A Fixed Point Approach to the Stability of Quintic and Sextic Functional Equations in Quasi--Normed Spaces

Abstract

We achieve the general solution of the quintic functional equation and the sextic functional equation . Moreover, we prove the stability of the quintic and sextic functional equations in quasi--normed spaces via fixed point method.

1. Introduction and Preliminaries

A basic question in the theory of functional equations is as follows: when is it true that a function, which approximately satisfies a functional equation, must be close to an exact solution of the equation? If the problem accepts a unique solution, we say the equation is stable (see [1]). The first stability problem concerning group homomorphisms was raised by Ulam [2] in 1940 and affirmatively solved by Hyers [3]. The result of Hyers was generalized by Rassias [4] for approximate linear mappings by allowing the Cauchy difference operator to be controlled by . In 1994, a generalization of Rassias' theorem was obtained by Găvruţa [5], who replaced by a general control function in the spirit of Rassias' approach. 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, e.g., [6–30] and references therein).

In 1996, Isac and Rassias [30] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. The stability problems of several various functional equations have been extensively investigated by a number of authors using fixed point methods (see [8, 10, 17, 22]).

The functional equation

(1.1)

is said to be a quadratic functional equation because the quadratic function is a solution of the functional equation (1.1). Every solution of the quadratic functional equation is said to be a quadratic mapping. A quadratic functional equation was used to characterize inner product spaces.

In 2001, Rassias [25] introduced the cubic functional equation

(1.2)

and established the solution of the Ulam-Hyers stability problem for these cubic mappings. It is easy to show that the function satisfies the functional equation (1.2), which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic mapping. The quartic functional equation

(1.3)

was introduced by Rassias [26]. It is easy to show that the function is a solution of (1.3). Every solution of the quartic functional equation is said to be a quartic mapping.

In this paper, we achieve the general solutions of the quintic functional equation

(1.4)

and the sextic functional equation

(1.5)

Moreover, we prove the stability of the quintic and sextic functional equations in quasi--normed spaces via fixed point method, and also using Gajda's example to give two counterexamples for a singular case. Since is a solution of (1.4), we say that it is a quintic functional equation. Similarly, is a solution of (1.5), and we say that it is a sextic functional equation. Every solution of the quintic or sextic functional equation is said to be a quintic or sextic mapping, respectively.

For the sake of convenience, we recall some basic concepts concerning quasi--normed spaces (see [24]).

Definition 1.1.

Let be a fix real number with , and let denote either or . Let be a linear space over . A quasi--norm is a real-valued function on satisfying the following:

(1) for all and if and only if ;

(2) for all and all ;

(3)there is a constant such that for all .

A quasi--normed space is a pair , where is a quasi--norm on . The smallest possible is called the modulus of concavity of . A quasi--Banach space is a complete quasi--normed space.

A quasi--norm is called a -norm () if

(1.6)

for all . In this case, a quasi--Banach space is called a -Banach space. We can refer to [7, 17] for the concept of quasinormed spaces and -Banach spaces.

Given a -norm, the formula gives us a translation invariant metric on . By the Aoki-Rolewicz theorem, each quasi-norm is equivalent to some -norm. Since it is much easier to work with -norms than quasi-norms, henceforth we restrict our attention mainly to -norms.

2. General Solutions to Quintic and Sextic Functional Equations

In this section, let and be vector spaces. In the following theorem, we investigate the general solutions of the functional equation (1.4) and (1.5). Some basic facts on -additive symmetric mappings can be found in [29].

Theorem 2.1.

A function is a solution of the functional equation (1.4) if and only if is of the form for all , where is the diagonal of the 5-additive symmetric map .

Proof.

Assume that satisfies the functional equation (1.4). Replacing in (1.4), one gets . Replacing with and in (1.4), respectively, and adding the two resulting equations, we obtain . Replacing with and in (1.4), respectively, and subtracting the two resulting equations, we get

(2.1)

Replacing with in (1.4), we have

(2.2)

for all . Subtracting (2.1) and (2.2), we find

(2.3)

for all . Replacing with in (1.4), and multiplying the result by 11, we obtain

(2.4)

for all . Subtracting (2.3) and (2.4), one gets

(2.5)

for all . Replacing with in (1.4), and multiplying the result by 15, one finds

(2.6)

for all . Subtracting (2.5) and (2.6), we arrive at

(2.7)

for all .

On the other hand, one can rewrite the functional equation (1.4) in the form

(2.8)

for all . By [29, Theorems 3.5 and 3.6], is a generalized polynomial function of degree at most 6, that is, is of the form

(2.9)

where is an arbitrary element of and is the diagonal of the -additive symmetric map for . By and for all , we get , and the function is odd. Thus we have . It follows that . By (2.7) and whenever and , we obtain . Hence , and so for all . Therefore, .

Conversely, assume that for all , where is the diagonal of the 5-additive symmetric map . From , , , , , and , we see that satisfies (1.4), which completes the proof of Theorem 2.1.

Theorem 2.2.

A function is a solution of the functional equation (1.5) if and only if is of the form for all , where is the diagonal of the 6-additive symmetric map .

