Fixed Points, Inner Product Spaces, and Functional Equations

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias [4] has provided a lot of influence on the development of what we call the generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. A generalization of the Rassias theorem was obtained by G vruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach.

A square norm on an inner product space satisfies the parallelogram equality

(1.1)

The functional equation

(1.2)

is called a quadratic functional equation. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [6] for mappings , where is a normed space and is a Banach space. Cholewa [7] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [8] proved the generalized Hyers-Ulam stability of the quadratic functional equation. The generalized Hyers-Ulam stability of the above quadratic functional equation and of two functional equations of quadratic type was obtained by C dariu and Radu [9].

By a square norm on an inner product space, Rassias [10] introduced the following equality:

(1.3)

for a fixed integer . By the above equality, we can define the following functional equation:

(1.4)

for all with , where is a real vector space.

A square norm on an inner product space satisfies

(1.5)

for all with (see [10]).

From the above equality we can define the following functional equation:

(1.6)

which is called a functional equation of quadratic type. In fact, in satisfies the functional equation of quadratic type. In particular, every solution of the functional equation of quadratic type is said to be a quadratic-type mapping. One can easily show that if satisfies the quadratic functional equation then satisfies the functional equation of quadratic type. 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 [11–24]).

Let be a set. Then, a function is called a generalized metric on if satisfies the following:

(1) if and only if ,

(2) for all ,

(3) for all .

We recall a fundamental result in fixed point theory.

Theorem 1.1 (see [25, 26]).

Let be a complete generalized metric space, and let be a strictly contractive mapping with the, Lipschitz constant . Then, for each given element , either

(1.7)

for all nonnegative integers or there exists a positive integer such that

(1) for all ,

(2)the sequence converges to a fixed point of ,

(3) is the unique fixed point of in the set ,

(4) for all .

In 1996, Isac and Rassias [27] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using the fixed point method, the stability problems of several functional equations have been extensively investigated by a number of authors (see [17, 28–31]).

Throughout this paper, assume that is a fixed integer greater than . Let be a real normed vector space with norm , and let be a real Banach space with norm .

In this paper, we investigate the functional equation (1.4). Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation (1.4) in real Banach spaces.

We investigate the functional equation (1.4).

Lemma 2.1.

Let and be real vector spaces. If a mapping satisfies

(2.1)

for all with , then the mapping is realized as the sum of an additive mapping and a quadratic-type mapping.

Proof.

Let and for all . Then, is an odd mapping and is an even mapping satisfying and (2.1).

Letting , and in (2.1) for the mapping , we get

(2.2)

for all . So, is an additive mapping.

Letting , and in (2.1) for the mapping , we get

(2.3)

for all . So, is a quadratic-type mapping.

For a given mapping , we define

(2.4)

for all .

Let and for all . Then, is an odd mapping and is an even mapping satisfying . If , then and .

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in real Banach spaces.

Theorem 2.2.

Let be a mapping with for which there exists a function such that there exists an such that

(2.5)

(2.6)

for all with . Then, there exists a unique quadratic-type mapping satisfying

(2.7)

for all .

Proof.

Consider the set

(2.8)

and introduce the generalized metric on :

(2.9)

By the same method given in [17, 28, 32], one can easily show that is complete.

Now we consider the linear mapping such that

(2.10)

for all .

It follows from the proof of Theorem of [25] that

(2.11)

for all .

Letting , and in (2.6), we get

(2.12)

for all . It follows from (2.12) that

(2.13)

for all . Hence, .

By Theorem 1.1, there exists a mapping satisfying the following.

1. (1)
   is a fixed point of ; that is,

   

   (2.14)
    

for all . The mapping is a unique fixed point of in the set

(2.15)

This implies that is a unique mapping satisfying (2.14) such that there exists a satisfying

(2.16)

for all .

1. (2)
   One has as . This implies the equality

   

   (2.17)
    

for all . Since is an even mapping, is an even mapping.

Moreover,one has (3) , which implies the inequality

(2.18)

This implies that inequality (2.7) holds.

It follows from (2.5), (2.6), and (2.17) that

(2.19)

for all with . So, for all with . By Lemma 2.1, the mapping is a quadratic-type mapping.

