Open Access

A fixed-point approach to the stability of a functional equation on quadratic forms

Journal of Inequalities and Applications20112011:82

https://doi.org/10.1186/1029-242X-2011-82

Received: 15 April 2011

Accepted: 10 October 2011

Published: 10 October 2011

Abstract

Using the fixed-point method, we prove the generalized Hyers-Ulam stability of the functional equation

f ( x + y , z + w ) + f ( x - y , z - w ) = 2 f ( x , z ) + 2 f ( y , w ) .

The quadratic form f : × given by f(x, y) = ax2 + bxy + cy2 is a solution of the above functional equation.

Keywords

alternative of fixed point functional equation quadratic form stability

1. Introduction

In 1940, S. M. Ulam [1] gave a wide-ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms:

Let G1 be a group and let G2 be a metric group with the metric d(· , ·). Given ε > 0, does there exist a δ > 0 such that if a function h : G1 → G2 satisfies the inequality d(h(xy), h(x)h(y)) < δ for all x, y G1, then there is a homomorphism H : G1 → G2 with d(h(x), H(x)) < ε for all x G1?

The case of approximately additive mappings was solved by D. H. Hyers [2] under the assumption that G1 and G2 are Banach spaces. Thereafter, many authors investigated solutions or stability of various functional equations (see [311]).

Let X be a set. A function d : X × X → [0, ∞] is called a generalized metric on X if d satisfies
  1. (1)

    d(x, y) = 0 if and only if x = y;

     
  2. (2)

    d(x, y) = d(y, x) for all x, y X;

     
  3. (3)

    d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z X.

     

Note that the only substantial difference of the generalized metric from the metric is that the range of generalized metric includes the infinity.

Throughout this paper, let X and Y be two real vector spaces and let φ : X × X × X × X → [0, ∞) be a function. For a mapping f : X × X → Y, consider the functional equation:
f ( x + y , z + w ) + f ( x - y , z - w ) = 2 f ( x , z ) + 2 f ( y , w ) .
(1.1)

The quadratic form f : × given by f(x, y) : = ax2 + bxy + cy2 is a solution of the Equation 1.1.

The authors [12] acquired the general solution and proved the stability of the functional Equation 1.1 for the case that X and Y are real vector spaces as follows.

Theorem A. A mapping f : X × X → Y satisfies the Equation 1.1 for all x, y, z, w X if and only if there exist two symmetric bi-additive mappings S, T : X × X → Y and a bi-additive mapping B : X × X → Y such that
f ( x , y ) = S ( x , x ) + B ( x , y ) + T ( y , y )

for all x, y X.

From now on, let Y be a complete normed space.

Theorem B. Assume that φ satisfies the condition
φ ̃ ( x , y , z , w ) : = j = 0 1 4 j + 1 φ ( 2 j x , 2 j y , 2 j z , 2 j w ) <
for all x, y, z, w X. Let f : X × X → Y be a mapping such that
f ( x + y , z + w ) + f ( x - y , z - w ) - 2 f ( x , z ) - 2 f ( y , w ) φ ( x , y , z , w )
for all x, y, z, w X. Then, there exists a unique mapping F : X × X → Y satisfying the Equation 1.1 such that
f ( x , y ) - F ( x , y ) φ ̃ ( x , x , y , y )
(1.2)
for all x, y X. The mapping F is given by
F ( x , y ) : = lim j 1 4 j f ( 2 j x , 2 j y )

for all x, y X.

In this paper, we prove the stability of the Equation 1.1 using the fixed-point method.

2. Stability using the alternative of fixed point

In this section, we investigate the stability of the functional Equation 1.1 using the alternative of fixed point. Before proceeding the proof, we will state the theorem, the alternative of fixed point.

Theorem 2.1. (The alternative of fixed point [13, 14]). Suppose that we are given a complete generalized metric space (Ω, d) and a strictly contractive mapping T : Ω Ω with Lipschitz constant L. Then, for each given x Ω, either
d ( T n x , T n + 1 x ) = f o r a l l n 0 ,

or

there exists a positive integer n 0 such that

  • d(T n x, Tn+1x) <for all nn0;

  • the sequence (T n x) is convergent to a fixed point y* of T;

  • y* is the unique fixed point of T in the set Δ = { y Ω d ( T n 0 x , y ) < } ;

  • d ( y , y * ) 1 1 - L d ( y , T y ) for all y Δ.