Proof.

Assume that satisfies the functional equation (1.5). Replacing in (1.5), one gets . Substituting by in (1.5) and subtracting the resulting equation from (1.5) and then by , we obtain . Replacing with and in (1.5), respectively, we get

(2.10)

(2.11)

for all . Subtracting (2.10) and (2.11), we find

(2.12)

for all . Replacing with in (1.5) and from and and then multiplying by 6, we obtain

(2.13)

for all . Subtracting (2.12) and (2.13), one gets

(2.14)

for all . Replacing with (and then multiplying by 10) and with (and then multiplying by 15) in (1.5), respectively, we find

(2.15)

by and , as well as

(2.16)

by and . Subtracting (2.14) and (2.16), one gets

(2.17)

for all . Subtracting (2.15) and (2.17), we have

(2.18)

for all . Hence

(2.19)

for all .

On the other hand, one can rewrite the functional equation (1.5) in the form

(2.20)

for all . By [29, Theorems 3.5 and 3.6], is a generalized polynomial function of degree at most 6, that is is of the form,

(2.21)

where is an arbitrary element of and is the diagonal of the -additive symmetric map for . By and for all , we get and the function is even. Thus we have , , and . It follows that . By (2.19) and whenever and , we obtain . Hence , and so for all . Therefore, .

Conversely, assume that for all , where is the diagonal of the 6-additive symmetric map . From , , , , , , and , we see that satisfies (1.5), which completes the proof of Theorem 2.2.

3. Stability of the Quintic Functional Equation

Throughout this section, unless otherwise explicitly stated, we will assume that is a linear space, is a ()-Banach space with ()-norm . Let be the modulus of concavity of . We will establish the following stability for the quintic functional equation in quasi--normed spaces. For notational convenience, given a function , we define the difference operator

(3.1)

for all .

Lemma 3.1.

Let be fixed, with and a function such that there exists an with for all . Let be a mapping satisfying

(3.2)

for all , then there exists a uniquely determined mapping such that and

(3.3)

for all .

Proof.

Consider the set

(3.4)

and introduce the generalized metric on ,

(3.5)

It is easy to show that is a complete generalized metric space (see [8–10]).

Define a function by for all . Let be given such that , by the definition,

(3.6)

Hence

(3.7)

for all . By definition, . Therefore,

(3.8)

This means that is a strictly contractive self-mapping of with Lipschitz constant .

It follows from (3.2) that

(3.9)

for all . Therefore, by [10, Theorem 1.3], has a unique fixed point in the set . This implies that and

(3.10)

for all . Moreover,

(3.11)

This implies that the inequality (3.3) holds.

To prove the uniqueness of the mapping , assume that there exists another mapping which satisfies (3.3) and for all . Fix . Clearly, and for all . Thus

(3.12)

Since, for every , , we get . This completes the proof.

Theorem 3.2.

Let be fixed, and let be a function such that there exists an with for all . Let be a mapping satisfying

(3.13)

for all . Then there exists a unique quintic mapping such that

(3.14)

for all , where

(3.15)

for all .

Proof.

Replacing in (3.13), we get

(3.16)

Replacing and by 0 and in (3.13), respectively, we get

(3.17)

for all . Replacing and by and in (3.13), respectively, we have

(3.18)

for all . By (3.17) and (3.18), we obtain

(3.19)

for all . Replacing and by and in (3.13), respectively, we get

(3.20)

for all . Replacing and by 0 and in (3.13), respectively, we find

(3.21)

for all . By (3.20) and (3.21), we obtain

(3.22)

for all . By (3.16), (3.19), and (3.22), we have

(3.23)

for all . Replacing and by and in (3.13), respectively, we get

(3.24)

for all . Using (3.16), we have

(3.25)

for all . Hence

(3.26)

for all . By (3.23) and (3.26), we get

(3.27)

for all . Replacing and by and in (3.13), respectively, we have

(3.28)

for all . By (3.16), (3.19), and (3.28), we have

(3.29)

for all . Thus

(3.30)

for all . By (3.27) and (3.30), we obtain

(3.31)

for all . By (3.16), (3.17), and (3.19), we have

(3.32)

for all . Hence

(3.33)

for all . By (3.31) and (3.33), we get

(3.34)

for all . By Lemma 3.1, there exists a unique mapping such that and

(3.35)

for all . It remains to show that is a quintic map. By (3.13), we have

(3.36)

for all and . So

(3.37)

for all . Thus the mapping is quintic, as desired.

Corollary 3.3.

Let be a quasi--normed space with quasi--norm , and let be a -Banach space with ( )-norm . Let , , be positive numbers with and a mapping satisfying

(3.38)

for all . Then there exists a unique quintic mapping such that

(3.39)

for all , where

(3.40)

Corollary 3.4.

Let be a quasi--normed space with quasi--norm , and let be a -Banach space with ( )-norm . Let , be positive numbers with and a mapping satisfying

(3.41)

for all . Then there exists a unique quintic mapping such that

(3.42)

for all , where

(3.43)

Corollary 3.5.

