Kil-Woung Jun,^1,2 Yang-Hi Lee,³ and Juri Lee²

Abstract

We establish the generalized Hyers-Ulam stability of a Pexider-type functonal equation f1(x+y+z)+f2(x−y)+f3(x−z)−f4(x−y−z)−f5(x+y)−f6(x+z)=0, which is mixed of a quadratic and an additive functional equations. Also, we obtain its general solution from the stability results.

1. Introduction

In 1940, Ulam [1] raised the following question. Under what conditions does there exist an additive mapping near an approximately additive mapping?

In 1941, Hyers [2] proved that if is a mapping satisfying(1.1)for all , where and are Banach spaces and is a given positive number, then there exists a unique additive mapping such that(1.2)for all . In 1978, Rassias [3] gave a significant generalization of Hyers' result. Rassias [4] during the 27th International Symposium on Functional Equations, that took place in Bielsko-Biala, Poland, in 1990, asked the question whether such a theorem can also be proved for a more general setting. Gadja [5] following Rassias's approach [3] gave an affirmative solution to the question. Recently, Găvruţa [6] obtained a further generalization of Rassias' theorem, the so-called generalized Hyers-Ulam-Rassias stability (see also [4, 7–10]). Jun et al. [11–13] also obtained the Hyers-Ulam-Rassias stability of the Pexider equation of . Quadratic functional equation was used to characterize inner product spaces [14]. Several other functional equations were also to characterize inner product spaces. A square norm on an inner product space satisfies the important parallelogram equality(1.3)The functional equation(1.4)is related to a symmetric biadditive function [14]. It is natural that each equation is called a quadratic functional equation. A stability problem for the quadratic functional equation was proved by Skof [15] for a function , where is a normed space and a Banach space. Cholewa [16] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [17] proved the Hyers-Ulam-Rassias stability of the quadratic functional equation. Jun and Lee [13, 18–22] proved the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equation(1.5)Now, we introduce the following new Pexider type functional equation:(1.6)which is mixed of a quadratic and an additive functional equations. In this paper, we establish the generalized Hyers-Ulam-Rassias stability for (1.6) on the punctured domain and obtain its general solution from the stability results. Throughout this paper, let and be a normed space and a Banach space, respectively. For convenience, we employ the operators as follows: for a given function , let , be functions defined by(1.7)for all .

2. Generalized Hyers-Ulam-Rassias Stability

We need the following lemma to prove our main results.

Lemma 2.1.1. Let be a positive real number. Let be a map such that (2.1) or (2.2) Suppose that the function satisfies the inequality (2.3) for all and . Then, there exists exactly one function satisfying (2.4)

Proof. First we assume that satisfies(2.5)for all . Replacing by and dividing it by in (2.3), we have(2.6)for all and . Induction argument implies that(2.7)for all and . Hence(2.8)for all positive integers and . This shows that is a Cauchy sequence for and thus converges. Therefore, we can define such that(2.9)for all . From (2.7) and the definition of , we obtain(2.10)for all . Now, let be another mapping satisfying the above inequality and equality. Then, it follows that(2.11)which tends to zero by the definition of as for all . So we can conclude that for all . This proves the uniqueness of .

Next we assume that satisfies(2.12)for all . Replacing by and multiplying it by in (2.3), we have(2.13)for all and . Induction argument implies that(2.14)for all and . Hence(2.15)for all positive integers and . This shows that is a Cauchy sequence for and thus converges. Therefore we can define such that(2.16)for all . From (2.14) and the definition of , we obtain(2.17)for all .

The uniqueness of is proved similarly as the first case. This completes the proof.

We establish the stability results for the even functions in Theorems 2.2 and 2.3.

Theorem 2.2. Let be a function such that (a) holds for all . If the even functions satisfy the inequality (2.18) for all , then there exists exactly one quadratic function satisfying the inequalities (2.19) (2.20) for all , where (2.21) for all . Moreover, the function is given by (2.22) for all and for .

Proof. Replace by in (2.18) to obtain(2.23)for all . From (2.18) and (2.23), we get(2.24)for all . Let the functions be defined by(2.25)for all . Then, it follows from (2.24) that(2.26)for all , where . Replace and by and in (2.26) to get(2.27)for all .

Replacing , by in (2.26) and using (2.27), we get(2.28)for all . Replacing , , by , , in (2.26) and using (2.27), one obtains(2.29)for all . From (2.28) and the above inequality, we have(2.30)for all . Replacing by and dividing it by in the above inequality, we get(2.31)for all . By Lemma 2.1, there exists for all satisfying(2.32)for all , where(2.33)By the similar method in obtaining inequality (2.32), we get(2.34)for all , where(2.35)From (2.27), we have(2.36)for all . From (2.36), we can define a map by(2.37)for all . It follows from (2.26), (2.32), and (2.37) that(2.38)for all . Replacing by , dividing it by in the above inequality and taking the limit in the resulted inequality as , we have(2.39)for all . Using (2.26), (2.36), (2.37), and (2.39), we obtain(2.40)for all . Replacing and by in (2.40) and using the fact , we have(2.41)for all . Replace and by and in (2.40) to have(2.42)for all . Subtracting (2.41) from (2.42) and using the evenness of , we lead to(2.43)for all .