Lemma 2.2. Let ψ : X × X → [0, ∞) be a function given by
ψ ( x , y ) : = φ x 2 , x 2 , y 2 , y 2
for all x, y X. Consider the set Ω : = {g | g : X × X → Y, g(0, 0) = 0} and the generalized metric d on Ω given by
d ( g , h ) = d ψ ( g , h ) : = inf S ψ ( g , h ) ,

where S ψ (g, h) : = {K [0, ∞] | ||g(x, y) - h(x, y) || ≤ (x, y) for all x, y X} for all g, h Ω. Then, (Ω, d) is complete.

Proof. Let {g n } be a Cauchy sequence in (Ω, d). Then, given ε > 0, there exists N such that d(g n , g k ) < ε if n, kN. Let n, kN. Since d(g n , g k ) = inf S ψ (g n , g k ) < ε, there exists K [0, ε) such that
g n ( x , y ) - g k ( x , y ) K ψ ( x , y ) ε ψ ( x , y )
(2.1)
for all x, y X. So, for each x, y X, {g n (x, y)} is a Cauchy sequence in Y. Since Y is complete, for each x, y X, there exists g(x, y) Y such that g n (x, y) → g(x, y) as n → ∞ and g(0, 0) = 0. Thus, we have g Ω. By (2.1), we obtain that
n N g n ( x , y ) - g ( x , y ) ε ψ ( x , y ) for all x , y X ε S ψ ( g n , g ) d ( g n , g ) = inf S ψ ( g n , g ) ε .

Hence, g n → g Ω as n → ∞.

By using an idea of Cădariu and Radu (see [15]), we will prove the Hyers-Ulam stability of the functional equation related to quadratic forms.

Theorem 2.3. Assume that φ satisfies the condition
lim n 1 4 n φ ( 2 n x , 2 n y , 2 n z , 2 n w ) = 0
for all x, y, z, w X. Suppose that a mapping f : X × X → Y satisfies the functional inequality
f ( x + y , z + w ) + f ( x - y , z - w ) - 2 f ( x , z ) - 2 f ( y , w ) φ ( x , y , z , w )
(2.2)
for all x, y, z, w X and f(0, 0) = 0. If there exists L < 1 such that the function ψ given in Lemma 2.2 has the property
ψ ( x , y ) 4 L ψ x 2 , y 2
(2.3)
for all x, y X, then there exists a unique mapping F : X × X → Y satisfying (1.1) such that the inequality
f ( x , y ) - F ( x , y ) L 1 - L ψ ( x , y )
(2.4)

holds for all x, y X.

Proof. Consider the complete generalized metric space (Ω, d) given in Lemma 2.2. Now we define a mapping T : Ω Ω by
T g ( x , y ) : = 1 4 g ( 2 x , 2 y )
for all g Ω and all x, y X. Observe that, for all g, h Ω,
K S ψ ( g , h ) and K < K g ( x , y ) - h ( x , y ) K ψ ( x , y ) K ψ ( x , y ) for all x , y X K S ψ ( g , h ) .
Let g, h Ω and ε (0, ∞]. Then, there is a K' S ψ (g, h) such that K' < d(g, h) + ε. By the above observation, we gain d(g, h) + ε S ψ (g, h). So we get ||g(x, y) - h(x, y)|| ≤ (d(g, h) + ε) ψ (x, y) for all x, y X. Thus, we have
1 4 g ( 2 x , 2 y ) - 1 4 h ( 2 x , 2 y ) 1 4 ( d ( g , h ) + ε ) ψ ( 2 x , 2 y )
for all x, y X. By (2.3), we obtain that
1 4 g ( 2 x , 2 y ) - 1 4 h ( 2 x , 2 y ) L ( d ( g , h ) + ε ) ψ ( x , y )
for all x, y X. Hence, d(Tg, Th) ≤ L (d(g, h) + ε). Now we obtain that
d ( T g , T h ) L ( d ( g , h ) + ε )
for all ε (0, ∞]. Taking the limit as ε → 0+ in the above inequality, we get
d ( T g , T h ) L d ( g , h )

for all g, h Ω, that is, T is a strictly contractive mapping of Ω with Lipschitz constant L.