Let be a quasi--normed space with quasi--norm , and let be a -Banach space with -norm . Let be positive numbers with and a mapping satisfying

(3.44)

for all . Then there exists a unique quintic mapping such that

(3.45)

for all , where and are defined as in Corollaries 3.3 and 3.4.

The following example shows that the assumption cannot be omitted in Corollary 3.4. This example is a modification of the example of Gajda [6] for the additive functional inequality (see also [7]).

Example 3.6.

Let be defined by

(3.46)

Consider that the function is defined by

(3.47)

for all . Then satisfies the functional inequality

(3.48)

for all , but there do not exist a quintic mapping and a constant such that for all .

Proof.

It is clear that is bounded by on . If or , then

(3.49)

Now suppose that . Then there exists a nonnegative integer such that

(3.50)

Hence , and for all . Hence, for ,

(3.51)

From the definition of and the inequality (3.50), we obtain that

(3.52)

Therefore, satisfies (3.48) for all . Now, we claim that the functional equation (1.4) is not stable for in Corollary 3.4 (). Suppose on the contrary that there exists a quintic mapping and constant such that for all . Then there exists a constant such that for all rational numbers (see [7]). So we obtain that

(3.53)

for all . Let with . If is a rational number in , then for all , and for this we get

(3.54)

which contradicts (3.53).

4. Stability of the Sextic Functional Equation

Throughout this section, unless otherwise explicitly stated, we will assume that is a linear space, is a ()-Banach space with ()-norm . Let be the modulus of concavity of . We will establish the following stability for the sextic functional equation in quasi--normed spaces. For notational convenience, given a function , we define the difference operator

(4.1)

for all .

Theorem 4.1.

Let be fixed, and let be a function such that there exists an with for all . Let be a mapping satisfying

(4.2)

for all . Then there exists a unique sextic mapping such that

(4.3)

for all , where

(4.4)

for all .

Proof.

Replacing in (4.2), we get

(4.5)

Replacing by in (4.2), we have

(4.6)

for all . By (4.2) and (4.6), we get

(4.7)

for all . Replacing and by 0 and in (4.2), respectively, we get

(4.8)

for all . By (4.5), (4.7), and (4.8), we have

(4.9)

for all . Replacing and by and in (4.2), respectively, we have

(4.10)

for all . Subtracting (4.9)-(4.10) and using (4.5), we obtain

(4.11)

for all . Replacing and by and in (4.2), respectively, we have

(4.12)

for all . Using (4.5) and (4.7), we get

(4.13)

for all . Hence

(4.14)

for all . Subtracting (4.11)–(4.14), we have

(4.15)

for all . Replacing and by 0 and in (4.2), respectively, we get

(4.16)

for all . By (4.5), (4.7), and (4.16), we have

(4.17)

for all . Thus

(4.18)

for all . Replacing and by and in (4.2), respectively, and then using (4.5) and (4.7), we have

(4.19)

for all . Multiply each side of (4.19) by , we have

(4.20)

for all . By (4.15) and (4.20), we get

(4.21)

for all . By (4.18) and (4.21), we obtain

(4.22)

for all . Therefore,

(4.23)

for all .

By Lemma 3.1, there exists a unique mapping such that and

(4.24)

for all . It remains to show that is a sextic map. By (4.2), we have

(4.25)

for all and . So

(4.26)

for all . Thus the mapping is sextic, as desired.

Corollary 4.2.

Let be a quasi--normed space with quasi--norm , and let be a -Banach space with -norm . Let be positive numbers with , and be a mapping satisfying

(4.27)

for all . Then there exists a unique sextic mapping such that

(4.28)

for all , where

(4.29)

Corollary 4.3.

Let be a quasi--normed space with quasi--norm , and let be a ( )-Banach space with ( )-norm . Let , be positive numbers with and a mapping satisfying

(4.30)

for all . Then there exists a unique sextic mapping such that

(4.31)

for all , where

(4.32)

Corollary 4.4.

Let be a quasi--normed space with quasi--norm , and let be a ( )-Banach space with ( )-norm . Let , , be positive numbers with and a mapping satisfying

(4.33)

for all . Then there exists a unique sextic mapping such that

(4.34)

for all , where and are defined as in Corollaries 4.2 and 4.3.

For the case , similar to Example 3.6, we have the following counterexample.

Example 4.5.

Let be defined by

(4.35)

Consider that the function is defined by

(4.36)

for all . Then satisfies the functional inequality

(4.37)

for all , but there do not exist a sextic mapping and a constant such that for all .

Proof.

It is clear that is bounded by on . If or , then

(4.38)

Now suppose that . Then there exists a non-negative integer such that

(4.39)

Similar to the proof of Example 3.6 we obtain that

(4.40)

Therefore, satisfies (4.37) for all . Now, we claim that the functional equation (1.5) is not stable for in Corollary 4.3 (). Suppose on the contrary that there exists a sextic mapping and constant such that for all . Then there exists a constant such that for all rational numbers (see [7]). So we obtain that

(4.41)

for all . Let with . If is a rational number in , then for all , and for this we get

(4.42)

which contradicts (4.41).