1. Introduction
Let X be a linear space on the field 𝔽 and Y be a linear space on the field 𝕂. A function is called 2-linear if it satisfies
with some and . If and in the above definition, we obtain a 2-additive (2-Cauchy) function. n-linear (n-additive) functions are defined similarly (see [1]).Let and be given scalars. In this paper, we deal with the following general functional equation:
(1)
It is clear that the bi-Jensen functional equation:
is a special case of (1) (see [2]). Another particular case of (1) is the functional equation:(2)
which characterizes 2-additive mappings (see [3]).The function defined by , where are real constants, is bi-Jensen.
The function defined by , where a is a real constant, fulfills (2).
The stability problem of homomorphisms between groups was introduced by Ulam [4] in 1940. A year later, Hyers [5] presented his solution for Banach spaces. We recall that a functional equation is said to be Hyers–Ulam stable in a class of functions provided each function from fulfilling approximately in is near to its actual solution.
In recent years, various functional equations have been introduced by many researchers and their stability has been studied. For more information on the concept of Hyers–Ulam stability and its applications, we refer the reader to [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20].
In [1], the Hyers–Ulam stability of functional Equation (1) was shown in Banach spaces and m-Banach spaces. However, for the study of an asymptotic behavior of Equation (1), we must prove its stability on an unbounded set.
In this paper, we prove the Hyers–Ulam stability of (1) on some restricted unbounded domains. Then, we apply the obtained results to obtain some asymptotic behaviors of functions fulfilling (1). We also study the Hyers–Ulam stability and superstability of the functional Equation (1) in 2-Banach spaces. Our results improve Theorem 7 of [1] and its consequences.
2. Stability
We now prove the Ulam stability of functional Equation (1) in Banach spaces on some restricted domains. For convenience, we set:
for a given function .Assume that is a Banach space, and is a function satisfying
(3)
for some . Suppose that . Then, there is a unique function fulfilling Equation (1) and(4)
where and is a constant and is dependent on .Put and . Letting and in (3), we obtain
Then,
(5)
Thus, for each with , the sequence is Cauchy. It is easy to infer that the sequence is Cauchy for each . From the fact that is a Banach space, we conclude that this sequence is convergent. Define
From the definition of and (3), we obtain
Since , we conclude
Hence, fulfills Equation (1).
Putting and letting in (5), we obtain
(6)
Now, we extend (6) to the whole . We may assume without loss of generality that . Then, two cases arise according to whether or . First take the case . Let . If , we choose such that
It is clear that . For the case , we can choose such that
In this case, . Then, (6) yields that
Adding these inequalities and (3) (by using the triangle inequality), we obtain
Since satisfies (1), we obtain
(7)
Now, consider the case . Then, . We distinguish two cases according to whether or . First, suppose . We may assume . Using the argument above, we conclude
(8)
Now, assume . Then, , and in this case, (3) changes as follows:
Therefore,
(9)
By (7)–(9), we obtain (4). The uniqueness of follows easily from (4). □
One can apply a similar argument as in the proof of Theorem 1 and prove the following theorem.
Assume that is a Banach space, , and is a function satisfying (3) for all . Suppose that for some and
Then, there is a unique mapping fulfilling Equation (1) and
where .Since yields (), Theorems 1 and 2 are still valid if we use instead of the condition .
Suppose z is a fixed point of Y and . For a function , the following conditions are equivalent:
-
;
-
;
-
, .
It is clear that and are equivalent. To prove , let f satisfy . Define
Then,
Let be an arbitrary real number. Then, there exists such that
Let be the completion of Y. In view of Theorem 1, there exists unique function fulfilling Equation (1) and
where and is a constant dependent on . Then,Since is arbitrary, we obtain for all . Then,
This implies . The implication is obvious. Hence, the proof is complete. □
Let . Then,
if and only if
Assume that is a Banach space, , and is a function satisfying
(10)
for some . Suppose that , and . Then, there is a unique function fulfilling Equation (1) for all , and(11)
for all , where .Letting and in (10), we obtain
where and . Then, for integers , we obtain(12)
Thus, for each with , the sequence is Cauchy. It is easy to infer that the sequence is Cauchy for and for each with , and then, it is convergent since is Banach. We now show that and are convergent. Let be an arbitrary element, be a fixed element, and m be an integer. We take
If m is large enough, then
Since the sequences (for ) are convergent, (10) implies that is convergent. The convergence of the sequence is similarly proven. Define
From the definition of and (10), we obtain
Therefore, fulfills Equation (1) for all .
Putting and letting in (12), we obtain
(13)
Now, we extend (13) to the whole . Let be an arbitrary element, be a fixed element, and n be an integer. We take
If n is large enough, then
By (13), we have
Adding these inequalities and (10) (by using the triangle inequality), we obtain
Since satisfies (1) and , we obtain
The uniqueness of follows easily from (11). □
Since implies , Theorem 3 is still valid if we use instead of the condition . It should be noted that in this case, Theorem 3 is somewhat different from Theorems 1 and 2, and fulfilling (1) for all .
