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A Fixed Point Approach to the Stability of an Additive-Quadratic-Cubic-Quartic Functional Equation

Abstract

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation in Banach spaces.

1. Introduction and Preliminaries

The stability problem of functional equations is 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 byAoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [4] has provided a lot of influence in the development of what we call generalized Hyers-Ulam stability or as Hyers-Ulam-Rassias stability of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Găvruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach.

The functional equation

(1.1)

is called a quadratic functional equation. In particular, every solution of the 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 [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 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 [919]).

In [20], Jun and Kim considered the following cubic 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.

In [21], Lee et al. considered the following quartic functional equation

(1.3)

which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic mapping. Quartic functional equations have been investigated in [22, 23].

Let be a set. A function is called a generalized metric on if satisfies

(1) if and only if ;

(2) for all ;

(3) for all .

We recall a fundamental result in fixed point theory.

Theorem 1.1 (see [24, 25]).

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

(1.4)

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 Th. M. Rassias [26] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [2732]).

This paper is organized as follows. In Section 2, we prove the generalized Hyers-Ulam stability of the additive-quadratic-cubic-quartic functional equation

(1.5)

in Banach spaces for an odd case. In Section 3, we prove the generalized Hyers-Ulam stability of the additive-quadratic-cubic-quartic functional equation (1.5) in Banach spaces for an even case.

Throughout this paper, assume that is a vector space and that is a Banach space.

2. Generalized Hyers-Ulam Stability of the Functional Equation (1.5): An Odd Case

For a given mapping , we define

(2.1)

for all .

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

Note that the fundamental ideas in the proofs of the main results in Sections 2 and 3 are contained in [24, 27, 28].

Theorem 2.1.

Let be a function such that there exists an with

(2.2)

for all . Let be an odd mapping satisfying

(2.3)

for all . Then there is a unique cubic mapping such that

(2.4)

for all .

Proof.

Letting in (2.3), we get

(2.5)

for all .

Replacing by in (2.3), we get

(2.6)

for all .

By (2.5) and (2.6),

(2.7)

for all . Letting and for all , we get

(2.8)

for all .

Consider the set

(2.9)

and introduce the generalized metric on :

(2.10)

where, as usual, . It is easy to show that is complete (see the proof of Lemma of [33]).

Now we consider the linear mapping such that

(2.11)

for all .

Let be given such that . Then

(2.12)

for all . Hence

(2.13)

for all . So implies that . This means that

(2.14)

for all .

It follows from (2.8) that

(2.15)

for all . So .

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

  1. (1)

    is a fixed point of , that is,

    (2.16)

for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set

(2.17)

This implies that is a unique mapping satisfying (2.16) such that there exists a satisfying

(2.18)

for all .

  1. (2)

    as . This implies the equality

    (2.19)

for all .

  1. (3)

    , which implies the inequality

    (2.20)

This implies that the inequality (2.4) holds.

By (2.3),

(2.21)

for all and all . So

(2.22)

for all and all . So

(2.23)

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

Corollary 2.2.

Let and let be a real number with . Let be a normed vector space with norm . Let be an odd mapping satisfying

(2.24)

for all . Then there is a unique cubic mapping such that

(2.25)

for all .

Proof.

The proof follows from Theorem 2.1 by taking

(2.26)

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

Theorem 2.3.

Let be a function such that there exists an with

(2.27)

for all . Let be an odd mapping satisfying (2.3). Then there is a unique cubic mapping such that

(2.28)

for all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 2.1.

Consider the linear mapping such that

(2.29)

for all .

It follows from (2.8) that

(2.30)

for all . So .

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.4.

Let and let be a real number with . Let be a normed vector space with norm . Let be an odd mapping satisfying (2.24). Then there is a unique cubic mapping such that

(2.31)

for all .

Proof.

The proof follows from Theorem 2.3 by taking

(2.32)

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

Theorem 2.5.

Let be a function such that there exists an with

(2.33)

for all . Let be an odd mapping satisfying (2.3). Then there is a unique additive mapping such that

(2.34)

for all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 2.1.

Letting and for all in (2.7), we get

(2.35)

for all .

Now we consider the linear mapping such that

(2.36)

for all .

It follows from (2.35) that

(2.37)

for all . So .

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.6.

Let and let be a real number with . Let be a normed vector space with norm . Let be an odd mapping satisfying (2.24). Then there is a unique additive mapping such that

(2.38)

for all .

Theorem 2.7.

Let be a function such that there exists an with

(2.39)

for all . Let be an odd mapping satisfying (2.3). Then there is a unique additive mapping such that

(2.40)

for all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 2.1.

Consider the linear mapping such that

(2.41)

for all .

It follows from (2.35) that

(2.42)

for all . So .

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.8.

Let and let be a real number with . Let be a normed vector space with norm . Let be an odd mapping satisfying (2.24). Then there is a unique additive mapping such that

(2.43)

for all .

