Hyers-Ulam-Rassias stability of the additive-quadratic mappings in non-Archimedean Banach spaces

Using the fixed point and direct methods, we prove the generalized Hyers-Ulam stability of the following additive-quadratic functional equation in non-Archimedean normed spaces

where r, s, γ are positive real numbers.

MSC: 39B55, 46S10.

A classical question in the theory of functional equations is the following: ‘When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of the equation?’ If the problem accepts a solution, we say that the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [44] in 1940. In the next year, Hyers [23] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces. In 1978, Rassias [39] proved a generalization of Hyers’ theorem for additive mappings. This new concept is known as generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. Furthermore, in 1994, a generalization of Rassias’ theorem was obtained by Gǎvruta [21] by replacing the bound by a general control function .

In 1983, a generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [43] for mappings , where X is a normed space and Y is a Banach space. In 1984, Cholewa [11] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group and, in 2002, Czerwik [13] proved the generalized Hyers-Ulam stability of the quadratic functional equation. The reader is referred to [1-42] and references therein for detailed information on stability of functional equations.

In 1897, Hensel [22] has introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications (see [16,25-27,33]).

Definition 1.1 By a non-Archimedean field, we mean a field equipped with a function (valuation) such that for all , the following conditions hold:

(1) if and only if ;

(2) ;

(3) .

Definition 1.2 Let X be a vector space over a scalar field with a non-Archimedean non-trivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions:

(1) if and only if ;

(2) (, );

(3) The strong triangle inequality (ultrametric); namely,

Then is called a non-Archimedean space.

Due to the fact that

Definition 1.3 A sequence is Cauchy if and only if converges to zero in a non-Archimedean space. By a complete non-Archimedean space we mean one in which every Cauchy sequence is convergent.

Definition 1.4 Let X be a set. A function is called a generalized metric on X if d satisfies

(1) if and only if ;

(2) for all ;

(3) for all .

We recall a fundamental result in fixed point theory.

Theorem 1.5 ([13,17])

Letbe a complete generalized metric space and letbe a strictly contractive mapping with Lipschitz constant. Then for each given element, either

for all nonnegative integersnor there exists a positive integersuch that

(1) , ;

(2) the sequenceconverges to a fixed pointofJ;

(3) is the unique fixed point ofJin the set;

(4) for all.

In 1996, G. Isac and Th. M. Rassias [24] 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 [14,15,35,36,40]).

This paper is organized as follows: In Section 2, using the fixed-point method, we prove the Hyers-Ulam stability of the following additive-quadratic functional equation:

where , in non-Archimedean normed space. In Section 3, using direct methods, we prove the Hyers-Ulam stability of the additive-quadratic functional equation (1.1) in non-Archimedean normed spaces.

It is easy to see that a mapping f with is a solution of equation (1.1) if and only if f is of the form for all .

In this section, we deal with the stability problem for the additive-quadratic functional equation (1.1). In the rest of the present article, let .

Theorem 2.1LetXis a non-Archimedean normed space and thatYbe a complete non-Archimedean space. Letbe a function such that there exists anwith

for all. Letbe an odd mapping satisfying

for all. Then there exists a unique additive mappingsuch that

for all.

Proof Putting in (2.2) and replacing x by 2x, we get

for all . Putting and in (2.2), we have

for all . By (2.4) and (2.5), we get

Consider the set and introduce the generalized metric on S:

where, as usual, . It is easy to show that is complete (see [30]). 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 (2.6) that . By Theorem 1.5, there exists a mapping satisfying the following:

(1) A is a fixed point of J, i.e.,

for all . The mapping A is a unique fixed point of J in the set . This implies that A is a unique mapping satisfying (2.7) such that there exists a satisfying for all ;

(2) as . This implies the equality

(3) , which implies the inequality . This implies that the inequalities (2.3) holds.

It follows from (2.1) and (2.2) that

for all . So

for all . Hence, satisfying (1.1). This completes the proof. □

Corollary 2.2Letθbe a positive real number andqis a real number with. Letbe an odd mapping satisfying

for all. Then there exists a unique additive mappingsuch 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 2.3LetXis a non-Archimedean normed space and thatYbe a complete non-Archimedean space. Letbe a function such that there exists anwith

for all. Letbe an odd mapping satisfying (2.2). Then there exists a unique additive mappingsuch that

for all.

Proof Let be the generalized metric space defined in the proof of Theorem 2.1.

Now we consider the linear mapping such that

for all .

Replacing x by in (2.6) and using (2.9), we have

So .

