Abstract
In this paper, we investigate the general solution of a new quadratic functional equation of the form We prove that a function admits, in appropriate conditions, a unique quadratic mapping satisfying the corresponding functional equation. Finally, we discuss the Ulam stability of that functional equation by using the directed method and fixed-point method, respectively.
1. Introduction
The stability problem of functional equations originated from a question of Ulam [1] concerning about the stability. Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. In addition, various generalizations of Ulam’s problem and Hyer’s theorem have been extensively studied and many elegant results have been obtained [3–9]. The theory of nonlinear analysis has become a fast-developing field during the past decades. Functional equations have substantially grown to become an important branch of this field. In [10], the authors deal with a comprehensive illustration of the stability of functional equations, and in [11], the authors studied functional equations and inequalities in several variables. Very recently, most classical results on the Hyers-Ulam-Rassias stability have been offered in an integrated and self-contained version in [12]. It is worth noting that among the stability problem of functional equations, the study of the Ulam stability of different types of quadratic functional equations is an important and interesting topic, and it has attracted many scholars [13–18]. In addition, very recently, authors studied various types of stability results and have been discussed with differential equation [19–29]. To the best of the author’s knowledge, a new approach to Hyers-Ulam stability of -variable quadratic functional equations has not been studied so far, which motivates the present study.
Consider the functional equation as follows: is called a quadratic functional equation. Every solution of the quadratic functional equation is a quadratic mapping. In this paper, we investigate the general solution of a new quadratic functional equation of the form
Motivated by the above discussion, we prove that a function admits in appropriate conditions and a unique quadratic mapping satisfying the corresponding functional equation. Finally, we discuss the Ulam stability of that functional equation by using the directed method and fixed-point method, respectively.
2. Preliminaries
Definition 1. Letbe a real linear space. A functionis said to be a fuzzy norm onif for alland all
(N1) for
(N2) if and only if for all
(N3) if
(N4)
(N5) is a nondecreasing function on and
(N6) for is continuous on
The pair is called fuzzy normed linear space one may regard as the truth value of the statement; the norm of is less than or equal to the real number .
Definition 2. Letbe a fuzzy normed linear space. Letbe a sequence in. Then,is said to be convergent if there existssuch thatfor all. In that case,is called the limit of the sequenceand we denote it by.
Definition 3. A sequencebe inis called Cauchy if for eachand each, there existssuch that for alland all, we have
Definition 4. Every convergent sequence in fuzzy normed space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed space is called a fuzzy Banach space.
Theorem 5. (Banach’s contraction principle). Let be a complete metric space and consider a mapping which is strictly contractive mapping, that is,
(A1) for some (Lipchitz constant) , then
(i)The mapping has one and only fixed point (ii)The fixed point for each given element is globally attractive that is(A2) , for any starting point (iii)One has the following estimation inequalities:(A3) for all
(A4) , with respect to
Theorem 6. (the alternative of fixed point). Suppose that for a complete generalized metric space and a strictly contractive mapping with Lipschitz constant . Then, for each given element , either
(B1) or
(B2) there exists natural number such that:
(i) for all (ii)The sequence is convergent to a fixed point of (iii) is the unique fixed point of in the set (iv) for all
3. General Solution of the Functional Equation (2)
In this sector, the authors obtain the general solution of the functional equation (2). All over this sector, let and be real vector space.
Theorem 7. Letandbe a real vector spaces. The mappingsatisfies the functional equation ((2)) for all, thensatisfies the functional equation ((1)) for all.
Proof. We first assume that the mapping satisfies (1). Setting in (1), we get . Replacing , in (1), then for all . Therefore, is even. If we choose , , and , in (1), we get for all . In general for any positive integer such that for all . Conversely, replacing by in (2), we get Replacing by in (2), we have Setting by in (2), we get Adding (6), (7), and (8) up to -times attack, we get It follows from (9), and using evenness of , we get Replacing by in (2), we obtain for all . Switching by in (2), we get for all . Setting by in (2) that for all . Adding (11), (12), and (13), we obtain for all . It follows from (14); it reduces that for all . Replacing by in (2), we get for all . Substituting by in (2), we arrive for all . Replacing by in (2), we get for all . Adding (16), (17), and (18) using evenness of , then we get for all . It follows from (19); we get for all . In general for any positive integer , then can be written as for all . Replacing by , we arrive for all . Setting by for all . Replacing by in (2), we get for all . Substituting by in (2) that for all . Adding (23), (24), and (25) and using evenness of , we get for all . Replace by in (26), we get for all . Adding (26) and (27) and using evenness of , then for all . So the mapping is quadratic.