Therefore, there exists a unique quadratic-type mapping satisfying (2.7).

Corollary 2.3.

Let and be real numbers, and let be a mapping such that

(2.20)

for all with . Then, there exists a unique quadratic-type mapping satisfying

(2.21)

for all .

Proof.

The proof follows from Theorem 2.2 by taking

(2.22)

for all . Then, we can choose , and we get the desired result.

Remark 2.4.

Let be a mapping for which there exists a function satisfying (2.6) and such that there exists an such that

(2.23)

for all with . By a similar method to the proof of Theorem 2.2, one can show that there exists a unique quadratic-type mapping satisfying

(2.24)

for all .

For the case , one can obtain a similar result to Corollary 2.3: let and be positive real numbers, and let be a mapping satisfying (2.20). Then, there exists a unique quadratic-type mapping satisfying

(2.25)

for all .

Theorem 2.5.

Let be a mapping for which there exists a function such that there exists an such that

(2.26)

(2.27)

for all with . Then, there exists a unique additive mapping satisfying

(2.28)

for all .

Proof.

Consider the set

(2.29)

and introduce the generalized metric on :

(2.30)

By the same method given in [17, 28, 32], one can easily show that is complete.

Now we consider the linear mapping such that

(2.31)

for all .

It follows from the proof of Theorem of [25] that

(2.32)

for all .

Letting and in (2.27), we get

(2.33)

for all . It follows from (2.33) that

(2.34)

for all . Hence, .

By Theorem 1.1, there exists a mapping satisfying the following.

1. (1)
   is a fixed point of ; that is,

   

   (2.35)
    

for all . The mapping is a unique fixed point of in the set

(2.36)

This implies that is a unique mapping satisfying (2.35) such that there exists a satisfying

(2.37)

for all .

1. (2)
   One has as . This implies the equality

   

   (2.38)
    

for all . Since is an odd mapping, is an odd mapping;

1. (3)
   Moreover , which implies the inequality

   

   (2.39)
    

This implies that inequality (2.28) holds.

It follows from (2.26), (2.27), and (2.38) that

(2.40)

for all with . So, for all with . By Lemma 2.1, the mapping is an additive mapping.

Therefore, there exists a unique additive mapping satisfying (2.28), as desired.

Corollary 2.6.

Let and be real numbers, and let be a mapping satisfying (2.20). Then, there exists a unique additive mapping satisfying

(2.41)

for all .

Proof.

The proof follows from Theorem 2.5 by taking

(2.42)

for all . Then, we can choose , and we get the desired result.

Remark 2.7.

Let be a mapping for which there exists a function satisfying (2.27) such that there exists an such that

(2.43)

for all . By a similar method to the proof of Theorem 2.5, one can show that there exists a unique additive mapping satisfying

(2.44)

for all .

For the case , one can obtain a similar result to Corollary 2.6: let and be positive real numbers, and let be a mapping satisfying (2.20). Then, there exists a unique additive mapping satisfying

(2.45)

for all .

Since

(2.46)

Note that

(2.47)

Combining Theorems 2.2 and 2.5, we obtain the following result.

Theorem 2.8.

Let be a mapping satisfying for which there exists a function satisfying (2.5) and

(2.48)

for all with . Then, there exist an additive mapping and a quadratic type mapping such that

(2.49)

for all .

Corollary 2.9.

Let and be positive real numbers, and let be a mapping such that

(2.50)

for all with . Then, there exist an additive mapping and a quadratic-type mapping such that

(2.51)

for all .

Proof.

Define , and apply Theorem 2.8 to get the desired result.

Note that

(2.52)

Combining Remarks 2.4 and 2.7, we obtain the following result.

Remark 2.10.

Let be a mapping for which there exists a function satisfying (2.48) and such that there exists an such that

(2.53)

for all . By a similar method to the proof of Theorem 2.8, one can show that there exist an additive mapping and a quadratic-type mapping such that

(2.54)

for all .

For the case , one can obtain a similar result to Corollary 2.9: let and be positive real numbers, and let be a mapping satisfying (2.50). Then, there exist an additive mapping and a quadratic-type mapping satisfying

(2.55)

for all .