Abstract
We consider the functional equation for a fixed rational number with and a fixed real number . We study the solution of the equation between linear spaces and prove the generalized Hyers-Ulam stability for it when the target space is a fuzzy normed space.
1. Introduction and Preliminaries
In 1940, Ulam proposed the following stability problem (cf. [1]):
“Let be a group and a metric group with the metric . Given a constant , does there exist a constant such that if a mapping satisfies for all , then there exists a unique homomorphism with for all ?”
In the next year, Hyers [2] gave a partial solution of Ulam’s problem for the case of approximate additive mappings. Subsequently, his result was generalized by Aoki [3] and Moslehian and Rassias [4] for additive mappings, and by Rassias [5] for linear mappings, to consider the stability problem with unbounded Cauchy differences. During the last decades, the stability problems of functional equations have been extensively investigated by a number of mathematicians (see [6–10]).
Recently, the stability problems in the fuzzy spaces have been extensively studied (see [11–13]). The concept of fuzzy norm on a linear space was introduced by Katsaras [14] in 1984. Later, Cheng and Mordeson [15] gave a new definition of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [16]. In 2008, for the first time, Mirmostafaee and Moslehian [12, 13] used the definition of a fuzzy norm in [17] to obtain a fuzzy version of stability for the Cauchy functional equation and the quadratic functional equation
We call a solution of (1) an additive mapping and a solution of (2) is called a quadratic mapping. Also, the equation is called Drygas functional equation (see [18, 19] for details).
Najati and Moghimi [20] investigated the generalized Hyers-Ulam stability for functional equation derived from additive and quadratic functions on quasi-Banach spaces.
In this paper, we introduce the following functional equation for a fixed rational number with and a fixed real number : with . It is easy to see that the function is a solution of the functional equation (4), so we can expect that a solution of (4) is additive-quadratic type. We note that the left-hand side of (4) is essentially the sum of two Drygas functionals and . In Section 2, a complete characterization of the solution of (4) is given. In Section 3, we prove the stability for (4) in fuzzy Banach spaces. One can find some kinds of gaps for finding in Theorems 13 and 14. In Theorem 15, we resolve these gaps for special and practical case of . Also, we give an example related to Theorem 15. We list some definitions related to fuzzy normed spaces.
Definition 1. Let be a real vector space. A function is called a fuzzy norm on if for all and all ,(N1) for ;(N2) if and only if for all ;(N3) if ;(N4);(N5) is a nondecreasing function of and ;(N6)for any , is continuous on .
In this case, the pair is called a fuzzy normed space.
Definition 2. Let be a fuzzy normed space. A sequence in is said to be convergent if there exists an such that for all . In this case, is called the limit of the sequence in X and one denotes it by .
Definition 3. Let be a fuzzy normed space. A sequence in is said to be Cauchy if for any , , there is an such that for any and any positive integer , .
It is well known that every convergent sequence in a fuzzy normed space is Cauchy. A fuzzy normed space is said to be complete if each Cauchy sequence in it is convergent and the complete fuzzy normed space is called a fuzzy Banach space.
2. Solution of (4)
In this section, we investigate solutions of (4) between linear spaces and by separating cases into odd functions and even functions. And then, in Theorem 8, it can be concluded that any solution of (4) is additive-quadratic type. We start with the odd function case.
Lemma 4. Let be an odd mapping with satisfying (4). Suppose that . Then is an additive mapping.
Proof. Since is an odd mapping, the functional equation (4) can be written by
for all .
If , then it is easy to check that is additive. Suppose that . Letting in (5), we have
for all . Replacing by in (5), we have
for all ; letting in (7), we have
for all . Replacing and by and in (5), respectively, by (6), we have
for all . Letting in (9), we have
for all , and letting in (5), we have
for all . By (5), (7), (8), (9), (10), and (11), we have
for all . If , then by (12), we have
for all and is additive-cubic ([20]). Since for all , is additive.