Putting y = x and w = z in (2.2), by (2.3), we have the inequality
f ( x , z ) - 1 4 f ( 2 x , 2 z ) 1 4 φ ( x , x , z , z ) = 1 4 ψ ( 2 x , 2 z ) L ψ ( x , z )
(2.5)
for all x, z X. Thus, we obtain that
d ( f , T f ) L < .
(2.6)
Applying the alternative of fixed point, we see that there exists a fixed point F of T in Ω such that
F ( x , y ) = lim n 1 4 n f ( 2 n x , 2 n y )
for all x, y X. Replacing x, y, z, w by 2 n x, 2 n y, 2 n z, 2 n w in (2.2), respectively, and dividing by 4 n , we have
F ( x + y , z + w ) + F ( x y , z w ) 2 F ( x , z ) 2 F ( y , w ) = lim n 1 4 n f ( 2 n ( x + y ) , 2 n ( z + w ) ) + f ( 2 n ( x y ) , 2 n ( z w ) ) 2 f ( 2 n x , 2 n z ) 2 f ( 2 n y , 2 n w ) lim n 1 4 n φ ( 2 n x , 2 n y , 2 n z , 2 n w ) = 0
for all x, y, z, w X. Thus, the mapping F satisfies the Equation 1.1. By (2.3) and (2.5), we obtain that
T n f ( x , y ) T n + 1 f ( x , y ) = 1 4 n f ( 2 n x , 2 n y ) 1 4 f ( 2 n + 1 x , 2 n + 1 y ) L 4 n ψ ( 2 n x , 2 n y ) L 4 n ( 4 L ) n ψ ( x , y ) = L n + 1 ψ ( x , y )
for all x, y X and all n , that is, d(T n f, Tn+1f) ≤ Ln+1< ∞ for all n . By the fixed-point alternative, there exists a natural number n0 such that the mapping F is the unique fixed point of T in the set Δ = { g Ω | d ( T n 0 f , g ) < } . So we have d ( T n 0 f , F ) < . Since
d ( f , T n 0 f ) d ( f , T f ) + d ( T f , T 2 f ) + + d ( T n 0 - 1 f , T n 0 f ) < ,
we get f Δ. Thus, we have d ( f , F ) d ( f , T m 0 f ) + d ( T m 0 f , F ) < . Hence, we obtain
| | f ( x , y ) - F ( x , y ) | | K ψ ( x , y )
for all x X and some K [0, ∞). Again using the fixed-point alternative, we have
d ( f , F ) 1 1 - L d ( f , T f ) .
By (2.6), we may conclude that
d ( f , F ) L 1 - L ,

which implies the inequality (2.4).

Lemma 2.4. Let ψ : X × X → [0, ∞) be a function given by
ψ ( x , y ) : = φ ( 2 x , 2 x , 2 y , 2 y )
for all x, y X. Consider the set Ω : = {g | g : X × X → Y, g(0, 0) = 0} and the generalized metric d on Ω given by
d ( g , h ) = d ψ ( g , h ) : = inf S ψ ( g , h ) ,

where S ψ (g, h) : = K [0, ∞] | ||g(x, y) - h(x, y)|| ≤ (x, y) for all x, y X} for all g, h Ω. Then, (Ω, d) is complete.

Proof. The proof is similar to the proof of Lemma 2.2.

Theorem 2.5. Assume that φ satisfies the condition
lim n 4 n φ ( x 2 n , y 2 n , z 2 n , w 2 n ) = 0
for all x, y, z, w X. Suppose that a mapping f : X × X → Y satisfies the functional inequality (2.2) for all x, y, z, w X and f(0, 0) = 0. If there exists L < 1 such that the function ψ given in Lemma 2.4 has the property
ψ ( x , y ) L 4 ψ ( 2 x , 2 y )
(2.7)
for all x, y X, then there exists a unique mapping F : X × X → Y satisfying (1.1) such that the inequality
f ( x , y ) - F ( x , y ) L 2 1 6 ( 1 - L ) ψ ( x , y )
(2.8)

holds for all x, y X.

Proof. Consider the complete generalized metric space (Ω, d) given in Lemma 2.4. Now we define a mapping T : Ω Ω by
T g ( x , y ) : = 4 g x 2 , y 2

for all g Ω and all x, y X. By the same argument of the proof of Theorem 2.3, T is a strictly contractive mapping of Ω with Lipschitz constant L.

