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Stability of Pexiderized Quadratic Functional Equation in Random 2-Normed Spaces
The aim of this paper is to investigate the stability of Hyers-Ulam-Rassias type theorems by considering the pexiderized quadratic functional equation in the setting of random 2-normed spaces (RTNS), while the concept of random 2-normed space has been recently studied by Goleţ (2005).
1. Introduction and Preliminaries
In 1940, Ulam  proposed the famous “Ulam stability problem,” which was solved by Hyers , in 1941, for additive mappings. In 1950, Aoki  solved this Ulam problem for weaker additive mappings; for some historical comments regarding the work of Aoki we refer to . In 1978, Rassias  generalized the theorem of Hyers for linear mappings in which the Cauchy difference is allowed to be unbounded by replacing with a function depending on and in the Hyers theorem. The generalization of Hyers theorem was also presented by Rassias [6–9] in 1982–1989. Some important Ulam stability problems on Cauchy equation on semigroups, approximately additive mappings, and Jensen equation have been investigated by Gajda , Găvruta , and Jung , respectively. Until now, the stability problems for different types of functional equations in various spaces have been extensively studied, for instance, by Mirmostafaee and Moslehian [13, 14], Rassias , Chang et al. [16, 17], Xu et al. , Jun and Kim , Mursaleen et al. [20–22], and many others. Also very interesting results on additive, quadratic, and cubic functional equations have been achieved by Mohiuddine et al. [23–29]. This paper is inspired from the work of Alotaibi and Mohiuddine  in which they solved stability problem for cubic functional equation in random 2-normed spaces.
The pexiderized quadratic functional equation is of the form . For , it is called the quadratic functional equation.
A function is called a distribution function if it is nondecreasing and is left continuous with and . By , we denote the set of all distribution functions such that .
If , then , where It is obvious that for all .
A -norm is a continuous mapping such that is abelian monoid with unit one and if and for all . A triangle function is a binary operation on which is commutative and associative and for every .
Gähler  presented the following notion of 2-normed space.
Let be a linear space of a dimension (). A function is called 2-normed on if it satisfied the following conditions: for every , (i) if and only if and are linearly dependent; (ii) ; (iii) for every ; and (iv) for every . In this case, is called a 2-norm space.
Goleţ  defined and studied the notion of random 2-normed space with the help of 2-norm of Gähler . Recently, the notion of statistical convergence and lacunary statistical convergence have been studied by Mursaleen  and Mohiuddine and Aiyub , respectively, in random 2-normed spaces.
Let be a linear space of a dimension greater than one and let be a triangle function. A function is called a probabilistic 2-norm on if it satisfies the following conditions:(i) () if and are linearly dependent,(ii) if and are linearly independent,(iii),(iv) for all , and ,(v) whenever ,where denotes the value of at and the triple is called a probabilistic 2-normed space. If we replaced (v) by (v′), for all and ,then triple is called a random 2-normed space (RTNS).
Example A. Let be a 2-normed space with , , , and for . For all , , and nonzero , consider Then is a RTNS.
We remark that every 2-normed space can be made RTNS by considering , for every , , and , where .
The notions of convergence and Cauchy sequences have been recently studied by Alotaibi and Mohiuddine  in the setting of RTNS.
Let be a RTNS. Then, a sequence is said to be(i)convergent in (-convergent) to if for every and there exists such that whenever and nonzero . In this case we write -;(ii)Cauchy sequence in (-Cauchy) if for every , , and nonzero there exists a number such that for all . We say that RTNS is if every -Cauchy sequence is -convergent. A complete RTNS is called random 2-Banach space.
2. Main Results
Throughout the paper, by , , and , we denote linear space, random 2-normed space, and random 2-Banach space, respectively. Firstly, we prove the stability of the pexiderized quadratic functional equation in RTNS for an odd case.
Let be a function from to . A mapping is said to be -approximately pexiderized quadratic function if there exist mappings such that for all , , and nonzero .
Theorem 1. Suppose that and are odd functions from to satisfying (3). If for some real number with for all , then there exists a unique additive mapping such that where for all , , and nonzero .
Proof. Replacing by and by in (3), we obtain for all , , and nonzero . It follows from (3) and (7) that Substituting in (8), we get From (8) and (9), we conclude that for every and nonzero . Then, by our assumption, Taking in (10), for all , , and nonzero , we get Putting in (12), we haveThus, Therefore, for each , where . Let and be given. With the help of the definition of RTNS, we have and, therefore, we can find some such that . The convergence of the series gives some such that for each , . Therefore, It follows that is a Cauchy sequence in . Since is complete RTNS, this sequence converges to some point in ; that is, . Therefore, a mapping from to is defined by -. Fix and . From (10), we get thatfor all . Moreover, for all . From (17) and (18), we obtain Thus, . Now by taking (15) with , we getIt follows from (9) and (20) that Thus, we obtained (5). Now we will prove the uniqueness of . For this, we assume that is another additive mapping from into , which satisfies the required inequality. Since, for each , and , then We obtain with the help of the definition of RTNS that Therefore, , for all , , and nonzero . Hence, for all .
