Research

# On the Ulam stability of mixed type QA mappings in IFN-spaces

Abdulrahman S Al-Fhaid and Syed Abdul Mohiuddine*

Author Affiliations

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

For all author emails, please log on.

Advances in Difference Equations 2013, 2013:203  doi:10.1186/1687-1847-2013-203

 Received: 13 March 2013 Accepted: 15 June 2013 Published: 8 July 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

We give Ulam-type stability results concerning the quadratic-additive functional equation in intuitionistic fuzzy normed spaces.

### 1 Introduction

In 1940, Ulam [1] proposed the following stability problem: ‘When is it true that a function which satisfies some functional equation approximately must be close to one satisfying the equation exactly?’. Hyers [2] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Aoki [3] presented a generalization of Hyers results by considering additive mappings, and later on Rassias [4] did for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias has significantly influenced the development of what we now call the Hyers-Ulam-Rassias stability of functional equations. Various extensions, generalizations and applications of the stability problems have been given by several authors so far; see, for example, [5-24] and references therein.

The notion of intuitionistic fuzzy set introduced by Atanassov [25] has been used extensively in many areas of mathematics and sciences. Using the idea of intuitionistic fuzzy set, Saadati and Park [26] presented the notion of intuitionistic fuzzy normed space which is a generalization of the concept of a fuzzy metric space due to Bag and Samanta [27]. The authors of [28-34] defined and studied some summability problems in the setting of an intuitionistic fuzzy normed space.

In the recent past, several Hyers-Ulam stability results concerning the various functional equations were determined in [35-46], respectively, in the fuzzy and intuitionistic fuzzy normed spaces. Quite recently, Alotaibi and Mohiuddine [47] established the stability of a cubic functional equation in random 2-normed spaces, while the notion of random 2-normed spaces was introduced by Goleţ [48] and further studied in [49-51].

under the approximately even (or odd) condition were established by Jung [52] and the solution of the above functional equation where the range is a field of characteristic 0 was determined by Kannappan [53]. In this paper we determine the stability results concerning the above functional equation in the setting of intuitionistic fuzzy normed spaces. This work indeed presents a relationship between two various disciplines: the theory of fuzzy spaces and the theory of functional equations.

### 2 Definitions and preliminaries

We shall assume throughout this paper that the symbol ℕ denotes the set of all natural numbers.

A binary operation is said to be a continuoust-norm if it satisfies the following conditions:

(a) ∗ is associative and commutative, (b) ∗ is continuous, (c)  for all , (d)  whenever and for each .

A binary operation is said to be a continuoust-conorm if it satisfies the following conditions:

(a′) ♢ is associative and commutative, (b′) ♢ is continuous, (c′) for all , (d′) whenever and for each .

The five-tuple is said to be intuitionistic fuzzy normed spaces (for short, IFN-spaces) [26] if X is a vector space, ∗ is a continuous t-norm, ♢ is a continuous t-conorm, and μ, ν are fuzzy sets on satisfying the following conditions. For every and ,

(i) ,

(ii) ,

(iii) if and only if ,

(iv) for each ,

(v) ,

(vi) is continuous,

(vii) and ,

(viii) ,

(ix) if and only if ,

(x) for each ,

(xi) ,

(xii) is continuous,

(xiii) and .

In this case is called an intuitionistic fuzzy norm. For simplicity in notation, we denote the intuitionistic fuzzy normed spaces by instead of . For example, let be a normed space, and let and for all . For all and every , consider

Then is an intuitionistic fuzzy normed space.

The notions of convergence and Cauchy sequence in the setting of IFN-spaces were introduced by Saadati and Park [26] and further studied by Mursaleen and Mohiuddine [30].

Let be an intuitionistic fuzzy normed space. Then the sequence is said to be:

(i) Convergent to with respect to the intuitionistic fuzzy norm if, for every and , there exists such that and for all . In this case, we write or as .

(ii) Cauchysequence with respect to the intuitionistic fuzzy norm if, for every and , there exists such that and for all . An IFN-space is said to be complete if every Cauchy sequence in is convergent in the IFN-space. In this case, is called an intuitionistic fuzzy Banach space.

We shall assume the following abbreviation throughout this paper:

Theorem 3.1LetXbe a linear space andbe an IFN-space. Suppose thatfis an intuitionistic fuzzyq-almost quadratic-additive mapping fromto an intuitionistic fuzzy Banach spacesuch that

(3.1)

for alland, whereqis a positive real number with. Then there exists a unique quadratic-additive mappingsuch that

(3.2)

for alland allwith, where.

Proof Putting in (3.1), it follows that

and

for all . Using the definition of IFN-space, we have . Now we are ready to prove our theorem for three cases. We consider the cases as , and .

Case 1. Let . Consider a mapping to be such that

for all . Notice that and

(3.3)

for all and . Using the definition of IFN-space and (3.1), this equation implies that if , then

(3.4)

and

(3.5)

for all and , where , . Let and be given. Since and , there exists such that and for all . We observe that for some , the series converges for , there exists some such that for each and . Using (3.4) and (3.5), we have

and

for all and . Hence is a Cauchy sequence in the fuzzy Banach space . Thus, we define a mapping such that for all . Moreover, if we put in (3.4) and (3.5), we get

(3.6)

for all and . Now we have to show that T is quadratic additive. Let . Then

(3.7)

and

(3.8)

for all and . Taking the limit as in the inequalities (3.7) and (3.8), we can see that first seven terms on the right-hand side of (3.7) and (3.8) tend to 1 and 0, respectively, by using the definition of T. It is left to find the value of the last term on the right-hand side of (3.7) and (3.8). By using the definition of , write

(3.9)

and, similarly,

(3.10)

for all , and . Also, from (3.1), we have

(3.11)

and

(3.12)

for all , and . Since , therefore (3.9) tends to 1 as with the help of (3.11) and (3.12). Similarly, by proceeding along the same lines as in (3.11) and (3.12), we can show that (3.10) tends to 0 as . Thus, inequalities (3.7) and (3.8) become

for all and . Accordingly, for all . Now we approximate the difference between f and T in a fuzzy sense. Choose and . Since T is the intuitionistic fuzzy limit of such that

for all , and . From (3.6), we have

and

Since is arbitrary, we get the inequality (3.2) in this case.

To prove the uniqueness of T, assume that is another quadratic-additive mapping from X into Y, which satisfies the required inequality, i.e., (3.2). Then, by (3.3), for all and

(3.13)

Therefore

and

for all , and . Since and taking limit as in the last two inequalities, we get and for all and . Hence for all .

Case 2. Let . Consider a mapping to be such that

for all . Then and

for all and . Thus, for each , we have

where ∏ and ∐ are the same as in Case 1. Proceeding along a similar argument as in Case 1, we see that is a Cauchy sequence in . Thus, we define for all . Putting in the last two inequalities, we get

(3.14)

for all and . To prove that t is a quadratic-additive function, it is enough to show that the last term on the right-hand side of (3.7) and (3.8) tends to 1 and 0, respectively, as . Using the definition of and (3.1), we obtain

(3.15)

and

(3.16)

for each , and . Since and taking the limit as , we see that (3.15) and (3.16) tend to 1 and 0, respectively. As in Case 1, we have for all . Using the same argument as in Case 1, we see that (3.2) follows from (3.14). To prove the uniqueness of T, assume that is another quadratic-additive mapping from X into Y satisfying (3.2). Using (3.2) and (3.13), we have

(3.17)

and

(3.18)

for all , and . Letting in (3.17) and (3.18), and using the fact that together with the definition of IFN-space, we get and for all and . Hence for all .

Case 3. Let . Define a mapping by

for all . In this case, and

for all and . Thus, for each , we have

for all and . Proceeding along a similar argument as in the previous cases, we see that is a Cauchy sequence in . Thus, we define for all . Putting in the last two inequalities, we get

(3.19)

for all and . Write

(3.20)

and

(3.21)

for all , and . Since and taking the limit as , we see that (3.20) and (3.21) tend to 1 and 0, respectively. As in the previous cases, we have that for all . By the same argument as in previous cases, we can see that (3.2) follows from (3.19). To prove the uniqueness of T, assume that is another quadratic-additive mapping from X into Y satisfying (3.2). From (3.2) and (3.13), for all and , write

and, similarly,

for . Letting in (3.17) and (3.18), and using the fact that together with the definition of IFN-space, we get and for all and . Hence for all . □

Remark 3.2 Let be an IFN-space and be an intuitionistic fuzzy Banach space . Let be a mapping satisfying (3.1) with a real number and for all . If we choose a real number α with , then

for all , and . Since , we have . This implies that

Thus, we have and for all and . Hence for all . In other words, if f is an intuitionistic fuzzy q-almost quadratic-additive mapping for the case , then f is itself a quadratic-additive mapping.

Corollary 3.3Suppose thatfis an even mapping satisfying the conditions of Theorem 3.1. Then there exists a unique quadratic mappingsuch that

(3.22)

for alland, where.