The rest of the proof is similar to the proof of Theorem 2.1. □

Corollary 2.4Letθbe a positive real number andqis a real number with. Letbe an odd mapping satisfying (2.8). Then there exists a unique additive mappingsuch that

for all.

Proof The proof follows from Theorem 2.3 by taking for all . Then we can choose and we get the desired result. □

Theorem 2.5LetXis a non-Archimedean normed space and thatYbe a complete non-Archimedean space. Letbe a function such that there exists anwith

for all. Letbe an even mapping withand satisfying (2.2). Then there exists a unique quadratic mappingsuch that

for all.

Proof Consider the set and the generalized metric in defined by

where, as usual, . It is easy to show that is complete (see [30]). Now we consider the linear mapping such that

for all .

Putting and in (2.2), we have

for all .

Substituting and then replacing x by 2x in (2.2), we obtain

By (2.13) and (2.14), we get

The rest of the proof is similar to the proof of Theorem 2.1. □

Corollary 2.6Letθbe a positive real number andqis a real number with. Letbe an even mapping withand satisfying (2.8). Then there exists a unique quadratic mappingsuch that

for all.

Proof The proof follows from Theorem 2.5 by taking for all . Then we can choose and we get the desired result. □

Theorem 2.7LetXis a non-Archimedean normed space and thatYbe a complete non-Archimedean space. Letbe a function such that there exists anwith

for all. Letbe an even mapping withand satisfying (2.2). Then there exists a unique quadratic mappingsuch that

for all.

Proof It follows from (2.15) that

The rest of the proof is similar to the proof of Theorems 2.1 and 2.5. □

Corollary 2.8Letθbe a positive real number andqis a real number with. Letbe an even mapping withand satisfying (2.8). Then there exists a unique quadratic mappingsuch that

for all.

Proof The proof follows from Theorem 2.7 by taking for all . Then we can choose and we get the desired result. □

Let be a mapping satisfying and (1.1). Let and . Then is an even mapping satisfying (1.1) and is an odd mapping satisfying (1.1) such that .

On the other hand

and

for all , where is the difference operator of the functional equation (1.1). So we obtain the following theorem.

Theorem 2.9LetXis a non-Archimedean normed space and thatYbe a complete non-Archimedean space. Letbe a function such that there exists anwith

for all. Letbe a mapping withand satisfying (2.2). Then there exist a unique additive mappingand a unique quadratic mappingsuch that

for all.

Theorem 2.10LetXis a non-Archimedean normed space and thatYbe a complete non-Archimedean space. Letbe a function such that there exists anwith

for all. Letbe a mapping withand satisfying (2.2). Then there exist a unique additive mappingand a unique quadratic mappingsuch that

for all.

In this section, using direct method, we prove the generalized Hyers-Ulam stability of the additive-quadratic functional equation (1.1) in non-Archimedean space.

Theorem 3.1LetGbe a vector space and thatXis a non-Archimedean Banach space. Assume thatbe a function such that

for all. Suppose that, for any, the limit

exists andbe an odd mapping satisfying

Then the limit

exists for alland defines an additive mappingsuch that

Moreover, if

thenAis the unique additive mapping satisfying (3.4).

Proof By (2.10), we know

for all . Replacing x by in (3.5), we obtain

Thus, it follows from (3.1) and (3.6) that the sequence is a Cauchy sequence. Since X is complete, it follows that is convergent. Set

By induction on n, one can show that

for all and . By taking in (3.7) and using (3.2), one obtains (3.4). By (3.1) and (3.3), we get

for all . Therefore, the mapping satisfies (1.1).

To prove the uniqueness property of A, let L be another mapping satisfying (3.4). Then we have

for all . Therefore, . This completes the proof. □

Corollary 3.2Letbe a function satisfying

for all. Assume thatandbe a mapping withsuch that

for all. Then there exists a unique additive mappingsuch that

Proof Defining by , then we have

for all . The last equality comes form the fact that . On the other hand, it follows that

exists for all . Also, we have

Thus, applying Theorem 3.1, we have the conclusion. This completes the proof. □

Theorem 3.3LetGbe a vector space and thatXis a non-Archimedean Banach space. Assume thatbe a function such that

for all. Suppose that, for any, the limit

exists andbe an odd mapping satisfying (3.3). Then the limitexists for alland

for all. Moreover, if

thenAis the unique mapping satisfying (3.11).

