Solutions and Stability of Generalized Mixed Type QC Functional Equations in Random Normed Spaces

Abstract

We achieve the general solution and the generalized stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary t-norms: , where for any fixed integer with .

1. Introduction

In 1940, the stability problem of functional equations originated from a question given by Ulam [1], that is, the stability of group homomorphisms.

Let be a group and be a metric group with the metric . For any , does there exist , such that, if a mapping satisfies inequality .

For all , then there exists a homomorphism with for all ?

In other words, under what condition, does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation.

In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces.

Let be a mapping between Banach spaces and such that, for some

(1.1)

Then there exists a unique additive mapping such that

(1.2)

Moreover, if is continuous in for any fixed then is linear.

In 1978, Rassias [3] provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded. In 1991, Gajda [4] answered the question for the case , which was raised by Rassias. This new concept is known as the Hyers-Ulam-Rassias stability of functional equations (see [5–17]).

The functional equation

(1.3)

is related to symmetric biadditive mapping. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.3) is said to be a quadratic mapping. It is well known that a mapping between real vector spaces and is quadratic if and only if there exits a unique symmetric biadditive mapping such that for all (see [5, 18]). The biadditive mapping is given by

(1.4)

Hyers-Ulam-Rassias stability problem for the quadratic functional equation (1.3) was proved by Skof [19] for a mapping , where is a normed space and is a Banach space. Cholewa [20] noticed that the theorem of Skof is still true if the relevant domain is replaced by an abelian group. In [21], Czerwik proved the Hyers-Ulam-Rassias stability of (1.3) and Grabiec [22] generalized these results mentioned above.

Recently, Jun and Kim [23] introduced the following cubic functional equation:

(1.5)

where is a mapping from a real vector space into a real vector space and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.5). The function satisfies the functional equation (1.5) which is thus called a cubic functional equation. Every solution of the cubic functional equation is said to be a cubic function. Also, Jun and Kim proved that a function between real vector spaces and is a solution of (1.5) if and only if there exits a unique function such that for all and is symmetric for each fixed one variable and is additive for fixed two variables.

In the sequel, we adopt the usual terminology, notations and conventions of the theory of random normed spaces as in [24–28]. Throughout this paper, is the space of distribution functions, that is, the space of all mappings such that is left-continuous and nondecreasing on and .   is a subset of consisting of all functions for which , where denotes the left limit of the function at the point , that is, . The space is partially ordered by the usual pointwise ordering of functions, that is, if and only if for all . The maximal element for in this order is the distribution function given by

(1.6)

Definition 1.1 (see [27]).

A mapping is called a continuous triangular norm (briefly, a continuous-norm) if satisfies the following conditions:

(a) is commutative and associative;

(b) is continuous;

(c) for all ;

(d) whenever and for all .

Typical examples of continuous -norms are , and (the Lukasiewicz -norm).

Recall (see [29, 30]) that, if is a -norm and is a given sequence of numbers in , then is defined recurrently by

(1.7)

and is defined as

It is known [30] that, for the Lukasiewicz -norm, the following implication holds:

(1.8)

Definition 1.2 (see [28]).

A random normed space (briefly, RN-space) is a triple , where is a vector space, is a continuous -norm and is a mapping from into satisfying the following conditions:

for all if and only if ;

for all , and with ;

for all and

Every normed spaces defines a random normed space , where

(1.9)

and is the minimum -norm. This space is called the induced random normed space.

Definition 1.3.

Let be a RN-space.

(1)A sequence in is said to be convergent to a point if, for any and , there exists a positive integer such that for all .

(2)A sequence in is called a Cauchy sequence if, for any and , there exists a positive integer such that for all .

(3)A RN-space is said to be complete if every Cauchy sequence in is convergent to a point in .

Theorem 1.4 (see [27]).

If is a RN-space and is a sequence in such that , then almost everywhere.

The stability of different functional equations in fuzzy normed spaces and random normed spaces has been studied in [13, 31–42].

In this paper, we deal with the following functional equation for fixed integers with :

(1.10)

where on random normed spaces. It is easy to see that the mapping is a solution of the functional equation (1.10). In Section 2, we investigate the general solution of functional equation (1.10) when is a mapping between vector spaces and, in Section 3, we establish the stability of the functional equation (1.10) in RN-spaces.

2. General Solutions

Before proceeding to the proof of Theorem 2.3 which is the main result in this section, we need the following two lemmas.

Lemma 2.1.

If an even function with satisfies (1.10), then is quadratic.

Proof.

Setting in (1.10), by the evenness of , we obtain

(2.1)

Interchanging with in (2.1), we have

(2.2)

Letting in (1.10), we have

(2.3)

It follows from (2.2) and (2.3) that

(2.4)

According to (2.2) (2.4) and using the evenness of , (1.10) can be written as

(2.5)

Replacing by in (2.5) and then using (2.2), we obtain

(2.6)

Interchanging with in (2.5), by the evenness of , we have

(2.7)

But, since , it follows from (2.6) and (2.7) that

(2.8)

which shows that is quadratic. This completes the proof.

Lemma 2.2.

If an odd function satisfies (1.10), then is cubic.

Proof.