Lemma 5. Let be an odd mapping with satisfying (4). Suppose that . Then is an additive mapping.
Proof. Since is an odd mapping, the functional equation (4) can be written by
for all . Replacing by in (14), we have
for all , and interchanging and in (15), we have
for all . Replacing and by and in (5), respectively, we have
for all . By (16) and (17), we have
for all .
Replacing and by and in (5), respectively, we have
for all . Letting in (19), we have
for all . Replacing by in (14), we have
for all and by (20) and (21), we have
for all . Replacing by in (18), we have
for all and by (22) and (23), we have
for all . Since , we have
for all and letting in (25), we have
for all . By (22) and (23), we have
for all . Letting in (5), we have
for all , and by (27) and (28), we have
for all . Since , we have
for all , and hence is additive.
Combining Lemmas 4 and 5, we can get the following theorem.
Theorem 6. Let be an odd mapping with satisfying (4). Then is an additive mapping.
Now if we assume that is an even function, (4) turns into the following equation with : And in [21], the authors proved the following theorem.
Theorem 7 (see [21]). Let be a mapping with . Then is quadratic if and only if satisfies (31) for all , a fixed nonzero rational number , and fixed real numbers , with .
By Theorems 6 and 7, we have the following theorem which is the conclusion of this section.
Theorem 8. Let be a mapping with . Then satisfies (4) if and only if is an additive-quadratic mapping.
3. The Generalized Hyers-Ulam Stability for (4)
In this section, we prove the generalized Hyers-Ulam stability of functional equation (4) in fuzzy normed spaces. Throughout this section, we assume that is a linear space, is a fuzzy Banach space, and is a fuzzy normed space.
For any mapping , we define the difference operator by for all .
Theorem 9. Let be a function and let be a real number such that and for all and all . Let be an odd mapping such that and for all and all . Then there exists a unique additive mapping such that the inequality holds for all and all .
Proof. Since is an odd mapping, the inequality (34) is equivalent to the following inequality:
for all and all . By (33) and (N3), we have
for all and all , and so by (37), we have
for all and all . Letting in (36), by (N3), we have
for all and all . By (38), (39), and (N3), we have
for all , all , and all positive integers . Hence by (40) and (N4), for any , we have
for all , all , and all positive integers . So for any , we have
for all , all , all nonnegative integers , and all positive integers . Thus, by (42), for any , we have
for all , all , all nonnegative integers , and all positive integers . Since is convergent, , and so by the usual argument is a Cauchy sequence in . Since is a fuzzy Banach space, there is a mapping defined by
for all . Moreover by (41), we have
for all , all , and all positive integers . Let be a real number with . Then, by (43), (44), (N4), and (N5), we have
for sufficiently large positive integer , all , and all or . Since is continuous on for all from (N2) and (N6), by taking , we get
for all and all , and so we have (35).
By (33), (34), and (N3), we have
for all and all . Since and
for all and all , by (44), (48), and (N4), we have
for sufficiently large , all , and all or . Since , for all , and so, by (N2), for all . By Theorem 8, is additive.
To prove the uniqueness of , let be another additive mapping satisfying (35). Then for any and a positive integer , , and so by (45),
holds for all , all positive integers , and all . Since , , and so for all .
Now we deal with the even function case.
Theorem 10. Let be a function and let be a real number such that and for all and all . Let be an even mapping satisfying and (34). Then there exists a unique quadratic mapping such that the inequality holds for all and all .
Proof. Since is an even mapping, the inequality (34) is equivalent to the following inequality:
for all and all . By (52) and (N3), we have
for all and all , and so by (55), we have
for all and all . Letting in (54), by (N3), we have
for all and all . By (52), (56), (57), and (N3), we have
for all , all , and all positive integers . Hence by (58) and (N4), for any , we have
for all , all , and all positive integers . So for any , we have
for all , all , all nonnegative integers , and all positive integers . Thus, by (60) and (N3), for any , we have
for all , all , all nonnegative integers , and all positive integers . Since is convergent, , and so by the usual argument again, is a Cauchy sequence in . Since is a fuzzy Banach space, there is a mapping defined by
for all . Moreover by (59), we have
for all , all , and all positive integers . Let be a real number with . Then, by (62), (63), and (N4), we have
for sufficiently large positive integer , all , and all or . Since is continuous on for all from (N2) and (N6), we get
for all and all , and so we have (53). By (34) and (N5), we have
for all and all . Since
for all and all , and by (62), (66), and (N4), we have
for sufficiently large , all , and all or .