Proof By (2.6), we get

for all . Replacing x by in (3.12), we obtain

Thus, it follows from (3.9) and (3.13) that the sequence is convergent. Set

On the other hand, it follows from (3.13) that

for all and with . Letting , taking in the last inequality and using (3.10), we obtain (3.11).

The rest of the proof is similar to the proof of Theorem 3.1. This completes the proof. □

Theorem 3.4LetGbe a vector space and thatXis a non-Archimedean Banach space. Assume thatbe a function such that

for all. Suppose that, for any, the limit

exists andbe an even mapping withand satisfying (3.3). Then the limitexists for alland defines a quadratic mappingsuch that

Moreover, if

thenQis the unique additive mapping satisfying (3.16).

Proof It follows from (2.15) that

Replacing x by in (3.18), we have

It follows from (3.14) and (3.18) that the sequence is Cauchy sequence. The rest of the proof is similar to the proof of Theorem 3.1. □

Similarly, we can obtain the followings. We will omit the proof.

Theorem 3.5LetGbe a vector space and thatXis a non-Archimedean Banach space. Assume thatbe a function such that

for all. Suppose that, for any, the limit

exists andbe an even mapping withand satisfying (3.3). Then the limitexists for alland

for all. Moreover, if

thenQis the unique mapping satisfying (3.21).

Let be a mapping satisfying and (1.1). Let and . Then is an even mapping satisfying (1.1) and is an odd mapping satisfying (1.1) such that . On the other hand,

and

for all , where is the difference operator of the functional equation (1.1). So we obtain the following theorem.

Theorem 3.6LetGbe a vector space and thatXis a non-Archimedean Banach space. Assume thatbe a function such that

for all. Suppose that the limits

and

exist for allandbe a mapping withand satisfying (3.3). Then there exist an additive mappingand a quadratic mappingsuch that

for all. Moreover, if

and

thenA, Qare the unique mappings satisfying (3.22).

Theorem 3.7LetGbe a vector space and thatXis a non-Archimedean Banach space. Assume thatbe a function such that

for all. Suppose that the limits

and

exist for allandbe a mapping withand satisfying (3.3). Then there exist an additive mappingand a quadratic mappingsuch that

for all. Moreover, if

and

thenA, Qare the unique mappings satisfying (3.23).

We linked here two different disciplines, namely, the non-Archimedean normed spaces and functional equations. We established the generalized Hyers-Ulam stability of the functional equation (1.1) in non-Archimedean normed spaces.

The authors declare that they have no competing interests.

All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

1. Aoki, T: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn.. 2, 64–66 (1950). Publisher Full Text

2. Arriola, LM, Beyer, WA: Stability of the Cauchy functional equation over p-adic fields. Real Anal. Exch.. 31, 125–132 (2005/06)

3. Azadi Kenary, H: Stability of a pexiderial functional equation in random normed spaces. Rend. Circ. Mat. Palermo doi:10.1007/s12215-011-0027-5 (2011)

4. Azadi Kenary, H: Non-Archimedean stability of Cauchy-Jensen type functional equation. Int. J. Nonlinear Anal. Appl.. 2(2), 93–103 (2011)

5. Azadi Kenary, H: Approximate additive functional equations in closed convex cone. J. Math. Ext.. 5(2), 51–65 (2011)

6. Azadi Kenary, H: On the stability of a cubic functional equation in random normed spaces. J. Math. Ext.. 4(1), 105–113 (2009)

7. Azadi Kenary, H, Rezaei, H, Talebzadeh, S, Lee, SJ: Stabilities of cubic mappings in various normed spaces: direct and fixed point methods. J. Appl. Math.. 2012, Article ID 546819. doi:10.1155/2012/546819 (2012)

8. Azadi Kenary, H, Rezaei, H, Ebadian, A, Zohdi, AR: Hyers-Ulam-Rassias RNS approximation of Euler-Lagrange-type additive mappings. Math. Probl. Eng.. 2012, Article ID 672531. doi:10.1155/2012/672531 (2012)

9. Azadi Kenary, H, Shafaat, K, Shafiee, M, Takbiri, G: Hyers-Ulam-Rassias stability of the Apollonius type quadratic mapping in RN-spaces. J. Nonlinear Sci. Appl.. 4(1), 82–91 (2011)

10. Cădariu, L, Radu, V: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math.. 4(1), Article ID 4 (2003)

11. Cholewa, PW: Remarks on the stability of functional equations. Aequ. Math.. 27, 76–86 (1984). Publisher Full Text

12. Chung, J, Sahoo, PK: On the general solution of a quartic functional equation. Bull. Korean Math. Soc.. 40, 565–576 (2003)

