Abstract
In this paper, we introduce the -quadratic functional equation with three variables and obtain its general solution. The main aim of this work is to examine the Ulam-Hyers stability of this functional equation in non-Archimedean Banach space by using direct and fixed point techniques and examine the stability results in non-Archimedean random normed space.
1. Introduction
One of the most important areas of research in mathematics is the investigation of stability issues for functional equations, which has its origins in concerns of applied mathematics. The first question about the stability of homomorphisms was given by Ulam [1] as follows.
Given a group , a metric group with the metric , and a function from and , does there exists satisfying for all , then there exists a homomorphism such that for all ?
Ulam’s question on Banach spaces was partially answered affirmatively by Hyers [2]. By assuming an infinite Cauchy difference, Aoki [3] expanded Hyers’ and Rassias’ theorems for additive and linear mappings, respectively. Using the same method as Rassias [4], Gajda [5] discovered a positive solution to the question . Rassias and Šemrl [6], as well as Gajda [5], have proved that a Rassias’ type theorem cannot be formed for .
The functional equation is known as the Cauchy additive equation.
Since the function is the solution of the functional equation (3), every solution of the additive functional equation (3) is called as an additive function. Every solution of the functional equation (3), in particular, is called as an additive function.
The functional equation is known as the quadratic functional equation.
Since the function is the solution of the functional equation (4), each solution of the functional equation (4) is called as a quadratic function. Every solution of functional equation (4), in particular, is called as a quadratic mapping.
Skof [7] established the stability of the quadratic functional equation for the function between normed space and complete normed space. The authors [8–14] recently examined the Ulam-Hyers stability results for the following -functional equation in non-Archimedean Banach spaces.
The Skof theorem still applies when the relevant domain is replaced by an Abelian group, according to Cholewa [15]. See [15–21] for other functional equations. A survey of the Ulam-Hyers stability results of functional equations was conducted by Brillouët-Belluot [22]. Park and Kim [11] demonstrated the Ulam-Hyers stability of quadratic -functional equation.
In this paper, the authors present a new -quadratic functional equation with three variables as where is a fixed non-Archimedean number with , and its general solution was obtained. The motivation behind this study is to investigate the Ulam-Hyers stability results for the above functional equation (6) in non-Archimedean Banach space by using direct and fixed point methods and non-Archimedean random normed space.
The following is the structure of this paper: in Section 2, we recall some fundamental notions and definitions, in Section 3, we look at the general solution of the equation (6), where and are two vector spaces. We investigate the Ulam-Hyers stability in non-Archimedean Banach space by using fixed point method and direct method in Sections 4 and 5, where is a non-Archimedean normed space, is a non-Archimedean Banach space, and is a non-Archimedean Banach space. In Section 6, we recall some fundamental notions and results and investigate the Ulam-Hyers stability in non-Archimedean random normed space.
2. Preliminaries
To reach our major results, we use certain fundamental notations in [8, 10, 11].
A map is a valuation such that zero is the only one element having the zero valuation, , and the inequality of the triangle holds true, that is, , for all .
We call a field valued if holds a valuation. Examples of valuations include the typical absolute values of and .
Consider a valuation that satisfies a criterion that is stronger than the triangle inequality. A is called a non-Archimedean valuation if the triangle inequality is replaced by , for all , and a field is called a non-Archimedean field. Evidently, and are greater than or equal to , for all in . The map takes everything except for , and is a basic example of a non-Archimedean valuation.
Definition 1. Let be a linear space over with . A mapping is known as a non-Archimedean norm if it satisfies (i) if and only if .(ii), , and .(iii)the strong triangle inequality.Then, is called a non-Archimedean normed space. Every Cauchy sequence converges in a complete non-Archimedean normed space, which we call a complete non-Archimedean normed space.
