Research Article | Open Access

# Stabilities of Cubic Mappings in Various Normed Spaces: Direct and Fixed Point Methods

**Academic Editor:**Hui-Shen Shen

#### Abstract

In 1940 and 1964, Ulam proposed the general problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?”. In 1941, Hyers solved this stability problem for linear mappings. According to Gruber (1978) this kind of stability problems are of the particular interest in probability theory and in the case of functional equations of different types. In 1981, Skof was the first author to solve the Ulam problem for quadratic mappings. In 1982–2011, J. M. Rassias solved the above Ulam problem for linear and nonlinear mappings and established analogous stability problems even on restricted domains. The purpose of this paper is the generalized Hyers-Ulam stability for the following cubic functional equation: in various normed spaces.

#### 1. Introduction

A classical question in the theory of functional equations is the following: “When is it true that a function which approximately satisfies a functional equation must be close to an exact solution of ?”

If the problem accepts a solution, we say that the equation is stable. The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1964.

In the next year, Hyers [2] gave a positive answer to the above question for additive groups under the assumption that the groups are Banach spaces.

In 1978, Th. M. Rassias [3] proved a generalization of Hyers’ theorem for additive mappings.

Theorem 1.1 (Th. M. Rassias). *Let be a mapping from a normed vector space into a Banach space subject to the inequality
**
for all , where and are constants with and . Then, the limit
**
exists for all and is the unique additive mapping which satisfies
**
for all . If , then inequality (1.1) holds for and (1.3) for . Also, if for each the mapping is continuous in , then is -linear.*

The result of Th. M. Rassias has influenced the development of what is now called the Hyers-Ulam-Rassias stability theory for functional equations. In 1994, a generalization of Rassias’ theorem was obtained by Găvruţa [4] by replacing the bound by a general control function .

The functional equation
is called a *quadratic functional equation*. In particular, every solution of the above quadratic functional equation is said to be a *quadratic mapping*. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [5] for mappings , where is a normed space and is a Banach space. Cholewa [6] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [7] proved the generalized Hyers-Ulam stability of the quadratic functional equation.

The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [2–4, 8–48]).

On the other hand, J. M. Rassias [38] considered the Cauchy difference controlled by a product of different powers of norm.

Theorem 1.2 (J. M. Rassias). *Let be a mapping from a real normed vector space into a Banach space subject to the inequality
**
for all , where and are constants with and . Then, is the unique additive mapping which satisfies
**
for all .*

However, there was a singular case, for this singularity a counterexample was given by Găvruţa [19]. This stability phenomenon is called the Ulam-Gavruta-Rassias product stability (see also [13–17, 49]). In addition, J. M. Rassias considered the mixed product-sum of powers of norms control function. This stability is called JMRassias mixed product-sum stability (see also [44, 50–53]).

Jun and Kim [22] introduced the functional equation and they established the general solution and the generalized Hyers-Ulam-Rassias stability problem for the functional equation (1.7) in Banach spaces.

Park and Jung [35] introduced the functional equation and they established the general solution and the generalized Hyers-Ulam-Rassias stability problem for the functional equation (1.8) in Banach spaces.

It is easy to see that the function is a solution of the functional equations (1.7) and (1.8). Thus, it is natural that functional equations (1.7) and (1.8) are called cubic functional equations and every solution of these cubic functional equations is said to be a cubic mapping.

In this paper, we prove the generalized Hyers-Ulam stability of the following functional equation: where is a positive integer greater than 2, in various normed spaces.

#### 2. Preliminaries

In the sequel, we will adopt the usual terminology, notions, and conventions of the theory of random normed spaces as in [54]. Throughout this paper, the space of all probability distribution functions is denoted by . Elements of are functions , such that is left continuous and nondecreasing on , and . It is clear that the subset , where , is a subset of . The space is partially ordered by the usual pointwise ordering of functions, that is, for all , if and only if . For every , is the element of defined by One can easily show that the maximal element for in this order is the distribution function .