Replacing x, y, z, w by x 2 , x 2 , z 2 , z 2 in (2.2), respectively, and using (2.7), we have the inequality
f ( x , z ) - 4 f x 2 , z 2 φ x 2 , x 2 , z 2 , z 2 = ψ x 4 , z 4 L 4 ψ x 2 , z 2 L 2 1 6 ψ ( x , y )
(2.9)
for all x, z X. Thus, we obtain that
d ( f , T f ) L 2 1 6 < .
(2.10)
Applying the alternative of fixed point, we see that there exists a fixed point F of T in Ω such that
F ( x , y ) = lim n 4 n f ( x 2 n , y 2 n )
for all x, y X. Replacing x, y, z, w by x 2 n , y 2 n , z 2 n , w 2 n in (2.2), respectively, and multiplying by 4 n , we have
F ( x + y , z + w ) + F ( x y , z w ) 2 F ( x , z ) 2 F ( y , w ) = lim n 4 n f ( x + y 2 n , z + w 2 n ) + f ( x y 2 n , z w 2 n ) 2 f ( x 2 n , z 2 n ) 2 f ( y 2 n , w 2 n ) lim n 4 n φ ( x 2 n , y 2 n , z 2 n , w 2 n ) = 0
for all x, y, z, w X. Thus, the mapping F satisfies the Equation 1.1. By (2.7) and (2.9), we obtain that
| | T n f ( x , y ) T n + 1 f ( x , y ) | | = 4 n f ( x 2 n , y 2 n ) 4 f ( x 2 n + 1 , y 2 n + 1 ) 4 n 2 L 2 ψ ( x 2 n , y 2 n ) 4 n 3 L 3 ψ ( x 2 n 1 , y 2 n 1 ) L n + 2 16 ψ ( x , y )
for all x, y X and all n , that is, d ( T n f , T n + 1 f ) L n + 2 1 6 < for all n . By the same reasoning of the proof of Theorem 2.3, we have
d ( f , F ) 1 1 - L d ( f , T f ) .
By (2.10), we may conclude that
d ( f , F ) L 2 1 6 ( 1 - L ) ,

which implies the inequality (2.8).

Declarations

Acknowledgements

The authors would like to thank the referee for a number of valuable suggestions regarding a previous version of this paper.

Authors’ Affiliations

(1)
Graduate School of Education, Kyung Hee University
(2)
Department of Mathematics Education, College of Education, Mokwon University

References

  1. Ulam SM: A Collection of Mathematical Problems. Interscience Publishers, New York; 1968:63.Google Scholar
  2. Hyers DH: On the stability of the linear functional equation. Proc Natl Acad Sci USA 1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleGoogle Scholar
  3. Bae J-H, Park W-G: On stability of a functional equation with n variables. Nonlinear Anal TMA 2006, 64: 856–868. 10.1016/j.na.2005.06.028MathSciNetView ArticleGoogle Scholar
  4. Bae J-H, Park W-G: On a cubic equation and a Jensen-quadratic equation. Abstr Appl Anal 2007., 2007: Article ID 45179Google Scholar
  5. Găvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J Math Anal Appl 1994, 184: 431–436. 10.1006/jmaa.1994.1211MathSciNetView ArticleGoogle Scholar
  6. Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York; 2011.View ArticleGoogle Scholar
  7. Jung S-M, Kim T-S: A fixed point approach to the stability of the cubic functional equation. Bol Soc Mat Mexicana 2006,12(1):51–57. (3)MathSciNetGoogle Scholar
  8. Jung S-M, Kim T-S, Lee K-S: A fixed point approach to the stability of quadratic functional equation. Bull Korean Math Soc 2006, 43: 531–541.MathSciNetView ArticleGoogle Scholar
  9. Jung S-M, Lee Z-H: A fixed point approach to the stability of quadratic functional equation with involution. Fixed Point Theory Appl 2008, 2008: 11. Article ID 732086MathSciNetGoogle Scholar
  10. Park W-G, Bae J-H: A multidimensional functional equation having quadratic forms as solutions. J Inequal Appl 2007., 2007: Article ID 24716Google Scholar
  11. Rassias TM: On the stability of linear mappings in Banach spaces. Proc Amer Math Soc 1978, 72: 297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleGoogle Scholar
  12. Park W-G, Bae J-H: A functional equation originating from quadratic forms. J Math Anal Appl 2007, 326: 1142–1148. 10.1016/j.jmaa.2006.03.023MathSciNetView ArticleGoogle Scholar
  13. Margolis B, Dias JB: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull Amer Math Soc 1968, 74: 305–309. 10.1090/S0002-9904-1968-11933-0MathSciNetView ArticleGoogle Scholar
  14. Rus IA: Principles and Applications of Fixed Point Theory. In Edited by: Dacia, Cluj-Napoca. 1979.Google Scholar
  15. Cădariu L, Radu V: Fixed points and the stability of Jensen's functional equation. J Inequal Pure Appl Math 2003., 4: Article 4Google Scholar

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© Bae and Park; licensee Springer. 2011

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