Definition 2. Let be a non-Archimedean normed space and a sequence in . Then, (1)a sequence in is a Cauchy sequence if converges to 0.(2) is called convergent if, for any , there is an integer in and satisfies for every . Then, we called as is a limit of the sequence and denoted by (3)if every Cauchy sequence in a non-Archimedean normed space converges, it is called a non-Archimedean Banach space.
Theorem 3 (alternative fixed point theorem). Let be a generalized complete metric space and a strictly contractive mapping with Lipschitz constant . Then, for all , either or there exists a positive integer such that (i).(ii)the sequence converges to a fixed point of .(iii) is the unique fixed point of in .(iv), for all .
3. Solution
Lemma 4. If a mapping satisfies the functional equation (6) for all , then the function is quadratic.
Proof. A mapping satisfies the functional equation (6). Replacing by in (6), we obtain This implies that . Replacing by in (6), we obtain and so for all . Thus, equation (6) is reduced as for all . Now, replacing in (13), we get for all . Again, replacing by in (14), we have for all . From equalities (14) and (15), we can conclude that for any integer , we get for all . Now, replacing by in (13), we reach (3) for all . Hence, the function is quadratic.
For our notational simplicity, we use the following abbreviation:
4. Stability of (6) in Non-Archimedean Banach Space: Direct Method
Theorem 5. Let be a mapping and a mapping such that and for all . Then, there exists a unique quadratic mapping satisfying for all .
Proof. Setting in (19), we have Thus, from inequality (21), it implies that for all . Replacing by in (22), we obtain for all . Hence, for all and all . From inequality (24), the sequence is a Cauchy sequence for all . Since is complete, thus the sequence is convergent. Now, we can define a mapping by Taking and passing the limit in (24), we obtain (20). From inequalities (18) and (19), we have From above, we conclude that for all . By using Lemma 4, the function is quadratic. Consider another quadratic mapping satisfying (20). Then, we have for all . Thus, we can conclude that . Hence, the function is unique. Thus, the unique quadratic mapping satisfies (20). Hence, the proof of the theorem is now completed.
Theorem 6. Let be a mapping and a mapping such that and and (19) for all . Then, there exists a unique quadratic mapping satisfying for all .
Proof. Setting in (19), we have From inequality (30), we obtain Replacing by in (31), we get for all . Hence, for all and all . From inequality (33), the sequence is a Cauchy sequence for all . Since is complete, the sequence is convergent. Now, we can define a mapping by The remaining proof is the same as the proof of Theorem 5.
Corollary 7. Let be a mapping such that and for all , where and are in with . Then, there exists a unique quadratic mapping satisfying for all .
Corollary 8. Let be a mapping such that and for all , where and are in with . Then, there exists a unique quadratic mapping satisfying for all .
5. Stability of (6) in Non-Archimedean Banach Space: Fixed Point Method
Theorem 9. Let be a mapping such that there exists with for all . If a mapping such that and (19) for all , then there exists a unique quadratic mapping satisfying for all .
Proof. Setting in (19), we obtain for all . Consider and the generalized metric defined by here, as usual, . Clearly, is complete (see [23]). Next, consider a mapping defined by For all such that , then for all . Hence, for all . Thus, This concludes that From inequality (41), Therefore, By using Theorem 3, there exists a mapping satisfying the following conditions: (1) is a fixed point of , i.e.,In the set below, the function is the unique fixed point . This proves that the uniqueness of the function satisfies (51) such that there exists such that (2) tends to as taking the limit . This implies (3), which implies From (39) and (51), for all . Thus, for all . By using Lemma 4, the function is quadratic. Hence, the proof of the theorem is now completed.
Theorem 10. Let be a mapping such that there exists with If a mapping such that and (19) for all , then there exists a unique quadratic mapping satisfying for all .
Proof. Setting in (19), we have for all . From the inequality (60), we get The generalized metric space is defined in the proof of Theorem 9. Consider a mapping defined by From inequality (61), Hence, The remaining proof is the same as in the proof of Theorem 9.