*Definition 2.1. *A function is a continuous triangular norm (briefly a -norm) if satisfies the following conditions:(i) is commutative and associative;(ii) is continuous;(iii) for all ;(iv) whenever and for all .

Three typical examples of continuous -norms are , and . Recall that, if is a -norm and is a given group of numbers in , is defined recursively by and for .

*Definition 2.2. *A random normed space (briefly RN-space) is a triple , where is a vector space, is a continuous -norm and is a mapping such that the following conditions hold: for all if and only if ; for all , , and ;, for all and .

Every normed space defines a random normed space where, for every , and is the minimum -norm. This space is called the induced random normed space.

If the -norm is such that , then every RN-space is a metrizable linear topological space with the topology (called the -topology or the -topology) induced by the base of neighborhoods of , , where

*Definition 2.3. *Let be an RN-space*. *A sequence in is said to be convergent to in if, for all , .A sequence in is said to be Cauchy sequence in if, for all , . The RN-space is said to be complete if every Cauchy sequence in is convergent.

Theorem 2.4. *If is RN-space and is a sequence such that , then .*

A valuation is a function from a field into such that 0 is the unique element having the 0 valuation, , and the triangle inequality holds, that is,
A field is called a *valued field* if carries a valuation. The usual absolute values of and are examples of valuations.

Let us consider a valuation which satisfies a stronger condition than the triangle inequality. If the triangle inequality is replaced by
for all , then the function is called a *non-Archimedean valuation* and the field is called a *non-Archimedean field*. Clearly, and for all . A trivial example of a non-Archimedean valuation is the function taking everything except for 0 into 1 and .

*Definition 2.5. *Let be a vector space over a field with a non-Archimedean valuation *. *A function is called a non-Archimedean norm if the following conditions hold:(a) if and only if for all ;(b) for all and ;(c)the strong triangle inequality holds:
for all . Then is called a non-Archimedean normed space.

*Definition 2.6. *Let be a sequence in a non-Archimedean normed space *. *(a)A sequence in a non-Archimedean space is a *Cauchy sequence* if and only if, the sequence converges to zero.(b)The sequence is said to be convergent if, for any , there are a positive integer and such that
for all . Then, the point is called the *limit* of the sequence , which is denoted by .(c)If every Cauchy sequence in converges, then the non-Archimedean normed space is called a *non-Archimedean Banach space*.

*Definition 2.7. *Let be a set. A function is called a generalized metric on if satisfies the following conditions:(a) if and only if for all ;(b) for all ;(c) for all .

Theorem 2.8. *Let be a complete generalized metric space and a strictly contractive mapping with Lipschitz constant . Then, for all , either
**
for all nonnegative integers or there exists a positive integer such that*(a)* for all ;*(b)*the sequence converges to a fixed point of ;*(c)* is the unique fixed point of in the set ;*(d)* for all .*

#### 3. Random Stability of Functional Equation (1.9): A Direct Method

In this section, using direct method, we prove the generalized Hyers-Ulam stability of cubic functional equation (1.9) in random normed spaces.

Lemma 3.1. *Let and be real vector spaces. A function satisfies the functional equation (1.7) if and only if satisfies the functional equation (1.9). Therefore, every solution of functional equation (1.9) is also cubic function.*

*Proof. *Let satisfy the equation (1.7). Putting in (1.7), we get . Set in (1.7) to get . By induction, we lead to for all positive integer . Replacing by in (1.7), we have
for all . Once again replacing by in (1.7), we have
for all . Adding (3.1) to (3.2) and using (1.7), we obtain
for all . By using the previous method, by induction, we infer that
for all and each positive integer .

