Applied Mathematics
Vol.10 No.10(2019), Article ID:95432,9 pages
10.4236/am.2019.1010058
Approximation of Functions by Quadratic Mapping in (β, p)-Banach Space
Xiujiao Chi, Longyin Bao, Liguang Wang*
School of Mathematical Sciences, Qufu Normal University, Qufu, China
Copyright © 2019 by author(s) and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: August 5, 2019; Accepted: September 24, 2019; Published: September 27, 2019
ABSTRACT
In this paper, we study the functions with values in (β, p)-Banach spaces which can be approximated by a quadratic mapping with a given error.
Keywords:
Hyers-Ulam-Rassias Stability, Quadratic Mapping, (β, p)-Banach Space
1. Introduction
The stability problem of functional equations originated from a question of Ulam [1] in 1940 concerning the stability of group homomorphisms.
Give a group and a metric group with the metric . Given , does there exist a such that if satisfies for all , then there is a homomorphism with for all ?
Hyers [2] gave the first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’s 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 Th. M. Rassias has provided a lot of influence in the development of what we call generalized Hyers-Ulam-Rassias stability of functional equations. Beginning around 1980, the stability problems of several functional equations and approximate homomorphisms have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [5] - [18] ).
The functional equation
is called the quadratic functional equation. Every solution of the quadratic functional equation is said to be a quadratic mapping. The Hyers-Ulam stability for quadratic functional equation was first proved by Skof [5] for mappings acting between a normed space and a Banach space. P. W. Cholewa [6] showed that Skof’s Theorem is also valid if the normed space is replaced with an abelian group.
Now we recall some basic facts concerning -Banach spaces. We fixed real numbers with and p with . Let or . Let X be linear space over . A quasi-β-norm is a real-valued function on X satisfying the following conditions:
(i) ; if and only if ;
(ii) ;
(iii) There is a constant such that .
The pair is called a quasi-β-normed space if is a quasi-β-norm on X. The smallest possible K is called the module of concavity of . A quasi-β-Banach space is a complete quasi-β-normed space.
A quasi-β-norm is called a -norm if for all . In this case, a quasi- -Banach space is called a -Banach space. For more details and related stability results on -Banach spaces, we refer to [19] [20] . Recently, L. Gǎvruta and P. Gǎvruta [21] studied the approximation of functions in Banach space. In this paper, we will consider this problem in -Banach spaces and extend previous result for quadratic functional equations.
2. Main Results
Given and . Throughout this paper we always assume that X is a linear space, Y is a -Banach space and is a mapping.
Definition 2.1. Let be a mapping. We say f is Φ-approximable by a quadratic map if there exists a quadratic mapping such that
(1)
for all . In this case, we say that Q is the quadratic Φ-approximation of f.
The following result is our main result in this paper.
Theorem 2.2. Let and suppose . Then f is Φ-approximable by a quadratic map if and only if the following two condition hold:
(i) , ;
(ii) There exists such that
In this case, the quadratic Φ-approximation of f is unique and is given by
for all .
Proof. We first assume that f is Φ-approximable by a quadratic map. Then for , we have
and
It follows that
for all . Hence
for all . By letting , we obtain condition (i) since . Since Q is quadratic, we have
for all . We take in the first position, then for all , we have
and the condition (ii) holds.
Conversely we suppose that (i) and (ii) hold. It follows from condition (ii) that for all , we have
(2)
Then is a Cauchy sequence. Indeed, by using replace x, we get
and by multipling , for all , we have
Hence, for all ,
as . Since Y is a -Banach space, the limit exists. Let in relation (2), we get condition (1).
Now we show that Q satisfies the required conditions. From the hypothesis, for all ,
Hence for all ,
Therefore
and Q is a quadratic map. Now we show the uniqueness of Q. We suppose that Q satisfies
for all and there exists a satisfying
Since Q and are quadratic mappings, we have
for all . Hence for all ,
Since , for all , we have
Hence for all , . This completes the proof. ,
Corollary 2.3. Let be a mapping satisfying
and
for all where . Suppose a function with and satisfying
(3)
for all . Then there exists a unique quadratic function such that
which is defined
for all .
Proof. Replace x and y by in (3), we have
Dividing by , we have
(4)
Replacing x by in (4), we get
(5)
Then we have
for all . We claim that
(6)
holds for all and . When , this is obviously by (4). Suppose (6) holds when , i.e. for all ,
Then for , we have
for all . By induction, (6) is true for all and . Replacing by in (3) and multiplying both side by , we have
Since
we have
for all . Hence for all ,
It follows from Theorem 2.2 (with there) that there exists a unique quadratic function Q such that
for all . ,
Theorem 2.4. Let . Suppose . Then f is Φ-approximable by a quadratic map if and only if the following two condition
(i) ;
(ii) There exists a such that
hold for all . In this case, the quadratic Φ-approximation of f is unique and is given by
Proof. The proof is similar to that of Theorem 2.2 and we omit it. ,
Corollary 2.5. Let be a mapping such that
for all . Let . Suppose all . Let a function with and satisfying
for all . Then there exists a unique quadratic function such that
for all .
Proof. The proof is similar to that of Corollary 2.3 and we omit it. ,
Funding
This article is partially supported by NSFC (11871303 and 11671133) and NSF of Shandong Province (ZR2019MA039).