Corollary 11. Let be a mapping such that and for all , where and are in with ; then there exists a unique quadratic mapping satisfying for all .
Corollary 12. Let be a mapping such that and for all , where and are in with ; then there exists a unique quadratic mapping satisfying for all .
6. Stability of (6) in Non-Archimedean Random Normed Space
Definition 13 [24]. A random normed space is triple , where is a vector space, is a continuous norm, and a mapping satisfies
(RN1) if and only if .
(RN2) for all , .
(RN3) for all and .
Definition 14 [25]. A random normed space is said to be non-Archimedean random normed space if it satisfies
(NAR1) for all if and only if .
(NAR2) for all .
(NAR3) for all and .
It is clear that if (NAR3) holds, then so
Example 1 [25]. Let a non-Archimedean normed space and we define for all and all . Then, the triple is a non-Archimedean random normed space.
Definition 15 [25]. Let be a non-Archimedean random normed space and a sequence in . Then, the sequence is called as convergent if there exist such that for all . In particular, is called the limit of the sequence .
Here, let be a vector space over a non-Archimedean field and be a non-Archimedean random Banach space over . And consider that in .
Next, we define a random approximately quadratic function. Let a distribution mapping satisfies which is symmetric and nondecreasing and for all and all .
Definition 16. A function is called as a -approximately quadratic if for all and .
Theorem 17. Let a function be a -approximately quadratic mapping. If for some real number , and some integer , with , for all and , and for all and every ; then there exists a unique quadratic mapping such that where for all and all .
Proof. First, we demonstrate by induction on that for all , and ,
Setting in (73), we obtain
for all and all . This proves that (78) for . Suppose that (78) holds for some . Replacing by in (73), we get
for all and all . Since ,
for all . Thus, condition (78) holds for all . In particular,
for all and all . Replacing by in (82) and using the inequality (74), we have
for all and all . Then,
for all and all . Hence,
Since for all and all , is a Cauchy sequence in . Hence, we can define a mapping such that
for all and all . Now, for all ,
for all and . Thus,
By taking the limit , we have
This shows that (76) holds. Since is continuous, by a well-known result in probabilistic metric space (see, e.g., [[26], Chapter 12]), that
for all .
On the other hand, replacing by , respectively, in (73) and using (NAR2) and (74), we get
Since , we can conclude that the function is quadratic. Consider another quadratic mapping such that for all and all ; then for all and and all ,
From condition (86), we arrive at the conclusion that .
Corollary 18. Let a function be a -approximately quadratic. If for some real number and some integer , , with , for all and , then there exists a unique quadratic mapping satisfying where for all and all .
Proof. Since for all and all and is of Hadzic type, from Proposition 2.1 in [25], it follows that for all and . Now, we can obtain our needed result by using Theorem 17
Example 2. Let a non-Archimedean random normed space , in which for all and every , and let be a complete non-Archimedean random normed space (see Example 1). Now, we can define It is obvious that (74) holds for . Furthermore, We obtain for all and all .
7. Conclusion
In this paper, we introduced -quadratic functional equation and obtained its general solution. In Section 4 and Section 5, we investigated Ulam-Hyers stability of equation (6) by using direct method and fixed point method in non-Archimedean Banach space, and also in Section 6, we investigated the Ulam-Hyers stability results in non-Archimedean random normed space. The direct method requires us to find the Cauchy sequence and prove that every Cauchy sequence is convergent, as well as prove the uniqueness of the function; this method was introduced by Hyers [2], and the fixed point method requires us to use the Banach contraction principle and Lipschitz constant to obtain the stability results of the functional equation; this method was introduced by Radu [27]. The fixed point method gives more accurate stability results when compared with the direct method. Finally, these stability results generalized the findings of [11].
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflict interests.
Authors’ Contributions
All authors contributed equally to this work. And all the authors have read and approved the final version of the manuscript.