Let satisfy the functional equation (1.9) with the positive integer . Putting in (1.9), we get . Setting , we get . Let be a positive integer. Replacing by in (1.9), we have
for all . Replacing by in (1.9), we have
for all . Adding (3.5) to (3.6), we obtain
for all and for all integer . Let for each integer . Then, (3.7) means that
for all and for all integer . For and in (3.8), we obtain
for all . By the proof of the first part, since satisfies the functional equation (1.9) with the positive integer , then satisfies the functional equation (1.9) with the positive integer . It follows from (3.9) that satisfies the functional equation (1.7) and

Theorem 3.2. *Let be a real linear space, an RN-space, and a function such that, for some ,
**
and, for all and , . Let be a complete RN-space. If is a mapping with such that for all and **
then the limit exists for all and defines a unique cubic mapping such that
*

*Proof. *Putting in (3.12) we see that, for all ,
Replacing by in (3.14) and using (3.11), we obtain
So
This implies that
Replacing by in (3.17), we obtain
As
is a Cauchy sequence in complete RN-space , so there exists some point such that . Fix and put in (3.18). Then, we obtain
and so, for every , we have
Taking the limit as and using (3.21), we get
Since was arbitrary by taking in (3.22), we get
Replacing and by and in (3.12), respectively, we get, for all and for all ,
Since , we conclude that . To prove the uniqueness of the cubic mapping , assume that there exists another cubic mapping which satisfies (3.13).

By induction one can easily see that, since is a cubic functional equation, so, for all and every , , and , we have
so
Since , it follows that, for all , and so . This completes the proof.

Corollary 3.3. *Let be a real linear space, an RN-space, and a complete RN-space. Let and , and let be a mapping with and satisfying
**
for all and . Then, the limit exists for all and defines a unique cubic mapping such that
**
for all and .*

*Proof. *Let , and let be defined as .

*Remark 3.4. *In Corollary 3.3, if we assume that or *, *then we get Ulam-Gavruta-Rassias product stability and JMRassias mixed product-sum stability, respectively. But, since we put in this functional equation, the Ulam-Gavruta-Rassias product stability and JMRassias mixed product-sum stability corollaries will be obvious. Meanwhile, the JMRassias mixed product-sum stability when is an open question.

Corollary 3.5. *Let be a real linear space, an RN-space, and a complete RN-space. Let , and let be a mapping with and satisfying
**
for all and . Then, the limit exists for all and defines a unique cubic mapping such that
**
for all and .*

*Proof. *Let , and let be defined by .

Theorem 3.6. *Let be a real linear space, an RN-space, and a function such that for some **
and, for all and ,
**
Let be a complete RN-space. If is a mapping with and satisfying (3.12), then the limit exists for all and defines a unique cubic mapping such that
*

*Proof. *Putting in (3.12) and replacing by , we obtain that for all
Replacing by in (3.34) and using (3.31), we obtain
So
This implies that
The rest of the proof is similar to the proof of Theorem 3.2.

Corollary 3.7. *Let be a real linear space, an RN-space, and a complete RN-space. Let and , and let be a mapping with and satisfying (3.27). Then, the limit exists for all and defines a unique cubic mapping such that
**
for all and .*

*Proof. *Let , and let be defined as .

Corollary 3.8. *Let be a real linear space, be an RN-space, and a complete RN-space. Let , and let be a mapping with and satisfying (3.29). Then, the limit exists for all and defines a unique cubic mapping such that
**
for all and .*

*Proof. *Let , and let be defined by .

#### 4. Random Stability of the Functional Equation (1.9): A Fixed Point Approach

In this section, using the fixed point alternative approach, we prove the generalized Hyers-Ulam stability of functional equation (1.9) in random normed spaces.

Theorem 4.1. *Let be a linear space, a complete RN-space, and a mapping from to is denoted by such that there exists such that
**
for all and . Let be a mapping with and satisfying
**
for all and . Then, for all **
exists and is a unique cubic mapping such that
**
for all and .*

*Proof. *Putting in (4.2) and replacing by , we have
for all and . Consider the set
and the generalized metric in defined by
where . It is easy to show that is complete (see [26], Lemma 2.1). Now, we consider a linear mapping such that
for all . First, we prove that is a strictly contractive mapping with the Lipschitz constant .