In the following corollaries, we suppose that
Suppose z is a fixed point of Y. For a function , the following conditions are equivalent:
-
;
-
, .
Let and . Suppose z is a fixed point of Y. If a function satisfies
then for all .
Let and be real numbers with . Suppose z is a fixed point of Y. If a function satisfies
then for all .
With a slight modification in the proof of Theorem 3, the following theorem is proven, and we leave the proof to the reader.
Assume that is a Banach space, , and is a function satisfying
(14)
for some . Suppose that , and with . Then, there is a unique function fulfilling Equation (1) for all , and(15)
for all , where .In the following corollaries, we suppose that with and
Suppose z is a fixed point of Y. For a function , the following conditions are equivalent:
-
;
-
, .
Let and . Suppose z is a fixed point of Y. If a function satisfies
then for all .
Let and be real numbers with . Suppose z is a fixed point of Y. If a function satisfies
then for all .
3. Superstability and Stability in 2-Banach Spaces
The concept of 2-normed spaces was introduced by Gähler [21]. First, we recall (see for instance [22]) some basic definitions and facts concerning 2-normed spaces.
Let X be a real linear space of dimension greater than one and a function satisfying the following conditions:
-
1.
if and only if are linearly dependent;
-
2.
;
-
3.
;
-
4.
.
By and we infer that
Hence, is non-negative.
A sequence of elements of a 2-normed space X is called a Cauchy sequence if there exist linearly independent with
A sequence of a 2-normed space X is said to be convergent if there exists such that for all . In this case, x is called the limit of and denoted by . It is easy to see that in a 2-normed space, a sequence has at most one limit, and every convergent sequence is Cauchy.
A 2-normed space X is called a 2-Banach space if every Cauchy sequence in X is convergent.
Let be a linear 2-normed space and be linearly independent. If , then . In particular, if for all , then .
Since , there exist such that and . Then, , and we conclude that . Hence, . □
Let be a linear 2-normed space and be linearly independent. It is easy to verify that the function given by is a norm on X.
The following theorem improves Theorem 7 of [1] and its consequences.
Let , be a linear space and a 2-normed space. If a function satisfies
(16)
then for all .Replacing z by in (16) and dividing the resultant inequality by n, we obtain
Allowing n to tend to infinity, we obtain for all and . Hence, by Lemma 1, for all . □
Assume that , is a normed space, and Y is a 2-Banach space. Let be a surjective function and
(17)
If is a function satisfying
(18)
for , then there is a unique mapping fulfilling Equation (1) andPut and . Letting and in (18), we obtain
Then,
(19)
for all and .Thus, the sequence is Cauchy. Since is a 2-Banach space, we conclude that this sequence is convergent. Define
Putting now and letting in (19), we see that
In view of (17), we obtain
Letting now and applying the definition of , we deduce that
Since g is surjective, we infer for all by Lemma 1. The uniqueness of is obvious. □
4. Conclusions
In this work, we studied the following 2-linear functional equation:
(20)
where is the unknown function. We established a new strategy to the study Hyers–Ulam stability of the functional Equation (20) on some restricted unbounded domains. As a consequence, we applied the obtained results to investigate several asymptotic behaviors of functions fulfilling (20). We also investigated the Hyers–Ulam stability and superstability of the functional Equation (20) in 2-Banach spaces.Conceptualization, J.-H.B., M.B.M., A.N. and B.N.; methodology, J.-H.B., M.B.M., A.N. and B.N.; software, J.-H.B., M.B.M., A.N. and B.N.; validation, J.-H.B., M.B.M., A.N. and B.N.; formal analysis, J.-H.B., M.B.M., A.N. and B.N.; investigation, J.-H.B., M.B.M., A.N. and B.N.; resources, J.-H.B., M.B.M., A.N. and B.N.; data curation, J.-H.B., M.B.M., A.N. and B.N.; writing—original draft preparation, A.N. and B.N.; project administration, A.N. and B.N.; funding acquisition, J.-H.B. and M.B.M. All authors have read and agreed to the published version of the manuscript.
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The authors declare no conflict of interest.
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Abstract
In this paper, we deal with a 2-linear functional equation. The Hyers-Ulam stability of this functional equation is shown on some restricted unbounded domains, and the obtained results are applied to get several hyperstability consequences. Moreover, some asymptotic behaviors of 2-linear functions are investigated. We also study the Hyers-Ulam stability and superstability of the 2-linear functional equation in 2-Banach spaces.
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1 Humanitas College, Kyung Hee University, Yongin 17104, Korea;
2 Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran;