Abstract
We give Ulamtype stability results concerning the quadraticadditive functional equation in intuitionistic fuzzy normed spaces.
Keywords:
tnorm; tconorm; quadraticadditive functional equation; intuitionistic fuzzy normed space; HyersUlam stability1 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 HyersUlamRassias stability of functional equations. Various extensions, generalizations and applications of the stability problems have been given by several authors so far; see, for example, [524] 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 [2834] defined and studied some summability problems in the setting of an intuitionistic fuzzy normed space.
In the recent past, several HyersUlam stability results concerning the various functional equations were determined in [3546], 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 2normed spaces, while the notion of random 2normed spaces was introduced by Goleţ [48] and further studied in [4951].
The HyersUlam stability problems of quadraticadditive functional equation
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 continuoustnorm 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 continuoustconorm if it satisfies the following conditions:
(a′) ♢ is associative and commutative, (b′) ♢ is continuous, (c′) for all , (d′) whenever and for each .
The fivetuple is said to be intuitionistic fuzzy normed spaces (for short, IFNspaces) [26] if X is a vector space, ∗ is a continuous tnorm, ♢ is a continuous tconorm, and μ, ν are fuzzy sets on satisfying the following conditions. For every 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 IFNspaces 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 IFNspace is said to be complete if every Cauchy sequence in is convergent in the IFNspace. In this case, is called an intuitionistic fuzzy Banach space.
3 Stability of a quadraticadditive functional equation in the IFNspace
We shall assume the following abbreviation throughout this paper:
Theorem 3.1LetXbe a linear space andbe an IFNspace. Suppose thatfis an intuitionistic fuzzyqalmost quadraticadditive mapping fromto an intuitionistic fuzzy Banach spacesuch that
for alland, whereqis a positive real number with. Then there exists a unique quadraticadditive mappingsuch that
Proof Putting in (3.1), it follows that
and
for all . Using the definition of IFNspace, 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 and . Using the definition of IFNspace and (3.1), this equation implies that if , then
and
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
for all and . Now we have to show that T is quadratic additive. Let . Then
and
for all and . Taking the limit as in the inequalities (3.7) and (3.8), we can see that first seven terms on the righthand 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 righthand side of (3.7) and (3.8). By using the definition of , write
and, similarly,
for all , and . Also, from (3.1), we have
and
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 quadraticadditive mapping from X into Y, which satisfies the required inequality, i.e., (3.2). Then, by (3.3), for all and
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 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
for all and . To prove that t is a quadraticadditive function, it is enough to show that the last term on the righthand side of (3.7) and (3.8) tends to 1 and 0, respectively, as . Using the definition of and (3.1), we obtain
and
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 quadraticadditive mapping from X into Y satisfying (3.2). Using (3.2) and (3.13), we have
and
for all , and . Letting in (3.17) and (3.18), and using the fact that together with the definition of IFNspace, we get and for all and . Hence for all .
Case 3. Let . Define a mapping by
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
and
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 quadraticadditive 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 IFNspace, we get and for all and . Hence for all . □
Remark 3.2 Let be an IFNspace 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 qalmost quadraticadditive mapping for the case , then f is itself a quadraticadditive mapping.
Corollary 3.3Suppose thatfis an even mapping satisfying the conditions of Theorem 3.1. Then there exists a unique quadratic mappingsuch that
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 quadraticadditive 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
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 quadraticadditive function satisfying (3.23). Here , T is odd and for all . Hence, we obtain
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

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

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

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

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

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

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

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

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

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

Isac, G, Rassias, TM: On the HyersUlam stability of ψadditive mappings. J. Approx. Theory. 72, 131–137 (1993). Publisher Full Text

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

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

Rassias, JM, Kim, HM: Generalized HyersUlam stability for general additive functional equations in quasiβnormed spaces. J. Math. Anal. Appl.. 356, 302–309 (2009). Publisher Full Text

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

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

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

Eskandani, GZ, Rassias, JM, Gavruta, P: Generalized HyersUlam stability for a general cubic functional equation in quasiβnormed spaces. AsianEur. J. Math.. 4, 413–425 (2011). Publisher Full Text

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

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

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

Jung, SM: HyersUlamRassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor (2001)

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

Saadati, R, Park, C: NonArchimedean ℒfuzzy normed spaces and stability of functional equations. Comput. Math. Appl.. 60, 2488–2496 (2010). Publisher Full Text

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Xu, TZ, Rassias, JM, Xu, WX: Stability of a general mixed additivecubic functional equation in nonArchimedean fuzzy normed spaces. J. Math. Phys.. 51, (2010) Article ID 093508

Xu, TZ, Rassias, JM: Stability of general multiEulerLagrange quadratic functional equations in nonArchimedean fuzzy normed spaces. Adv. Differ. Equ.. 2012, (2012) Article ID 119

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

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

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

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

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

Jung, SM: On the HyersUlam stability of the functional equations that have the quadratic property. J. Math. Anal. Appl.. 222, 126–137 (1998). Publisher Full Text

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