In fact, let be such that . Then, we have
for all and , and so
for all and . Thus, implies that
This means that
for all . It follows from (4.5) that
By Theorem 2.8, there exists a mapping satisfying the following.

(1) is a fixed point of , that is,
for all .

The mapping is a unique fixed point of in the set
This implies that is a unique mapping satisfying (4.14) such that there exists satisfying
for all and .

(2) as . This implies the equality
for all .

(3) with , which implies the inequality and so
for all and . This implies that inequality (4.4) holds. Now, we have
for all , , and , and so, from (4.1), it follows that
Since
for all and , we have
for all and . Thus, the mapping is cubic. This completes the proof.

Corollary 4.2. *Let be a real normed space, , and a real number with . Let be a mapping with and satisfying
**
for all and . Then, for all , the limit exists and is a unique cubic mapping such that
**
for all and .*

*Proof. *The proof follows from Theorem 4.1 if we take
for all and . In fact, if we choose , then we get the desired result.

Theorem 4.3. *Let be a linear space, a complete RN-space, and a mapping from to ( is denoted by such that for some **
for all and . Let be a mapping with and satisfying (4.2). Then, for all , the limit exists and is a unique cubic mapping such that
**
for all and .*

*Proof. *Let be the generalized metric space defined in the proof of Theorem 4.1. Now, we consider a linear mapping such that
for all .

Let be such that . Then, we have
for all and and so
for all and . Thus, implies that
This means that
for all .

Putting in (4.2), we see that, for all ,
It follows from (4.33) that
By Theorem 2.8, there exists a mapping satisfying the following.

(1) is a fixed point of , that is,
for all .

The mapping is a unique fixed point of in the set
This implies that is a unique mapping satisfying (4.35) such that there exists satisfying
for all and .

(2) as . This implies the equality
for all .

(3) with , which implies the inequality , and so
for all and . The rest of the proof is similar to the proof of Theorem 4.1.

Corollary 4.4. *Let be a real normed space, , and a real number with . Let be a mapping with and satisfying (4.23). Then, for all , the limit exists and is a unique cubic mapping such that
**
for all and .*

*Proof. *The proof follows from Theorem 6.3 if we take
for all and . In fact, if we choose , then we get the desired result.

*Remark 4.5. *In Corollaries 4.2 and 4.4, if we assume that * or **, *then we get Ulam-Gavruta-Rassias product stability and JMRassias mixed product-sum stability, respectively. But, since we put in this functional equation, the Ulam-Gavruta-Rassias product stability and JMRassias mixed product-sum stability corollaries will be obvious. Meanwhile, the JMRassias mixed product-sum stability when is an open question.

#### 5. Non-Archimedean Stability of Functional Equation (1.9): A Direct Method

In this section, using direct method, we prove the generalized Hyers-Ulam stability of cubic functional equation (1.9) in non-Archimedean normed spaces. Throughout this section, we assume that is an additive semigroup and is a complete non-Archimedean space.

Theorem 5.1. *Let be a function such that
**
for all and let for each the limit
**
exist. Suppose that a mapping with and satisfying the following inequality:
**
Then, the limit exists for all and defines a cubic mapping such that
**
Moreover, if
**
then is the unique cubic mapping satisfying (5.4).*

*Proof. * Putting in (5.3), we get
for all . Replacing by in (5.6), we obtain
It follows from (5.1) and (5.7) that the sequence is a Cauchy sequence. Since is complete, is convergent. Set .

Using induction, one can show that
for all and all . By taking to approach infinity in (5.8) and using (5.2), one obtains (5.4). By (5.1) and (5.3), we get
for all . Therefore, the function satisfies (1.9). To prove the uniqueness property of , let be another function satisfying (5.4). Then,
for all . Therefore, , and the proof is complete.

Corollary 5.2. *Let be a function satisfying
**
Let , and let be a mapping with and satisfying the following inequality:
**
for all . Then there exists a unique cubic mapping such that
*