Now, we are going to prove the stability of the pexiderized quadratic functional equation in RTNS for an even case.
Theorem 2. If (4) holds for , let , , and be three even functions from to such that and satisfies (3). Then there is a unique quadratic mapping such that, for every , , and nonzero , where is defined by (6).
Proof. Substitute by and by in (3). Then, for all , , and nonzero , we obtain Again substituting in (3), we get Putting in (3), we get For , (3) becomes It follows from (25), (27), and (28) that By substituting in (29), we get From (4), we obtain for every , nonzero and for each . It follows from (30) and (31) that From (32), we obtain or, equivalently, Therefore, for all , , and nonzero and for each where is the same as Theorem 1. Given and , since , there is some such that . By the convergence of , we can find some such that for each . This gives thatWe see that is a Cauchy sequence in and so it is convergent to some point . Therefore, a mapping from to is defined by -. Fix and . Thus, (29) gives thatfor all . Furthermore,Equations (37) and (38) give that for all , , and nonzero . Thus, . Using (35) with , we get for sufficiently large . From (28) and (40), we conclude that Thus, Similarly, one can show that the above inequality also holds for . We obtain the uniqueness assertion of this theorem by proceeding the same lines as in Theorem 1.
Theorem 3. Suppose that (4) holds with . If a map satisfies for all , , and nonzero with . Then, there are unique mappings such that is additive, is quadratic, and for all , , and nonzero , where
Proof. Passing to the odd part and even part of , we deduce from (43) that On the other hand,With the help of the proofs of Theorems 1 and 2, we obtain unique additive and quadratic mappings and , respectively, satisfying Therefore, for all , , and nonzero .
Remark 4. Let be a 2-inner product space. We can define a 2-norm on by for all . In this case, parallelogram law is given by for all (for more details of 2-inner product space we refer to ).
Now we give the following illustrative example.
Example 5. Let be a 2-inner product space. Let be a 2-normed space such that , where and . Suppose that for all . Suppose that and are two random 2-norms on and , respectively, which are given by Example A. Suppose that the random 2-norm makes into an random 2-Banach space. Fixing and , we define for each . Using parallelogram law, one can easily verify thatfor all . Therefore,for each , , and nonzero . Moreover, for each . We can see that the conditions of Theorems 1 and 2 for and are satisfied. It follows that odd and even parts of can be approximated by linear and quadratic functions, respectively. In fact , the odd part of and , is linear. The even part of is , and contains a quadratic . Also
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
- S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publishers, New York, NY, USA, 1960.
- 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.
- T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, no. 1-2, pp. 64–66, 1950.
- M. S. Moslehian and T. M. Rassias, “Stability of functional equations in non-Archimedean spaces,” Applicable Analysis and Discrete Mathematics, vol. 1, no. 2, pp. 325–334, 2007.
- T. 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.
- J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Journal of Functional Analysis, vol. 46, no. 1, pp. 126–130, 1982.
- J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,” Bulletin des Sciences Mathématiques, vol. 108, no. 2, pp. 445–446, 1984.
- J. M. Rassias, “On a new approximation of approximately linear mappings by linear mappings,” Discussiones Mathematicae, vol. 7, pp. 193–196, 1985.
- J. M. Rassias, “Solution of a problem of Ulam,” Journal of Approximation Theory, vol. 57, no. 3, pp. 268–273, 1989.
- Z. Gajda, “On stability of the Cauchy equation on semigroups,” Aequationes Mathematicae, vol. 36, no. 1, pp. 76–79, 1988.
- P. Găvruta, “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.
- S.-M. Jung, “Hyers-Ulam-Rassias stability of Jensen's equation and its application,” Proceedings of the American Mathematical Society, vol. 126, no. 11, pp. 3137–3143, 1998.
- A. K. Mirmostafaee and M. S. Moslehian, “Fuzzy almost quadratic functions,” Results in Mathematics, vol. 52, no. 1-2, pp. 161–177, 2008.
- A. K. Mirmostafaee and M. S. Moslehian, “Fuzzy versions of Hyers-Ulam-Rassias theorem,” Fuzzy Sets and Systems, vol. 159, no. 6, pp. 720–729, 2008.
