doi:10.1155/2009/503724
Research Article
On the Superstability Related with
the Trigonometric Functional Equation
Gwang Hui Kim
Department of Mathematics, Kangnam University, Youngin, Gyeonggi 446-702, South Korea
Correspondence should be addressed to Gwang Hui Kim,[email protected]
Received 22 August 2009; Accepted 6 November 2009
Recommended by Patricia J. Y. Wong
We will investigate the superstability of thehyperbolictrigonometric functional equation from the following functional equations:fxy±gx−y λfxgy,fxy±gx−y λgxfy, fxy±gx−y λfxfy,fxy±gx−y λgxgy, which can be considered the mixed functional equations of the sine function and cosine function, of the hyperbolic sine function and hyperbolic cosine function, and of the exponential functions, respectively.
Copyrightq2009 Gwang Hui Kim. 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
Baker et al. in 1 introduced the following: if f satisfies the inequality |E1f−E2f| ≤ ε, then either f is bounded or E1f E2f. This is frequently referred to as
super-stability.
The superstability of the cosine functional equation also called the d’Alembert equation:
fxyfx−y2fxfy C
and the sine functional equation
fxfyf xy
2
2
−f x−y
2
2
were investigated by Baker2and Cholewa3, respectively. Their results were improved by Badora4, Badora and Ger5, Forti6, and G˘avruta7, as well as by Kim8,9and Kim and Dragomir10. The superstability of the Wilson equation
fxyfx−y2fxgy, Cfg
was investigated by Kannappan and Kim11.
The superstability of the trigonometric functional equation with the sine and the cosine equation
fxy−fx−y2fxfy, T
fxy−fx−y2fxgy Tfg
was investigated by Kim12.
The hyperbolic cosine function, hyperbolic sine function, hyperbolic trigonometric function, and some exponential functions satisfy the aforementioned equations; thus they can be called by the hyperboliccosinesine, trigonometric, exponentialfunctional equation, respectively.
The aim of this paper is to investigate the superstability of the hyperbolic sine functional equationSfrom the following functional equations:
fxygx−yλfxgy, Cfgfg
fxygx−yλgxfy, Cfggf
fxy−gx−yλfxgy, Tfgfg
fxy−gx−yλgxfy, Tfggf
on the abelian group. Consequently, we obtain the superstability of Sfrom the following functional equations:
fxygx−yλfxfy, Cfgff
fxygx−yλgxgy, Cfggg
fxy−gx−yλfxfy, Tfgff
fxy−gx−yλgxgy. Tfggg
We may assume that f and g are nonzero functions, and ε is a nonnegative real constant,ϕ:G → R. For the notation of the equation,
fxyfx−yλfxfy, Cλ
fxyfx−yλfxgy, Cλfg
gxygx−yλgxgy, Cλg
gxygx−yλgxfy. Cλgf
2. Superstability of the Functional Equations
In this section, we will investigate the superstability of the hyperbolic sine functional equationSfrom the functional equationsCfgfg,Cfggf,Cfgff,Cfggg,Tfgfg,Tfggf, Tfgff, andTfggg.
Theorem 2.1. Suppose thatf, g:G → Csatisfy the inequality
f
xygx−y−λfxgy≤ε ∀x, y∈G. 2.1
Ifg or ffails to be bounded, then
ifwithf0 0satisfiesS, iigwithg0 0satisfiesS,
iiiparticularly, if g satisfiesCλ, thenf and g are solutions of the Wilson-type equation Cλ
fg; iffsatisfiesCλ, thenfandgare solutions of Cλgf.
Proof. Takingy0 in the2.1, then it implies that
gx≤fx−λfxg0ε,
fx≤ gxε 1−λg0.
2.2
From2.2, we can know thatfis bounded if and only ifgis bounded.
Letgbe the unbounded solution of2.1. Then, there exists a sequence{yn}inGsuch
that 0/|gyn| → ∞asn → ∞.
iTakingyynin2.1, dividing both sides by|λgyn|,and passing to the limit as
n → ∞, we obtain the following:
fx lim
n→ ∞
fxyngx−yn
Using2.1, we have
We conclude that, for everyy∈G, there exists a limit function
k1
where the functionk1:G → Csatisfies
valid for allx, y∈G, which, in the light of the unique 2-divisibility ofG, states nothing else butS.