Since , for all , and so by (N2), for all . By Theorem 8, is quadratic.
To prove the uniqueness of , let be another quadratic mapping satisfying (53). Then for any and a positive integer , , and so by (63),
holds for all , all positive integers , and all . Since , , and so for all .
Now we consider the next two theorems which are similar to Theorems 9 and 10. The proofs are straightforward and similar to those of Theorems 9 and 10.
Theorem 11. Let be a function and let be a real number such that such that for all and all . Let be an odd mapping with satisfying (34). Then there exists a unique additive mapping such that the inequality holds for all and all .
Theorem 12. Let be a function and let be a real number such that such that for all and all . Let be an even mapping with satisfying (34). Then there exists a unique additive mapping such that the inequality holds for all and all .
By combining Theorems 9 and 10, we can have the following theorem which is the main theorem of the paper.
Theorem 13. Let be a function and let be a real number such that such that for all and all . Let be a mapping with satisfying (34). Then there exists a unique additive-quadratic mapping such that the inequality holds for all and all , where
Proof. By (34), we have
for all and . By Theorem 9, there is a unique additive mapping such that
holds for all and all .
By (34), we have
for all and . By Theorem 10, there is a unique quadratic mapping such that
holds for all and all . Let . Then by (78) and (80), we have (75).
The uniqueness of satisfying (75) is trivial.
Also, if we combine Theorems 11 and 12, we have the following theorem.
Theorem 14. Let be a function and let be a real number such that such that for all and all . Let be a mapping with satisfying (34). Then there exists a unique additive-quadratic mapping such that the inequality holds for all and all , where
Among the examples of the function , there are lots of meaningful ones satisfying for all real numbers and for some real number . The following theorem says that a strong and useful result can be obtained in such cases.
Theorem 15. Let be a function and let be a real number such that such that for all and all . Let be a mapping with satisfying (34). Then there exists a unique additive-quadratic mapping such that the inequality holds for all and all , where
We can use Theorem 15 to get a classical result in the framework of normed spaces. For example, it is well known that for any normed space , mappings , defined by are fuzzy norms on X. In [12, 13, 22], some examples are provided for the fuzzy norms , and other fuzzy norms. Here especially using the fuzzy norm and taking , we have the following example.
Example 16. Let be a mapping such that and for all , a fixed rational number , and a fixed positive number such that . Then there exists a unique additive-quadratic mapping such that the inequality holds for all .
The condition in Example 16 is indispensable. The following example shows that the inequality (88) is not stable for , especially in the case of . We will give the proof when , and the proof when is similar.
Example 17. Let be a real number with . Define mappings by
and a mapping by
We will show that satisfies the following inequality:
for all . Here,
But there do not exist an additive-quadratic mapping and a nonnegative constant such that
for all .
Proof. Note that , , and for all . First, suppose that . Then . Now suppose that . Then there is a positive integer such that
and so
Hence, we have
and so for any ,
for all . Thus,
Note that for all , , and . First, suppose that . Then for all . Now suppose that . Then there is a positive integer such that
and so
Hence, we have
and so for any ,
for all . Thus,
Hence, we have
for all , and so we have (92).
Suppose that there exist an additive mapping , a quadratic mapping , and a nonnegative constant such that satisfies (94). Since , by (94), we have
for all and all positive integers , and so for all . Since is additive,
for all . Take a positive integer such that , and pick with . Then
which is in contradiction with (107).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.