In Sections 4 and 5, using be a normed space and be a Banach space. For notational handiness, we define a function by for all
4. Stability of the Functional Equation (2): Direct Method
In this section, we establish the stability of (2) in a fuzzy Banach space using a direct method.
Theorem 8. Let. Letbe a mapping withfor all and all and for all and all . Suppose that a function satisfies the inequality for all and Then, the limit exists for all and the mapping is a unique quadratic mapping such that for all and for all .
Proof. First, assume that . Replacing by , in (32), we have
for all and for all . Replacing by in (35), we obtain
for all and for all . Using (30) and (N3) in (36), we have
for all and for all ; it is easy to verify from (37) that
holds for all and for all . Replacing by in (38), we get
for all and for all ; it is easy to see that
for all . From equations (39) and (40), we get
for all and for all . Replacing by in (41) and using (30) and (N3), we obtain
for all and for all . And all . Replacing by in (42), we get
for all and for all . And all . Using (N3) in (42), we have
for all and for all . And all . Since and , the Cauchy criterion for convergence and (N5) implies that is a Cauchy sequence in is a fuzzy Banach space. This sequence converges to some point so one can define the mapping by
for all . Letting in (44), we receive
for all . Letting in (46) and using (N6), we have
for all and for all . To prove satisfies (2), replacing by in (32), we get
for all and all , since .
Hence, satisfies the quadratic functional equation (2), in order to prove is unique.
We let be another quadratic functional equation satisfying (2) and (34). Hence,
for all and for . Since
we obtain
Thus, for all and for . Hence, . Therefore, is unique. For , we can prove the result by a similar method. This completes the proof of the theorem.
The following Corollary 9 is an immediate consequence of Theorem 8 concerning the stability of (2).
Corollary 9. Suppose that the functionsatisfies the inequalityfor all and all , where are constants. Then, there exists a unique quadratic mapping such that for all and for .
5. Stability of the Functional Equation (2): Fixed-Point Method
In this section, the authors investigate the generalized Ulam-Hyers stability of the functional equation (2) in fuzzy normed space using the fixed-point method.
To prove the stability result, we define the following is a constant such that and is the set such that .
Theorem 10. Letbe a mapping for which there exists a functionwith conditionfor all and satisfying the inequality
for all and .
If there exists such that the function
has the property
for all and . Then, there exists a unique quadratic function satisfying the functional equation (2) and
for all and .
Proof. Let be a general metric on , such that
It is easy to see that is complete.
Define by .
For , we get
Therefore, is strictly contractive mapping on with Lipschitz constant , replacing by in (56), we get
for all and . Using (N3) in (62), we have
for all and with the help of (58), when . It follows from (63) that
Replacing by in (62), we get
for all and when ; it follows from (65); we arrive
Then from (64) and (66), we get
Now from the fixed-point alternative in both cases, it follows that there exists a fixed point of in such that
for all and . Replacing by in (56), we get
for all and all . Utilizing the same procedure in Theorem 8, we can prove the function is quadratic and it satisfies the functional equation (2) by a fixed-point alternative, since is a unique fixed point of in the set . Therefore, is a unique function such that
for all and . Again using the fixed-point alternative, we get
This completes the proof.
The following Corollary 11 is an immediate consequence of Theorem 10 concerning the stability of (2).
Corollary 11. Suppose that the functionsatisfies the inequalityfor all and all , where are constants. Then, there exists a unique quadratic mapping such that for all and for .
Proof. Setting
for all . Then,
i.e., (55) holds. We have that has the property for all and . Hence,
Now,
for all . The following cases hold with the below conditions:
if and if .
for if and if for if .
for if and if for if .
for if and if for if
Case 1. if
Case 2. if
Case 3. for if
Case 4. for if
Case 5. for if
Case 6. for if
Hence, the proof is completed.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
The second and fifth authors would like to express their sincere thanks to the financial and research support of Phuket Rajabhat University, 83000, Phuket, Thailand.