Due to the necessary and sufficient conditions for the boundedness of f and g,the unboundedness off is assumed. For the unboundedf of 2.1, we can choose a sequence {xn}inGsuch that 0/|fxn| → ∞asn → ∞.
iiTakingxxnin2.1, dividing both sides by|λfxn|,and passing to the limit as
n → ∞,we obtain
gy lim
n→ ∞
fxnygxn−y
λfxn , x∈G. 2.11
Replacingxbyxn xand xn −xin2.1, dividing by|λfxn|,it then gives us the
existence of a limit function
k2x: lim
n→ ∞
fxnx fxn−x
λfxn , 2.12
where the functionk2:G → Csatisfies
gyxgy−xλk2xg
y ∀x, y∈G. 2.13
Applying the caseg0 0 in2.13, it implies thatgis odd.
A similar procedure to that applied iniin2.13allows us to show thatg satisfies
S.
iii In the case g satisfies Cλ, the limit k1 states nothing else but g; thus, 2.7
validates the required equationCλfg. Also in the casefsatisfiesCλ, since the limitk2states
nothing else butf, the functionsgandfare solutions ofCλgffrom2.13.
Corollary 2.2. Suppose thatf, g:G → Csatisfy the inequality
fxygx−y−λfxfy≤ε ∀x, y∈G. 2.14
Then, eitherfwithf0 0is bounded orfsatisfiesS.
Proof. Substitutingfyforgyin the stability inequality2.1ofTheorem 2.1, the process of the proof is the same asiofTheorem 2.1.
Namely, forfbe unbounded, there exists a sequence{yn}inGsuch that 0/|fyn| →
∞asn → ∞. Takingyynin2.1, dividing both sides by|λfyn|,and passing to the limit
asn → ∞,we obtain
fx lim
n→ ∞
fxyngx−yn
λfyn , x∈G. 2.15
Theorem 2.3. Suppose thatf, g:G → Csatisfy the inequality
fxygx−y−λgxfy≤ε ∀x, y∈G. 2.16
Iff or gfails to be bounded, then
igwithg0 0satisfiesS, iifwithf0 0satisfiesS,
iiiparticularly, ifg satisfiesCλ, thenf andg are solutions of the Wilson equationCλfg, and also iffsatisfiesCλ, thengandfare solutions of Cλgf.
Proof. The process of the proof is similar asTheorem 2.1. Therefore, we will only write an brief proof for the casei. Indeed, the necessary and sufficient conditions for the boundedness of
fandgare same.
iFor the unboundedf, we can choose a sequence{yn}inGsuch that 0/|fyn| →
∞asn → ∞.
A similar reasoning as the proof applied inTheorem 2.1for2.16withyyngives us
gx lim
n→ ∞
fxyngx−yn
λfyn , x∈G. 2.17
Substitutingyynand−yynforyin2.16, and dividing by|λfyn|,it then gives
us the existence of a limit function
k3y: lim
n→ ∞
fyynf−yyn
λfyn , 2.18
where the functionk3:G → Csatisfies the equation
gxygx−yλgxk3
y ∀x, y∈G. 2.19
Applying the caseg0 0 in2.19, it implies thatgis odd.
A similar procedure to that applied iniofTheorem 2.1in2.19allows us to show thatgsatisfiesS.
The proofs foriiandiiialso run along those ofTheorem 2.1.
Corollary 2.4. Suppose thatf, g:G → Csatisfy the inequality
fxygx−y−λgxgy≤ε ∀x, y∈G. 2.20
Then, eithergwithg0 0is bounded orgsatisfiesS.
Since the proofs of the functional equationsTfgfg,Tfggf,Tfgff, andTfgggare very similar to above mentioned proofs, we will give a brief proof forTheorem 2.5.
Theorem 2.5. Suppose thatf, g:G → Csatisfy the inequality f
xy−gx−y−λfxgy≤ε ∀x, y∈G. 2.21
Ifg or ffails to be bounded, then
ifwithf0 0satisfiesS, iigwithg0 0satisfiesS,
iiiparticularly, ifg satisfiesCλ, thenf andg are solutions of the Wilson equationCλfg, and also iffsatisfiesCλ, thenfandgare solutions of Cλgf.