3. Generalized Hyers-Ulam Stability of the Functional Equation (1.5): An Even Case

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

Theorem 3.1.

Let be a function such that there exists an with

(3.1)

for all . Let be an even mapping satisfying and (2.3). Then there is a unique quartic mapping such that

(3.2)

for all .

Proof.

Letting in (2.3), we get

(3.3)

for all .

Replacing by in (2.3), we get

(3.4)

for all .

By (3.4) and (3.5),

(3.5)

for all . Letting for all , we get

(3.6)

for all .

Let be the generalized metric space defined in the proof of Theorem 2.1.

It follows from (3.16) that

(3.7)

for all . So .

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 3.2.

Let and let be a real number with . Let be a normed vector space with norm . Let be an even mapping satisfying and (2.24). Then there is unique quartic mapping such that

(3.8)

for all .

Theorem 3.3.

Let be a function such that there exists an with

(3.9)

for all . Let be an even mapping satisfying and (2.3). Then there is a unique quartic mapping such that

(3.10)

for all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 2.1.

Consider the linear mapping such that

(3.11)

for all .

It follows from (3.16) that

(3.12)

for all . So .

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 3.4.

Let and let be a real number with . Let be a normed vector space with norm . Let be an even mapping satisfying and (2.24). Then there is a unique quartic mapping such that

(3.13)

for all .

Theorem 3.5.

Let be a function such that there exists an with

(3.14)

for all . Let be an even mapping satisfying and (2.3). Then there is a unique quadratic mapping such that

(3.15)

for all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 2.1.

Letting for all in (3.6), we get

(3.16)

for all .

Now we consider the linear mapping such that

(3.17)

for all .

It follows from (3.16)that

(3.18)

for all . So .

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 3.6.

Let and let be a real number with . Let be a normed vector space with norm . Let be an even mapping satisfying and (2.24). Then there is a unique quadratic mapping such that

(3.19)

for all .

Theorem 3.7.

Let be a function such that there exists an with

(3.20)

for all . Let be an even mapping satisfying and (2.3). Then there is a unique quadratic mapping such that

(3.21)

for all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 2.1.

Consider the linear mapping such that

(3.22)

for all .

It follows from (3.16) that

(3.23)

for all . So .

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 3.8.

Let and let be a real number with . Let be a normed vector space with norm . Let be an even mapping satisfying and (2.24). Then there is a unique quadratic mapping such that

(3.24)

for all .

4. Generalized Hyers-Ulam Stability of the Functional Equation (1.5)

One can easily show that an odd mapping satisfies (1.5) if and only if the odd mapping is an additive-cubic mapping, that is,

(4.1)

It was shown in of [34, Lemma ] that and are cubic and additive, respectively, and that .

One can easily show that an even mapping satisfies (1.5) if and only if the even mapping is a quadratic-quartic mapping, that is,

(4.2)

It was shown in of [35, Lemma ] that and are quartic and quadratic, respectively, and that . Functional equations of mixed type have been investigated in [36, 37].

Let and . Then is odd and is even. and satisfy the functional equation (1.5). Let and . Then . Let and . Then . Thus

(4.3)

So we obtain the following results.

Theorem 4.1.

Let be a function such that there exists an with

(4.4)

for all . Let be a mapping satisfying and (2.3). Then there exist an additive mapping , a quadratic mapping , a cubic mapping and a quartic mapping such that

(4.5)

for all .

Proof.

Since , , and . The result follows from Theorems 2.1, 2.5, 3.1, and 3.5.

Corollary 4.2.

Let and let be a real number with . Let be a mapping satisfying and (2.24). Then there exist an additive mapping , a quadratic mapping , a cubic mapping and a quartic mapping such that

(4.6)

for all .

Theorem 4.3.

Let be a function such that there exists an with

(4.7)

for all . Let be a mapping satisfying and (2.3). Then there exist an additive mapping , a quadratic mapping , a cubic mapping , and a quartic mapping such that

(4.8)

for all .

Proof.

Since , , and . The result follows from Theorems 2.3, 2.7, 3.3, and 3.7.

Corollary 4.4.

Let and let be a real number with . Let be a mapping satisfying and (2.24). Then there exist an additive mapping , a quadratic mapping , a cubic mapping and a quartic mapping such that

(4.9)

for all .

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Acknowledgments

The first and third authors were supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0071229) and (NRF-2009-0070788), respectively.

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Lee, J., Kim, Jh. & Park, C. A Fixed Point Approach to the Stability of an Additive-Quadratic-Cubic-Quartic Functional Equation. Fixed Point Theory Appl 2010, 185780 (2010). https://doi.org/10.1155/2010/185780

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Keywords

  • Banach Space
  • Functional Equation
  • Quadratic Mapping
  • Additive Mapping
  • Normed Vector Space