. 305–309, 1968.
30. G. Isac and Th. M. Rassias, “Stability of $\psi$-additive mappings: applications to nonlinear analysis,” International Journal of Mathematics and Mathematical Sciences, vol. 19, no. 2, pp. 219–228, 1996. View at: Publisher Site | Google Scholar | MathSciNet
31. L. Cădariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,” in Iteration Theory (ECIT '02), vol. 346 of Grazer Mathematische Berichte, pp. 43–52, Karl-Franzens-Universitaet Graz, Graz, Austria, 2004.
32. L. Cădariu and V. Radu, “Fixed point methods for the generalized stability of functional equations in a single variable,” Fixed Point Theory and Applications, vol. 2008, Article ID 749392, 15 pages, 2008.
33. M. Mirzavaziri and M. S. Moslehian, “A fixed point approach to stability of a quadratic equation,” Bulletin of the Brazilian Mathematical Society, vol. 37, no. 3, pp. 361–376, 2006.
34. C. Park, “Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras,” Fixed Point Theory and Applications, vol. 2007, Article ID 50175, 15 pages, 2007.
35. C. Park, “Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach,” Fixed Point Theory and Applications, vol. 2008, Article ID 493751, 9 pages, 2008.
36. V. Radu, “The fixed point alternative and the stability of functional equations,” Fixed Point Theory, vol. 4, no. 1, pp. 91–96, 2003.
37. M. Eshaghi-Gordji, S. Kaboli-Gharetapeh, C. Park, and S. Zolfaghari, “Stability of an additive-cubic-quartic functional equation,” Advances in Difference Equations, vol. 2009, Article ID 395693, 20 pages, 2009. View at: Google Scholar | MathSciNet
38. M. Eshaghi Gordji, S. Abbaszadeh, and C. Park, “On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces,” Journal of Inequalities and Applications, vol. 2009, Article ID 153084, 26 pages, 2009. View at: Publisher Site | Google Scholar | MathSciNet
39. D. Miheţ and V. Radu, “On the stability of the additive Cauchy functional equation in random normed spaces,” Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 567–572, 2008.

Copyright © 2010 Choonkil Park. 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.