On the other hand, it follows from (2.18) and (2.23) that(2.44)for all . Let the functions be defined by(2.45)for all .

From (2.44), we have(2.46)for all . Replace , by in (2.46) to get(2.47)for all . Replace , , by , , in (2.46) to get(2.48)for all . From (2.47) and the above inequality, we have(2.49)for all .

Replace , , by , , in (2.46) to get(2.50)for all . From (2.47) and the above inequality, we get(2.51)for all . It follows from (2.46) that(2.52)for all . By the definitions of , we have(2.53)for all . Hence by using (2.32), (2.34), (2.36), (2.37), (2.38), (2.49), (2.51), and (2.52), the inequalities in (2.19) can be shown. The uniqueness of follows from Lemma 2.1.

Theorem 2.3. Let be a function such that (a^') holds for all . If the even functions satisfy inequality (2.18) for all , then there exists exactly one quadratic function satisfying inequalities (2.19) for all , where (2.54) Moreover, the function is given by (2.55) for all and for .

Proof. The proof is similar to that of Theorem 2.2.

Applying Theorems 2.2 and 2.3, we get the following corollary in the sense of Rassias inequality.

Corollary 2.4.1. Let and . If the even functions , , satisfy (2.56) for all .

Then there exist exactly one quadratic function satisfying (2.57) for all and . Moreover, the function is given by (2.58) for all and

Proof. Apply Theorem 2.2 for and Theorem 2.3 for .

We establish Theorems 2.5 and 2.6 for the odd functions.

Theorem 2.5. Let be a function such that (b) holds for all . If the odd functions satisfy (2.59) for all , then there exist exactly three additive functions satisfying (2.60) (2.61) for all , where (2.62) Moreover, the functions are given by (2.63) for all .

Proof. Replace by in (2.59) to obtain(2.64)for all . Let the functions be defined by(2.65)for all . From (2.59) and (2.64), we get(2.66)for all . From (2.66), we have(2.67)for all . It follows from (2.66) and (2.67) that(2.68)for all . Replacing by and dividing it by in the above inequality, we obtain(2.69)for all . Applying Lemma 2.1, we obtain(2.70)for all . Similarly we have(2.71)for all . From (2.67), we get(2.72)for all and we can define a function by(2.73)for all . It follows from (2.66) and (2.70) that(2.74)for all . Replacing by , dividing it by in the above inequality and taking the limit in the resulted inequality as , we obtain(2.75)for all . From (2.73) and (2.75), we have(2.76)for all . Replace and by and in (2.76) to obtain(2.77)for all . Replace and by and in (2.76) to get(2.78)for all . Since and , using the above two equalities, we have(2.79)for all . Hence, is an additive function.

Let the functions be defined by(2.80)for all . From (2.59) and (2.64), we have(2.81)for all . It follows from (2.81) that(2.82)for all . Applying Lemma 2.1, we obtain an odd function defined by(2.83)and the inequality(2.84)holds for all . Similarly we have an odd function defined by(2.85)for all and the inequality(2.86)for all . Replace , , by , , in (2.81) to get(2.87)for all . Replacing by and dividing it by in the above inequality, we obtain(2.88)for all . Taking the limit in the above inequality as , we have(2.89)for all . It follows from (2.81) that(2.90)for all . Applying Lemma 2.1 and (2.89), we have(2.91)for all . From (2.81), (2.83), (2.85), and (2.89), we have(2.92)for all . Replace and by and in (2.92) to get(2.93)for all . Replace and by and in (2.92) to get(2.94)for all . From the above two equalities, we get(2.95)for all . Since , we have(2.96)for all . Hence is additive, that is,(2.97)for all . Replace by in (2.92) to obtain(2.98)for all . Since is additive, we have(2.99)for all . From this and (2.98), we get(2.100)for all . From this and , we have(2.101)for all . Since and are additive, is additive.

From (2.74), (2.91), and the definitions of , we have(2.102)for all . The rest of inequalities in (2.60) can be shown by the similar method.

Theorem 2.6. Let be a function such that (b^') holds for all . If the odd functions satisfy inequalities (2.59) for all , then there exist exactly three additive functions satisfying the inequalities (2.60) for all , where (2.103) Moreover, the functions are given by (2.104) for all .

Proof. The proof is similar to that of Theorem 2.5.

Applying Theorems 2.5 and 2.6, we get the following corollary in the sense of Rassias inequality.

Corollary 2.7.2.7. Let . If the odd functions , , satisfy (2.105) for all .

Then there exist exactly three additive functions satisfying (2.106) for all . Moreover, the functions are given by (2.107) for all .

Proof. Apply Theorem 2.5 for and Theorem 2.6 for .

We establish the following theorem for the general case from Theorems 2.2 and 2.5.

Theorem 2.8. Let be a function that satisfies conditions (a) and (b). Suppose that the functions , , satisfy the inequality (2.108) for all . Then there exist exactly one quadratic function and exactly three additive functions satisfying (2.109) for all , where (2.110) for all . Moreover, the function is given by (2.111) for and the functions are given by (2.112) for all, .