Proof Since f is an even mapping, we get

for all , where is defined as in Theorem 3.1. In this case, . For all and , we have

Proceeding along the same lines as in Theorem 3.1, we obtain that T is a quadratic-additive function satisfying (3.22). Notice that , T is even and for all . Hence, we get

for all . It follows that T is a quadratic mapping. □

Corollary 3.4Suppose thatfis an even mapping satisfying the conditions of Theorem 3.1. Then there exists a unique additive mappingsuch that

(3.23)

for alland, where.

Proof Since f is an odd mapping, we get

for all , where is defined as in Theorem 3.1. Here . For all and , we have

Proceeding along the same lines as in Theorem 3.1, we obtain that T is a quadratic-additive function satisfying (3.23). Here , T is odd and for all . Hence, we obtain

for all . It follows that T is an additive mapping. □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors contributed equally and significantly in writing this paper. Both the authors read and approved the final manuscript.

### Acknowledgements

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. (405/130/1433). The authors, therefore, acknowledge with thanks DSR technical and financial support.

### References

1. Ulam, SM: A Collection of the Mathematical Problems, Interscience, New York (1960)

2. 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

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

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

5. Rassias, TM: On the stability of functional equations and a problem of Ulam. Acta Appl. Math.. 62, 123–130 (2000)

6. Agarwal, RP, Xu, B, Zhang, W: Stability of functional equations in single variable. J. Math. Anal. Appl.. 288, 852–869 (2003). Publisher Full Text

7. Gajda, Z: On stability of additive mappings. Int. J. Math. Math. Sci.. 14, 431–434 (1991). Publisher Full Text

8. 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

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

10. Isac, G, Rassias, TM: On the Hyers-Ulam stability of ψ-additive mappings. J. Approx. Theory. 72, 131–137 (1993). Publisher Full Text

11. Najati, A, Park, C: On the stability of an n-dimensional functional equation originating from quadratic forms. Taiwan. J. Math.. 12, 1609–1624 (2008)

12. Lu, G, Park, C: Additive functional inequalities in Banach spaces. J. Inequal. Appl.. 2012, (2012) Article ID 294

13. Rassias, JM, Kim, H-M: Generalized Hyers-Ulam stability for general additive functional equations in quasi-β-normed spaces. J. Math. Anal. Appl.. 356, 302–309 (2009). Publisher Full Text

14. Rassias, JM: Solution of a problem of Ulam. J. Approx. Theory. 57, 268–273 (1989). Publisher Full Text

15. Rassias, JM: On the Ulam stability of mixed type mappings on restricted domains. J. Math. Anal. Appl.. 276, 747–762 (2002). Publisher Full Text

16. Dadipour, F, Moslehian, MS, Rassias, JM, Takahasi, S-E: Characterization of a generalized triangle inequality in normed spaces. Nonlinear Anal.. 75, 735–741 (2012). Publisher Full Text

17. Eskandani, GZ, Rassias, JM, Gavruta, P: Generalized Hyers-Ulam stability for a general cubic functional equation in quasi-β-normed spaces. Asian-Eur. J. Math.. 4, 413–425 (2011). Publisher Full Text

18. Faziev, V, Sahoo, PK: On the stability of Jensen’s functional equation on groups. Proc. Indian Acad. Sci. Math. Sci.. 117, 31–48 (2007). Publisher Full Text

19. Gordji, ME, Khodaei, H, Rassias, JM: Fixed point methods for the stability of general quadratic functional equation. Fixed Point Theory. 12, 71–82 (2011)

20. Jun, KW, Kim, HM: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J. Math. Anal. Appl.. 274, 867–878 (2002). Publisher Full Text

21. Jung, SM: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor (2001)

22. Ravi, K, Arunkumar, M, Rassias, JM: Ulam stability for the orthogonally general Euler-Lagrange type functional equation. Int. J. Math. Stat.. 3(A08), 36–46 (2008)

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

24. Xu, TZ, Rassias, MJ, Xu, WX, Rassias, JM: A fixed point approach to the intuitionistic fuzzy stability of quintic and sextic functional equations. Iranian J. Fuzzy Sys.. 9, 21–40 (2012)

25. Atanassov, K: Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia, June 1983 (Deposed in Central Science-Technical Library of Bulg. Academy of Science, 1697/84) (in Bulgarian)

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

27. Bag, T, Samanta, SK: Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math.. 11(3), 687–705 (2003)

28. Mohiuddine, SA, Danish Lohani, QM: On generalized statistical convergence in intuitionistic fuzzy normed space. Chaos Solitons Fractals. 42, 1731–1737 (2009). Publisher Full Text