Letting in (1.10), by the oddness of , we have

(2.9)

Setting in (1.10) and then using (2.9), we obtain

(2.10)

According to (2.9), (2.10) and using the oddness of , (1.10) can be written as

(2.11)

Letting in (2.11) and using (2.9), by the oddness of it follows that

(2.12)

Replacing by in (2.11), we have

(2.13)

Now, replacing by in (2.11) and using (2.12), we have

(2.14)

Substituting with in (2.11) and then with in (2.11), we obtain

(2.15)

(2.16)

If we subtract (2.16) from (2.15), we have

(2.17)

Interchanging with in (2.17) and using the oddness of , we get

(2.18)

Thus it follows from (2.14) and (2.18) that

(2.19)

Again, substituting with in (2.11) and then with in (2.11), we get, by the oddness of

(2.20)

(2.21)

Then, by adding (2.20) to (2.21) and then using (2.13), we have

(2.22)

Finally, if we compare (2.19) with (2.22), then we conclude that

(2.23)

Therefore, is a cubic function. This completes the proof.

Theorem 2.3.

A function with satisfies (1.10) for all if and only if there exist functions and such that for all where the function is symmetric for each fixed one variable and is additive for fixed two variables and is symmetric biadditive.

Proof.

Let be a mapping with satisfies (1.10). We decompose into the even part and odd part by putting

(2.24)

It is clear that for all and it is easy to show that the functions and satisfy (1.10). Hence, by Lemmas 2.1 and 2.2, we know that the functions and are quadratic and cubic, respectively. Thus there exist a symmetric biadditive function such that for all and the function such that for all where the function is symmetric for each fixed one variable and is additive for fixed two variables. Therefore, we get for all

Conversely, let for all where the function is symmetric for each fixed one variable and is additive for fixed two variables and is biadditive. By a simple computation, one can show that the functions and satisfy the functional equation (1.10). Thus the function satisfies (1.10). This completes the proof.

3. Stability Problems

From now on, we suppose that is a real linear space, is a complete RN-space and is a function with for which there exists a mapping ( is denoted by ) with the property:

(3.1)

where for all .

Theorem 3.1.

Let

(3.2)

Suppose that an even function with satisfies inequality

(3.3)

where for all . Then there exists a unique quadratic mapping such that

(3.4)

Proof.

It follows from (3.3) and the evenness of that

(3.5)

Setting in (3.5), we get

(3.6)

If we replace by in (3.6) and divide both sides of (3.6) by 2, we get

(3.7)

In other words, we have

(3.8)

Therefore, it follows that

(3.9)

Hence we have

(3.10)

This means that

(3.11)

Since by the triangle inequality, it follows that

(3.12)

In order to prove the convergence of the sequence , we replace with in (3.12) to find that

(3.13)

Since the right hand side of inequality (3.13) tends to as , the sequence is a Cauchy sequence. Therefore, we may define

(3.14)

Now, we show that is a quadratic mapping. In fact, replacing with and , respectively, in (3.3) and then taking the limit as , we find that satisfies (1.10) for all Therefore, the mapping is quadratic.

To prove (3.4) taking the limit as in (3.12), we get (3.4).

Finally, to prove the uniqueness of the quadratic function subject to (3.4), assume that there exists a quadratic function which satisfies (3.4). Since and for all and it follows from (3.4) that

(3.15)

for all and . Therefore, by letting in inequality (3.15), we find that . This completes the proof.

Theorem 3.2.

Let be an odd mapping with satisfies inequality

(3.16)

where for all . If

(3.17)

(3.18)

then there exists a unique cubic mapping such that

(3.19)

for all and .

Proof.

It follows from (3.15) and the oddness of that

(3.20)

By putting in (3.20) and using we obtain

(3.21)

If we replace by in (3.21) and divide both sides of (3.21) by we get

(3.22)

Letting in (3.20), we get

(3.23)

Therefore, we have

(3.24)

It follows from (3.22) and (3.24) that

(3.25)

Let

(3.26)

Then we get

(3.27)

It follows that

(3.28)

Hence we have

(3.29)

which implies that

(3.30)

Thus we have

(3.31)

Since by the triangle inequality, it follows that

(3.32)

In order to prove the convergence of the sequence , we replace with in (3.32) to find that

(3.33)

Since the right hand side of inequality (3.33) tends to as , the sequence is a Cauchy sequence. Therefore, we can define

(3.34)

Now, we show that is a cubic mapping. In fact, replacing with and in (3.15), respectively, and then taking the limit , we find that satisfies (1.10) for all Therefore, the mapping is cubic.

Letting the limit in (3.32), we get inequality (3.18) by (3.26).

Finally, to prove the uniqueness of the cubic function subject to inequality (3.19), assume that there exists a cubic function which satisfies inequality (3.19). Since and for all and it follows from (3.19) that

(3.35)

for all and . By letting in inequality (3.35), we know that . This completes the proof.

Theorem 3.3.

If

(3.36)

for all and and

(3.37)

then there exist a unique quadratic mapping and a unique cubic mapping such that

(3.38)

Proof.

If for all Then , and

(3.39)

where for all Hence, in view of Theorem 3.1, there exists a unique quadratic function such that

(3.40)

Let for all Then and

(3.41)

where for all From Theorem 3.2, it follows that there exists a unique cubic mapping such that

(3.42)

Obviously, (3.38) follows from (3.40) and (3.42). This completes the proof.