Proof. From (2.108), we obtain(2.113)for all . From (2.108) and this inequality, one gets(2.114)for all , where , for all , . Since is an even function, is an odd function, and , we can apply Theorems 2.2 and 2.5 to get the desired result.

We establish the following theorem for the general case from Theorems 2.2 and 2.6.

Theorem 2.9. Let be a function that satisfies conditions (a) and (b^'). If the functions satisfy inequalities (2.108) for all , then there exist exactly one quadratic function and exactly three additive functions satisfying the inequalities in Theorem 2.8 for all , where are as in Theorem 2.8 and (2.115) for all . Moreover, the function is given by (2.111) and the functions are given by (2.116) (2.117) for all .

We establish the following theorem for the general case from Theorems 2.3 and 2.6.

Theorem 2.10. Let be a function that satisfies conditions (a^') and (b^'). If the functions satisfy inequalities (2.108) for all , then there exist exactly one quadratic function and exactly three additive functions satisfying the inequalities in Theorem 2.8 for all , where are as in Theorem 2.9 and (2.118) for all . Moreover, the function is given by (2.119) for and the functions are given by (2.116) for all .

Corollary 2.11.3. Let and . Suppose that the functions , , satisfy (2.120) for all .

Then there exist exactly one quadratic function and three additive functions satisfying (2.121) for all . Moreover, the function is given by (2.111) for and (2.119) for and the functions are given by (2.112) for and (2.116) for .

Corollary 2.12.4. Let be a fixed real number. Suppose that the functions , , satisfy (2.122) for all .

Then there exist exactly one quadratic function and three additive functions satisfying (2.123) for all . Moreover, the function is given by (2.111) for and the functions are given by (2.112) for all .

Now we obtain the general solution of (1.6) from Corollary 2.12.

Corollary 2.13.5. Suppose that the functions , , satisfy (2.124) for all .

Then there exist exactly one quadratic function and three additive functions satisfying (2.125) for all . Moreover, the function is given by (2.126) for and the functions are given by (2.127) for all .

References

1. S. M. Ulam, Problems in Modern Mathematics, chapter VI, John Wiley & Sons, New York, NY, USA, 1960.
2. D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, no. 4, 222 pages, 1941.
3. Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, 297 pages, 1978.
4. Th. M. Rassias, “Problem 16; 2, Report of the 27th International Symposium on Functional Equations,” Aequationes Mathematicae, vol. 39, no. 2-3, 292 pages, 1990.
5. Z. Gajda, “On stability of additive mappings,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, 431 pages, 1991.
6. P. Găvruţa, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, 431 pages, 1994.
7. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, vol. 34 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, 1998.
8. Th. M. Rassias, “On the stability of functional equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, 264 pages, 2000.
9. Th. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Applicandae Mathematicae, vol. 62, no. 1, 23 pages, 2000.
10. Th. M. Rassias and P. Šemrl, “On the behavior of mappings which do not satisfy Hyers-Ulam stability,” Proceedings of the American Mathematical Society, vol. 114, no. 4, 989 pages, 1992.
11. K.-W. Jun and Y.-H. Lee, “On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic equation. II,” in Functional Equations, Inequalities and Applications, p. 39, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.
12. Y.-H. Lee and K.-W. Jun, “A generalization of the Hyers-Ulam-Rassias stability of the Pexider equation,” Journal of Mathematical Analysis and Applications, vol. 246, no. 2, 627 pages, 2000.
13. K.-W. Jun, D.-S. Shin, and B.-D. Kim, “On Hyers-Ulam-Rassias stability of the Pexider equation,” Journal of Mathematical Analysis and Applications, vol. 239, no. 1, 20 pages, 1999.
14. J. Aczél and J. Dhombres, Functional Equations in Several Variables, vol. 31 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1989.
15. F. Skof, “Local properties and approximation of operators,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, 113 pages, 1983.
16. P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1, 76 pages, 1984.
17. S. Czerwik, “On the stability of the quadratic mapping in normed spaces,” Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 62, 59 pages, 1992.
18. J. Wang, “Some further generalizations of the Hyers-Ulam-Rassias stability of functional equations,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, 406 pages, 2001.
19. K.-W. Jun, J.-H. Bae, and Y.-H. Lee, “On the Hyers-Ulam-Rassias stability of an n-dimensional Pexiderized quadratic equation,” Mathematical Inequalities & Applications, vol. 7, no. 1, 63 pages, 2004.
20. K.-W. Jun and H.-M. Kim, “On the stability of an n-dimensional quadratic and additive functional equation,” Mathematical Inequalities & Applications, vol. 9, no. 1, 153 pages, 2006.
21. K.-W. Jun and Y.-H. Lee, “On the Hyers-Ulam-Rassias stability of a generalized quadratic equation,” Bulletin of the Korean Mathematical Society, vol. 38, no. 2, 261 pages, 2001.
22. K.-W. Jun and Y.-H. Lee, “On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality,” Mathematical Inequalities & Applications, vol. 4, no. 1, 93 pages, 2001.