- T. M. Rassias, “On the stability of the quadratic functional equation and its applications,” Studia Universitatis Babes-Bolyai—Studia Mathematica, vol. 43, no. 3, pp. 89–124, 1998.
- I.-S. Chang, “Higher ring derivation and intuitionistic fuzzy stability,” Abstract and Applied Analysis, vol. 2012, Article ID 503671, 16 pages, 2012.
- J. Roh and I.-S. Chang, “On the intuitionistic fuzzy stability of ring homomorphism and ring derivation,” Abstract and Applied Analysis, vol. 2013, Article ID 192845, 8 pages, 2013.
- T. Z. Xu, J. M. Rassias, and W. X. Xu, “Intuitionistic fuzzy stability of a general mixed additive-cubic equation,” Journal of Mathematical Physics, vol. 51, no. 6, Article ID 063519, 21 pages, 2010.
- K.-W. Jun and H.-M. Kim, “On the Hyers-Ulam stability of a generalized quadratic and additive functional equation,” Bulletin of the Korean Mathematical Society, vol. 42, no. 1, pp. 133–148, 2005.
- M. Mursaleen and K. J. Ansari, “Stability results in intuitionistic fuzzy normed spaces for a cubic functional equation,” Applied Mathematics & Information Sciences, vol. 7, no. 5, pp. 1677–1684, 2013.
- M. Mursaleen and S. A. Mohiuddine, “On stability of a cubic functional equation in intuitionistic fuzzy normed spaces,” Chaos, Solitons & Fractals, vol. 42, no. 5, pp. 2997–3005, 2009.
- A. Alotaibi, M. Mursaleen, H. Dutta, and S. A. Mohiuddine, “On the Ulam stability of Cauchy functional equation in IFN-spaces,” Applied Mathematics & Information Sciences, vol. 8, no. 3, pp. 1135–1143, 2014.
- S. A. Mohiuddine and H. Şevli, “Stability of Pexiderized quadratic functional equation in intuitionistic fuzzy normed space,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2137–2146, 2011.
- S. A. Mohiuddine, M. Cancan, and H. Şevli, “Intuitionistic fuzzy stability of a Jensen functional equation via fixed point technique,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2403–2409, 2011.
- S. A. Mohiuddine, “Stability of Jensen functional equation in intuitionistic fuzzy normed space,” Chaos, Solitons & Fractals, vol. 42, no. 5, pp. 2989–2996, 2009.
- S. A. Mohiuddine and A. Alotaibi, “Fuzzy stability of a cubic functional equation via fixed point technique,” Advances in Difference Equations, vol. 2012, article 48, 2012.
- S. A. Mohiuddine and M. A. Alghamdi, “Stability of functional equation obtained through a fixed-point alternative in intuitionistic fuzzy normed spaces,” Advances in Difference Equations, vol. 2012, article 141, 2012.
- S. A. Mohiuddine, A. Alotaibi, and M. Obaid, “Stability of various functional equations in non-Archimedean intuitionistic fuzzy normed spaces,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 234727, 16 pages, 2012.
- A. S. Al-Fhaid and S. A. Mohiuddine, “On the Ulam stability of mixed type QA mappings in IFN-spaces,” Advances in Difference Equations, vol. 2013, article 203, 2013.
- A. Alotaibi and S. A. Mohiuddine, “On the stability of a cubic functional equation in random 2-normed spaces,” Advances in Difference Equations, vol. 2012, article 39, 10 pages, 2012.
- K. Menger, “Statistical metrics,” Proceedings of the National Academy of Sciences of the United States of America, vol. 28, pp. 535–537, 1942.
- B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, pp. 313–334, 1960.
- B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland, New York, NY, USA, 1983.
- S. Gähler, “2-metrische Räume und ihre topologische Struktur,” Mathematische Nachrichten, vol. 26, pp. 115–148, 1963.
- I. Goleţ, “On probabilistic 2-normed spaces,” Novi Sad Journal of Mathematics, vol. 35, no. 1, pp. 95–102, 2005.
- M. Mursaleen, “On statistical convergence in random 2-normed spaces,” Acta Scientiarum Mathematicarum (Szeged), vol. 76, no. 1-2, pp. 101–109, 2010.
- S. A. Mohiuddine and M. Aiyub, “Lacunary statistical convergence in random 2-normed spaces,” Applied Mathematics & Information Sciences, vol. 6, no. 3, pp. 581–585, 2012.
- Y. J. Cho, M. Matić, and J. Pečarić, “On Gram's determinant in 2-inner product spaces,” Journal of the Korean Mathematical Society, vol. 38, no. 6, pp. 1125–1156, 2001.
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