13. Czerwik, S: Functional Equations and Inequalities in Several Variables, World Scientific, River Edge (2002)

14. Cădariu, L, Radu, V: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber.. 346, 43–52 (2004)

15. Cădariu, L, Radu, V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl.. 2008, Article ID 749392 (2008)

16. Deses, D: On the representation of non-Archimedean objects. Topol. Appl.. 153, 774–785 (2005). Publisher Full Text

17. Diaz, J, Margolis, B: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc.. 74, 305–309 (1968). Publisher Full Text

18. Eshaghi-Gordji, M, Abbaszadeh, S, Park, C: On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces. J. Inequal. Appl.. 2009, Article ID 153084 (2009)

19. Eshaghi-Gordji, M, Kaboli-Gharetapeh, S, Park, C, Zolfaghri, S: Stability of an additive-cubic-quartic functional equation. Adv. Differ. Equ.. 2009, Article ID 395693 (2009)

20. Fechner, W: Stability of a functional inequality associated with the Jordan-von Neumann functional equation. Aequ. Math.. 71, 149–161 (2006). Publisher Full Text

21. Gǎvruta, P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl.. 184, 431–436 (1994). Publisher Full Text

22. Hensel, K: Ubereine news Begrundung der Theorie der algebraischen Zahlen. Jahresber. Dtsch. Math.-Ver.. 6, 83–88 (1897)

23. Hyers, DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA. 27, 222–224 (1941). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

24. Hyers, DH, Isac, G, Rassias, TM: Stability of Functional Equations in Several Variables, Birkhäuser, Basel (1998)

25. Katsaras, AK, Beoyiannis, A: Tensor products of non-Archimedean weighted spaces of continuous functions. Georgian Math. J.. 6, 33–44 (1999). Publisher Full Text

26. Khrennikov, A: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic, Dordrecht (1997)

27. Kominek, Z: On a local stability of the Jensen functional equation. Demonstr. Math.. 22, 499–507 (1989)

28. Lee, S, Im, S, Hwang, I: Quartic functional equations. J. Math. Anal. Appl.. 307, 387–394 (2005). Publisher Full Text

29. Mohammadi, M, Cho, YJ, Park, C, Vetro, P, Saadati, R: Random stability of an additive-quadratic-quartic functional equation. J. Inequal. Appl.. 2010, Article ID 754210 (2010)

30. Mihet, D, Radu, V: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl.. 343, 567–572 (2008). Publisher Full Text

31. Mursaleen, M, Mohiuddine, SA: On stability of a cubic functional equation in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals. 42, 2997–3005 (2009). Publisher Full Text

32. Najati, A, Park, C: The pexiderized Apollonius-Jensen type additive mapping and isomorphisms between -algebras. J. Differ. Equ. Appl.. 14, 459–479 (2008). Publisher Full Text

33. Nyikos, PJ: On some non-Archimedean spaces of Alexandrof and Urysohn. Topol. Appl.. 91, 1–23 (1999). Publisher Full Text

34. Park, C: Generalized Hyers-Ulam-Rassias stability of n-sesquilinear-quadratic mappings on Banach modules over -algebras. J. Comput. Appl. Math.. 180, 279–291 (2005). Publisher Full Text

35. Park, C: Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory Appl.. 2007, Article ID 50175 (2007)

36. Park, C: Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory Appl.. 2008, Article ID 493751 (2008)

37. Parnami, JC, Vasudeva, HL: On Jensen’s functional equation. Aequ. Math.. 43, 211–218 (1992). Publisher Full Text

38. Radu, V: The fixed point alternative and the stability of functional equations. Fixed Point Theory. 4, 91–96 (2003)

39. Rassias, TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc.. 72, 297–300 (1978). Publisher Full Text

40. Rätz, J: On inequalities associated with the Jordan-von Neumann functional equation. Aequ. Math.. 66, 191–200 (2003). Publisher Full Text

41. Saadati, R, Park, C: Non-Archimedean -fuzzy normed spaces and stability of functional equations. Comput. Math. Appl.. 60, 2488–2496 (2010). Publisher Full Text

42. Saadati, R, Park, JH: On the intuitionistic fuzzy topological spaces. Chaos Solitons Fractals. 27, 331–344 (2006). Publisher Full Text

43. Skof, F: Local properties and approximation of operators. Rend. Semin. Mat. Fis. Milano. 53, 113–129 (1983). Publisher Full Text

44. Ulam, SM: Problems in Modern Mathematics, Wiley, New York (1964)