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Cite this paper
Chi, X.J., Bao L.Y., and Wang, L.G. (2019) Approximation of Functions by Quadratic Mapping in (β, p)-Banach Space. Applied Mathematics, 10, 817-825. https://doi.org/10.4236/am.2019.1010058
References
- 1. Ulam, S.M. (1960) A Collection of Mathematical Problems. Interscience Publishers, New York.
- 2. Hyers, D.H. (1941) On the Stability of the Linear Functional Equation. Proceedings of the National Academy of Sciences of the United States of America, 27, 222-224. https://doi.org/10.1073/pnas.27.4.222
- 3. Aoki, T. (1950) On the Stability of the Linear Transformation in Banach Spaces. Journal of the Mathematical Society of Japan, 2, 64-66. https://doi.org/10.2969/jmsj/00210064
- 4. Rassias, Th.M. (1978) On the Stability of the Linear Mapping in Banach Space. Proceedings of the American Mathematical Society, 72, 297-300. https://doi.org/10.2307/2042795
- 5. Skof, F. (1983) Proprieta locali e approssimazione di operatori. Rendiconti del Seminario Matematico e Fisico di Milano, 53, 113-129. https://doi.org/10.1007/BF02924890
- 6. Cholewa, P.W. (1984) Remark on the Stability of Functional Equations. Aequationes Mathematicae, 27, 67-86. https://doi.org/10.1007/BF02192660
- 7. Wang, L.G., Liu, B. and Bai, R. (2010) Stability of a Mixed Type Functional Equation on Multi-Banach Spaces: A Fixed Point Approach. Fixed Point Theory and Applications, 2010, Article ID: 283827. https://doi.org/10.1155/2010/283827
- 8. Wang, L.G. (2010) The Fixed Point Method for Intuitionistic Fuzzy Stability of a Quadratic Functional Equation. Fixed Point Theory and Applications, 2010, Article ID: 107182. https://doi.org/10.1155/2010/107182
- 9. Wang, L.G. and Li, J. (2012) On the Stability of a Functional Equation Deriving from Additive and Quadratic Functions. Advances in Difference Equations, 2012, Article No. 98. https://doi.org/10.1186/1687-1847-2012-98
- 10. Wang, L.G., Xu, K.P. and Liu, Q.W. (2014) On the Stability a Mixed Functional Equation Deriving from Additive, Quadratic and Cubic Mappings. Acta Mathematica Sinica, 30, 1033-1049. https://doi.org/10.1007/s10114-014-3335-9
- 11. Badea, C. (1994) On the Hyers-Ulam Stability of Mappings: The Direct Method. In: Rassias, Th.M. and Tabor, J., Eds., Stability of Mapping of Hyers-Ulam Type, Hadronic Press, Palm Harbor, 7-13.
- 12. Czerwik, S. (1992) On the Stability of Quadratic Mapping in Normed Spaces. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 62, 59-64. https://doi.org/10.1007/BF02941618
- 13. Lee, S.H., Im, S.M. and Hawng, I.S. (2005) Quadratic Functional Equation. Journal of Mathematical Analysis and Applications, 307, 387-394. https://doi.org/10.1016/j.jmaa.2004.12.062
- 14. Czerwik, S. (2003) Stability of Functional Equations of Ulam-Hyers-Rassias Type. Hadronic Press, Palm Harbor.
- 15. Jung, S.-M., Popa, D. and Rassias, M.Th. (2014) On the Stability of the Linear Functional Equation in a Single Variable on Complete Metric Groups. Journal of Global Optimization, 59, 165-171. https://doi.org/10.1007/s10898-013-0083-9
- 16. Jung, S.-M. (1996) On the Hyers-Ulam-Rassias Stability of Approximately Additive Mappings. Journal of Mathematical Analysis and Applications, 204, 221-226. https://doi.org/10.1006/jmaa.1996.0433
- 17. Lee, Y.-H., Jung, S.-M. and Rassias, M.Th. (2014) On an n-Dimensional Mixed Type Additive and Quadratic Functional Equation. Applied Mathematics and Computation, 228, 13-16. https://doi.org/10.1016/j.amc.2013.11.091
- 18. Mortici, C., Rassias, M.Th. and Jung, S.-M. (2014) On the Stability of a Functional Equation Associated with the Fibonacci Numbers. Abstract and Applied Analysis, 2014, Article ID: 546046. https://doi.org/10.1155/2014/546046
- 19. Rassias, J.M. and Kim, H.M. (2009) Generalized Hyers-Ulam Stability for General Additive Functional Equation in Quasi-β-Normed Space. Journal of Mathematical Analysis and Applications, 356, 302-309. https://doi.org/10.1016/j.jmaa.2009.03.005
- 20. Wang, L.G. and Liu, B. (2010) The Hyers-Ulam Stability of a Functional Equation Deriving from Quadratic and Cubic Functions in Quasi-β-Normed Spaces. Acta Mathematica Sinica, English Series, 26, 2335-2348. https://doi.org/10.1007/s10114-010-9330-x
- 21. Gavruta, L. and Gavruta, P. (2016) Approximation of Functions by Additive and by Quadratic Mappings. In: Rassias, T.M. and Gupta, V., Eds., Mathematical Analysis, Approximation Theorem and Their Applications, Springer Optimization and Its Applications, Berlin, 281-292. https://doi.org/10.1007/978-3-319-31281-1_12