29. Mursaleen, M, Karakaya, V, Mohiuddine, SA: Schauder basis, separability, and approximation property in intuitionistic fuzzy normed space. Abstr. Appl. Anal.. 2010, (2010) Article ID 131868

30. Mursaleen, M, Mohiuddine, SA: Statistical convergence of double sequences in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals. 41, 2414–2421 (2009). Publisher Full Text

31. Mursaleen, M, Mohiuddine, SA: On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space. J. Comput. Appl. Math.. 233(2), 142–149 (2009). Publisher Full Text

32. Mursaleen, M, Mohiuddine, SA: Nonlinear operators between intuitionistic fuzzy normed spaces and Fréchet differentiation. Chaos Solitons Fractals. 42, 1010–1015 (2009). Publisher Full Text

33. Mursaleen, M, Mohiuddine, SA, Edely, OHH: On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces. Comput. Math. Appl.. 59, 603–611 (2010). Publisher Full Text

34. Yilmaz, Y: On some basic properties of differentiation in intuitionistic fuzzy normed spaces. Math. Comput. Model.. 52, 448–458 (2010). Publisher Full Text

35. Jin, SS, Lee, Y-H: Fuzzy stability of a mixed type functional equation. J. Inequal. Appl.. 2011, (2011) Article ID 70

36. Mohiuddine, SA, Alotaibi, A: Fuzzy stability of a cubic functional equation via fixed point technique. Adv. Differ. Equ.. 2012, (2012) Article ID 48

37. Mohiuddine, SA, Alotaibi, A, Obaid, M: Stability of various functional equations in non-Archimedean intuitionistic fuzzy normed spaces. Discrete Dyn. Nat. Soc.. 2012, (2012) Article ID 234727

38. Mohiuddine, SA, Alghamdi, MA: Stability of functional equation obtained through a fixed-point alternative in intuitionistic fuzzy normed spaces. Adv. Differ. Equ.. 2012, (2012) Article ID 141

39. Mohiuddine, SA, Şevli, H: Stability of pexiderized quadratic functional equation in intuitionistic fuzzy normed space. J. Comput. Appl. Math.. 235, 2137–2146 (2011). PubMed Abstract | Publisher Full Text

40. Mohiuddine, SA, Cancan, M, Şevli, H: Intuitionistic fuzzy stability of a Jensen functional equation via fixed point technique. Math. Comput. Model.. 54, 2403–2409 (2011). Publisher Full Text

41. Mohiuddine, SA: Stability of Jensen functional equation in intuitionistic fuzzy normed space. Chaos Solitons Fractals. 42, 2989–2996 (2009). Publisher Full Text

42. 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

43. Wang, Z, Rassias, TM: Intuitionistic fuzzy stability of functional equations associated with inner product spaces. Abstr. Appl. Anal.. 2011, (2011) Article ID 456182

44. Xu, TZ, Rassias, JM, Xu, WX: Intuitionistic fuzzy stability of a general mixed additive-cubic equation. J. Math. Phys.. 51, (2010) Article ID 063519

45. Xu, TZ, Rassias, JM, Xu, WX: Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces. J. Math. Phys.. 51, (2010) Article ID 093508

46. Xu, TZ, Rassias, JM: Stability of general multi-Euler-Lagrange quadratic functional equations in non-Archimedean fuzzy normed spaces. Adv. Differ. Equ.. 2012, (2012) Article ID 119

47. Alotaibi, A, Mohiuddine, SA: On the stability of a cubic functional equation in random 2-normed spaces. Adv. Differ. Equ.. 2012, (2012) Article ID 39

48. Goleţ, I: On probabilistic 2-normed spaces. Novi Sad J. Math.. 35(1), 95–102 (2005)

49. Mohiuddine, SA, Aiyub, M: Lacunary statistical convergence in random 2-normed spaces. Appl. Math. Inform. Sci.. 6(3), 581–585 (2012)

50. Mursaleen, M: On statistical convergence in random 2-normed spaces. Acta Sci. Math.. 76, 101–109 (2010)

51. Mohiuddine, SA, Alotaibi, A, Alsulami, SM: Ideal convergence of double sequences in random 2-normed spaces. Adv. Differ. Equ.. 2012, (2012) Article ID 149

52. Jung, S-M: On the Hyers-Ulam stability of the functional equations that have the quadratic property. J. Math. Anal. Appl.. 222, 126–137 (1998). Publisher Full Text

53. Kannappan, P: Quadratic functional equation and inner product spaces. Results Math.. 27, 368–372 (1995). Publisher Full Text