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Fixed Points, Inner Product Spaces, and Functional Equations
Fixed Point Theory and Applications volume 2010, Article number: 713675 (2010)
Abstract
Rassias introduced the following equality , , for a fixed integer . Let be real vector spaces. It is shown that, if a mapping satisfies the following functional equation for all with , which is defined by the above equality, then the mapping is realized as the sum of an additive mapping and a quadratic mapping. Using the fixed point method, we prove the generalized HyersUlam stability of the above functional equation in real Banach spaces.
1. Introduction and Preliminaries
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 HyersUlam stability or HyersUlamRassias stability of functional equations. A generalization of the Rassias theorem was obtained by Gvruţa [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
The functional equation
is called a quadratic functional equation. A generalized HyersUlam 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 HyersUlam stability of the quadratic functional equation. The generalized HyersUlam stability of the above quadratic functional equation and of two functional equations of quadratic type was obtained by Cdariu and Radu [9].
By a square norm on an inner product space, Rassias [10] introduced the following equality:
for a fixed integer . By the above equality, we can define the following functional equation:
for all with , where is a real vector space.
A square norm on an inner product space satisfies
for all with (see [10]).
From the above equality we can define the following functional equation:
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 quadratictype 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.
Let be a complete generalized metric space, and let be a strictly contractive mapping with the, Lipschitz constant . Then, for each given element , either
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 HyersUlam stability of the functional equation (1.4) in real Banach spaces.
2. Fixed Points and Functional Equations Associated with Inner Product Spaces
We investigate the functional equation (1.4).
Lemma 2.1.
Let and be real vector spaces. If a mapping satisfies
for all with , then the mapping is realized as the sum of an additive mapping and a quadratictype 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
for all . So, is an additive mapping.
Letting , and in (2.1) for the mapping , we get
for all . So, is a quadratictype mapping.
For a given mapping , we define
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 HyersUlam 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
for all with . Then, there exists a unique quadratictype mapping satisfying
for all .
Proof.
Consider the set
and introduce the generalized metric on :
By the same method given in [17, 28, 32], one can easily show that is complete.
Now we consider the linear mapping such that
for all .
It follows from the proof of Theorem of [25] that
for all .
Letting , and in (2.6), we get
for all . It follows from (2.12) that
for all . Hence, .
By Theorem 1.1, there exists a mapping satisfying the following.

(1)
is a fixed point of ; that is,
(2.14)
for all . The mapping is a unique fixed point of in the set
This implies that is a unique mapping satisfying (2.14) such that there exists a satisfying
for all .

(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
This implies that inequality (2.7) holds.
It follows from (2.5), (2.6), and (2.17) that
for all with . So, for all with . By Lemma 2.1, the mapping is a quadratictype mapping.
Therefore, there exists a unique quadratictype mapping satisfying (2.7).
Corollary 2.3.
Let and be real numbers, and let be a mapping such that
for all with . Then, there exists a unique quadratictype mapping satisfying
for all .
Proof.
The proof follows from Theorem 2.2 by taking
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
for all with . By a similar method to the proof of Theorem 2.2, one can show that there exists a unique quadratictype mapping satisfying
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 quadratictype mapping satisfying
for all .
Theorem 2.5.
Let be a mapping for which there exists a function such that there exists an such that
for all with . Then, there exists a unique additive mapping satisfying
for all .
Proof.
Consider the set
and introduce the generalized metric on :
By the same method given in [17, 28, 32], one can easily show that is complete.
Now we consider the linear mapping such that
for all .
It follows from the proof of Theorem of [25] that
for all .
Letting and in (2.27), we get
for all . It follows from (2.33) that
for all . Hence, .
By Theorem 1.1, there exists a mapping satisfying the following.

(1)
is a fixed point of ; that is,
(2.35)
for all . The mapping is a unique fixed point of in the set
This implies that is a unique mapping satisfying (2.35) such that there exists a satisfying
for all .

(2)
One has as . This implies the equality
(2.38)
for all . Since is an odd mapping, is an odd mapping;

(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
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
for all .
Proof.
The proof follows from Theorem 2.5 by taking
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
for all . By a similar method to the proof of Theorem 2.5, one can show that there exists a unique additive mapping satisfying
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
for all .
Since
Note that
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
for all with . Then, there exist an additive mapping and a quadratic type mapping such that
for all .
Corollary 2.9.
Let and be positive real numbers, and let be a mapping such that
for all with . Then, there exist an additive mapping and a quadratictype mapping such that
for all .
Proof.
Define , and apply Theorem 2.8 to get the desired result.
Note that
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
for all . By a similar method to the proof of Theorem 2.8, one can show that there exist an additive mapping and a quadratictype mapping such that
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 quadratictype mapping satisfying
for all .
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Acknowledgment
This paper was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF20090070788).
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Park, C. Fixed Points, Inner Product Spaces, and Functional Equations. Fixed Point Theory Appl 2010, 713675 (2010). https://doi.org/10.1155/2010/713675
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DOI: https://doi.org/10.1155/2010/713675
Keywords
 Banach Space
 Functional Equation
 Stability Problem
 Additive Mapping
 Positive Real Number