Fixed Point Theory and Applications Volume 2008, Article ID 493751,9pages doi:10.1155/2008/493751
Research Article
Generalized Hyers-Ulam Stability of Quadratic Functional Equations: A Fixed Point Approach
Choonkil Park
Department of Mathematics, Hanyang University, Seoul 133-791, South Korea
Correspondence should be addressed to Choonkil Park,[email protected] Received 6 September 2007; Revised 19 November 2007; Accepted 15 February 2008 Recommended by Thomas Bartsch
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic func- tional equationf2xy 4fx fy fxy−fx−yin Banach spaces.
Copyrightq2008 Choonkil Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The stability problem of functional equations originated from a question of Ulam1concern- ing the stability of group homomorphisms. Hyers2gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki3for addi- tive mappings and by Rassias4for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias 4has 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 Rassias theorem was obtained by G˘avrut¸a5 by replac- ing the unbounded Cauchy difference by a general control function in the spirit of Rassias’
approach.
The functional equation
fxy fx−y 2fx 2fy 1.1
is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic function. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof 6for mappings f : X → Y, whereX is a normed space andY is a Banach space.Cholewa 7noticed that the theorem of Skof is
still true if the relevant domainX is replaced by an Abelian group. Czerwik8proved the generalized Hyers-Ulam stability of the quadratic functional equation and Park9proved the generalized Hyers-Ulam stability of the quadratic functional equation in Banach modules over aC∗-algebra. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problemsee10–17.
LetX be a set. A function d : X ×X → 0,∞is called a generalized metric onX if d satisfies
1dx, y 0 if and only ifxy;
2dx, y dy, xfor allx, y∈X;
3dx, z≤dx, y dy, zfor allx, y, z∈X.
We recall the following theorem by Diaz and Margolis.
Theorem 1.1see18. Let X, dbe a complete generalized metric space and letJ : X → X be a strictly contractive mapping with Lipschitz constantL <1. Then for each given elementx∈X, either
d
Jnx, Jn1x
∞ 1.2
for all nonnegative integersnor there exists a positive integern0such that 1dJnx, Jn1x<∞for all n≥n0;
2the sequence{Jnx}converges to a fixed pointy∗ofJ;
3y∗is the unique fixed point ofJin the setY {y∈X|dJn0x, y<∞};
4dy, y∗≤1/1−Ldy, Jyfor ally∈Y.
In this paper, using the fixed point method, we prove the generalized Hyers-Ulam sta- bility of the following quadratic functional equation:
f2xy 4fx fy fxy−fx−y 1.3
in Banach spaces.
Throughout this paper, assume thatXis a normed vector space with norm·and that Yis a Banach space with norm·.
In 1996, Isac and Rassias19were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications.
2. Fixed points and generalized Hyers-Ulam stability of quadratic functional equations
For a given mappingf :X→Y, we define
Cfx, y:f2xy−4fx−fy−fxy fx−y 2.1 for allx, y∈X.
Proposition 2.1. LetXandYbe vector spaces. A mappingf:X→Ysatisfies
f2xy 4fx fy fxy−fx−y 2.2
if and only if the mappingf:X→Ysatisfies
fxy fx−y 2fx 2fy 2.3
for allx, y∈X.
Proof. Assume thatf:X→Ysatisfies2.2.
Lettingxy0 in2.2, we getf0 0.
Lettingy0 in2.2, we getf2x 4fxfor allx∈X.
Lettingx0 in2.2, we getf−y fyfor ally∈X.
Replacingyin2.2by−y, we get
f2x−y 4fx f−y fx−y−fxy 2.4
for allx, y∈X. It follows from2.2and2.4that
f2xy f2x−y 8fx fy f−y 2f2x 2fy 2.5
for allx, y∈X. So the mappingf:X→Y satisfies
fxy fx−y 2fx 2fy 2.6
for allx, y∈X.
Assume thatf:X→Ysatisfiesfxy fx−y 2fx 2fyfor allx, y∈X.
Since
f2xy fxyx
2fxy 2fx−fy
fxy fxy 2fx−fy
fxy 2fx 2fy−fx−y 2fx−fy 4fx fy fxy−fx−y
2.7
for allx, y∈X, the mappingf:X→Ysatisfies2.2.
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equationCfx, y 0.
Theorem 2.2. Letf :X→Ybe a mapping withf0 0 for which there exists a functionϕ:X2→ 0,∞such that there exists anL <1 such thatϕx,0≤4Lϕx/2,0for allx∈X, and
∞ j0
4−jϕ
2jx,2jy
<∞, 2.8
Cfx, y≤ϕx, y 2.9
for allx, y∈X. Then there exists a unique quadratic mappingQ:X→Y satisfying2.2and fx−Qx≤ 1
4−4Lϕx,0 2.10
for allx∈X.
Proof. Consider the set
S:{g:X−→Y} 2.11
and introduce the generalized metric onSas follows:
dg, h inf
K∈R:gx−hx≤Kϕx,0, ∀x∈X
. 2.12
It is easy to show thatS, dis complete.See the proof of Theorem 2.5 of20.
Now we consider the linear mappingJ:S→Ssuch that
Jgx: 1
4g2x 2.13
for allx∈X.
It follows from the proof of Theorem 3.1 of21that
dJg, Jh≤Ldg, h 2.14
for allg, h∈S.
Lettingy0 in2.9, we get
f2x−4fx≤ϕx,0 2.15 for allx∈X. So
fx−1 4f2x
≤1
4ϕx,0 2.16
for allx∈X. Hencedf, Jf≤1/4.
ByTheorem 1.1, there exists a mappingQ:X→Ysatisfying the following.
1Qis a fixed point ofJ, that is,
Q2x 4Qx 2.17
for allx∈X. The mappingQis a unique fixed point ofJin the set
M
g∈S:df, g<∞
. 2.18
This implies thatQ is a unique mapping satisfying2.17 such that there existsK ∈ 0,∞ satisfying
fx−Qx≤Kϕx,0 2.19 for allx∈X.
2dJnf, Q→0 asn→ ∞. This implies the equality
n→∞lim f
2nx
4n Qx 2.20
for allx∈X.
3df, Q≤1/1−Ldf, Jf, which implies the inequality df, Q≤ 1
4−4L. 2.21
This implies that the inequality2.10holds.
It follows from2.8,2.9, and2.20that CQx, yn→∞lim 1
4nCf
2nx,2ny
≤ lim
n→∞
1 4nϕ
2nx,2ny 0
2.22
for allx, y∈X. SoCQx, y 0 for allx, y∈X. ByProposition 2.1, the mappingQ:X →Yis quadratic.
Therefore, there exists a unique quadratic mappingQ:X→Ysatisfying2.2and2.10, as desired.
Corollary 2.3. Letp <2 andθ≥0 be real numbers, and letf :X→Y be a mapping such that Cfx, y≤θ
xpyp
2.23 for allx, y∈X. Then there exists a unique quadratic mappingQ:X→Y satisfying (2.2) and
fx−Qx≤ θ
4−2pxp 2.24
for allx∈X.
Proof. The proof follows from Theorem2.2by taking ϕx, y:θ
xpyp
2.25 for allx, y∈X. Then we can chooseL2p−2and we get the desired result.
Remark 2.4. Letf : X → Y be a mapping for which there exists a function ϕ : X2 → 0,∞ satisfying2.9andf0 0 such that
∞ j0
4jϕ x
2j, y
2j <∞ 2.26
for allx, y ∈ X. By a similar method to the proof of Theorem2.2, one can show that if there exists an L < 1 such thatϕx,0 ≤ 1/4Lϕ2x,0for allx ∈ X, then there exists a unique quadratic mappingQ:X→Y satisfying2.2and
fx−Qx≤ L
4−4Lϕx,0 2.27
for allx∈X.
For the casep >2, one can obtain a similar result to Corollary2.3
Theorem 2.5. Letf :X→Y be an even mappingf0 0 for which there exists a functionϕ:X2→ 0,∞satisfying2.8and2.9such that there exists anL <1 such thatϕx,−x≤4Lϕx/2,−x/2 for allx∈X. Then there exists a unique quadratic mappingQ:X→Ysatisfying2.2and
fx−Qx≤ 1
4−4Lϕx,−x 2.28
for allx∈X.
Proof. Consider the set
S:{g:X−→Y} 2.29
and introduce the generalized metric onSas follows:
dg, h inf
K∈R:gx−hx≤Kϕx,−x∀x∈X
. 2.30
It is easy to show thatS, dis complete.See the proof of Theorem 2.5 of20.
Now we consider the linear mappingJ:S→Ssuch that Jgx: 1
4g2x 2.31
for allx∈X.
It follows from the proof of Theorem 3.1 of21that
dJg, Jh≤Ldg, h 2.32
for allg, h∈S.
Lettingy−xin2.9, we get
f2x−4fx≤ϕx,−x 2.33 for allx∈X. So
fx−1 4f2x
≤1
4ϕx,−x 2.34
for allx∈X. Hencedf, Jf≤1/4.
ByTheorem 1.1, there exists a mappingQ:X→Ysatisfying the following.
1Qis a fixed point ofJ, that is,
Q2x 4Qx 2.35
for allx∈X. The mappingQis a unique fixed point ofJin the set
M
g∈S:df, g<∞
. 2.36
This implies thatQ is a unique mapping satisfying2.35 such that there existsK ∈ 0,∞ satisfying
fx−Qx≤Kϕx,−x 2.37 for allx∈X.
2dJnf, Q→0 asn→ ∞. This implies the equality
n→∞lim f
2nx
4n Qx 2.38
for allx∈X.
3df, Q≤1/1−Ldf, Jf, which implies the inequality df, Q≤ 1
4−4L. 2.39
This implies that the inequality2.38holds.
It follows from2.8,2.9, and2.38that CQx, y lim
n→∞
1
4nCf2nx,2ny
≤ lim
n→∞
1 4nϕ
2nx,2ny 0
2.40
for allx, y∈X. SoCQx, y 0 for allx, y∈X. ByProposition 2.1, the mappingQ:X →Yis quadratic.
Therefore, there exists a unique quadratic mappingQ:X→Ysatisfying2.2and2.28, as desired.
Corollary 2.6. Letp <1 andθ≥0 be real numbers, and letf :X→Y be an even mapping such that Cfx, y≤θ·xp·yp 2.41 for allx, y∈X. Then there exists a unique quadratic mappingQ:X→Y satisfying2.2and
fx−Qx≤ θ
4−4px2p 2.42
for allx∈X.
Proof. The proof follows from Theorem 2.5 by taking
ϕx, y:θ·||x||p·||y||p 2.43
for allx, y∈X. Then we can chooseL4p−1and we get the desired result.
Remark 2.7. Letf:X→Ybe an even mapping for which there exists a functionϕ:X2→0,∞ satisfying2.9,2.26, andf0 0. By a similar method to the proof of Theorem 2.5, one can show that if there exists anL < 1 such thatϕx,−x ≤ 1/4Lϕ2x,−2xfor allx ∈ X, then there exists a unique quadratic mappingQ:X→Ysatisfying2.2and
fx−Qx≤ L
4−4Lϕx,−x 2.44
for allx∈X.
For the casep >1, one can obtain a similar result toCorollary 2.6.
Acknowledgments
This work was supported by the research fund of Hanyang UniversityHY-2007-Sand the au- thor would like to thank the referees for a number of valuable suggestions regarding a previous version of this paper.
References
1 S. M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, New York, NY, USA, 1964.
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, pp. 222–224, 1941.
3 T. Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol. 2, pp. 64–66, 1950.
4 Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.
5 P. G˘avrut¸a, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive map- pings,” Journal of Mathematical Analysis and Applications, vol. 184, no. 3, pp. 431–436, 1994.
6 F. Skof, “Local properties and approximation of operators,” Rendiconti del Seminario Matematico e Fisico di Milano, vol. 53, pp. 113–129, 1983.
7 P. W. Cholewa, “Remarks on the stability of functional equations,” Aequationes Mathematicae, vol. 27, no. 1-2, pp. 76–86, 1984.
8 St. Czerwik, “On the stability of the quadratic mapping in normed spaces,” Abhandlungen aus dem Mathematischen Seminar der Universit¨at Hamburg, vol. 62, pp. 59–64, 1992.
9 C.-G. Park, “On the stability of the quadratic mapping in Banach modules,” Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 135–144, 2002.
10 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, Birkh¨auser, Boston, Mass, USA, 1998.
11 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, pp. 93–118, 2001.
12 S.-M. Jung, “On the Hyers-Ulam stability of the functional equations that have the quadratic prop- erty,” Journal of Mathematical Analysis and Applications, vol. 222, no. 1, pp. 126–137, 1998.
13 M. Mirzavaziri and M. S. Moslehian, “A fixed point approach to stability of a quadratic equation,”
Bulletin of the Brazilian Mathematical Society, vol. 37, no. 3, pp. 361–376, 2006.
14 K. Nikodem, “On some properties of quadratic stochastic processes,” Annales Mathematicae Silesianae, vol. 3, no. 15, pp. 58–69, 1990.
15 C. Park, “Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras,” Fixed Point Theory and Applications, vol. 2007, Article ID 50175, 15 pages, 2007.
16 C.-G. Park and Th. M. Rassias, “Hyers-Ulam stability of a generalized Apollonius type quadratic mapping,” Journal of Mathematical Analysis and Applications, vol. 322, no. 1, pp. 371–381, 2006.
17 Th. M. Rassias, “On the stability of functional equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 251, no. 1, pp. 264–284, 2000.
18 J. B. Diaz and B. Margolis, “A fixed point theorem of the alternative, for contractions on a generalized complete metric space,” Bulletin of the American Mathematical Society, vol. 74, pp. 305–309, 1968.
19 G. Isac and Th. M. Rassias, “Stability ofψ-additive mappings: applications to nonlinear analysis,”
International Journal of Mathematics and Mathematical Sciences, vol. 19, no. 2, pp. 219–228, 1996.
20 L. C˘adariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,”
in Iteration Theory, vol. 346 of Grazer Mathematische Berichte, pp. 43–52, Karl-Franzens-Universit¨aet Graz, Graz, Austria, 2004.
21 L. C˘adariu and V. Radu, “Fixed points and the stability of Jensen’s functional equation,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 4, 2003.