Proof. Using the same method as the proof ofTheorem 2.1, we can know thatfis bounded if and only ifgis bounded.
iFor the unboundedg, we can choose a sequence{yn}inGsuch that 0/|gyn| →
∞asn → ∞.
A similar reasoning as the proof applied inTheorem 2.1for2.21withyyngives us
fx lim
n→ ∞
fxyn−gx−yn
λgyn , x∈G. 2.22
Substitutingyynand−yynforyin2.21, and dividing by|λfyn|,it then gives
us the existence of a limit function
k4
y: lim
n→ ∞
λgyyng−yyn
λgyn , 2.23
where the functionk4:G → Csatisfies the equation
fxyfx−yλfxk4y ∀x, y∈G. 2.24
The next of the proof runs along the same procedure as before.
iiFor unboundedf,letxxnin2.21, dividing both sides by|λfxn|,and passing
to the limit asn → ∞,we obtain
gy lim
n→ ∞
fxny−gxn−y
λfxn , x∈G. 2.25
Replacingxbyxxnand−xxnin2.21and dividing it by|λfyn|, which gives us
the existence of a limit function
k5x:nlim→ ∞
fxxn f−xxn
where the functionk5:G → C,satisfy
gyxgy−xλk5xg
y ∀x, y∈G. 2.27
The next of the proof andiiialso run along the same procedure as before.
Corollary 2.6. Suppose thatf, g:G → Csatisfy the inequality
fxy−gx−y−λfxfy≤ε ∀x, y∈G. 2.28
Then, eitherfwithf0 0is bounded orfsatisfiesS.
Theorem 2.7. Suppose thatf, g:G → Csatisfy the inequality f
xy−gx−y−λgxfy≤ε ∀x, y∈G. 2.29
Ifg or ffails to be bounded, then
ifwithf0 0satisfiesS, iigwithg0 0satisfiesS,
iiiparticularly, ifg satisfiesCλ, thenf andg are solutions of the Wilson equationCλfg, and also iffsatisfiesCλ, thenfandgare solutions of Cλgf.
Proof. As inTheorem 2.5, the proof steps inTheorem 2.1should be followed.
Corollary 2.8. Suppose thatf, g:G → Csatisfy the inequality
fxy−gx−y−λgxgy≤ε ∀x, y∈G. 2.30
Then, eithergwithg0 0is bounded orgsatisfiesS.
Remark 2.9. Let us consider the caseλ2.
i Substituting f for g of the second term of the stability inequalities in the aforementioned results, which imply the hyperboliccosine type functional equationsC,
Cfg, and thehyperbolictrigonometric-type functional equationT, Tfg.Their stability was
founded in papers8,10,12,13.
ii Substituting f for g in the aforementioned results, Theorems 2.1 and 2.3 and Corollaries2.2and2.4imply thehyperboliccosine functional equationC, the stability of which is established in the work in4–7. Furthermore, Theorems2.5and2.7and Corollaries
2.6 and 2.8 imply the hyperbolic trigonometric functional equation T, the stability of which is established in14.
3. Extension to the Banach Space
Theorem 3.1. LetE, · be a semisimple commutative Banach space. Assume thatf, g :G → E
satisfy one of each inequalities
f
xy±gx−y−λfxgy ≤ε, 3.1
fxy±gx−y−λgxfy ≤ε, ∀x, y∈G. 3.2
For an arbitrary linear multiplicative functionalx∗∈E∗, ifx∗◦g or x∗◦ffails to be bounded, then
ifwithf0 0satisfiesS,
iigwithg0 0satisfiesS,
iiiparticularly, ifg satisfiesCλ, thenf andg are solutions of the Wilson equationCλfg, and also iffsatisfiesCλ, thenfandgare solutions of Cλgf.
Proof. Asand−have the same procedure, we will show only casein3.1.
iAssume that3.1holds and arbitrarily fixes a linear multiplicative functionalx∗∈
E∗. As is well known, we havex∗1, hence, for everyx, y∈G, we have
ε≥ fxygx−y−λfxgy
sup y∗1
y∗fxygx−y−λfxgy
≥x∗fxyx∗gx−y−λx∗fxx∗gy,
3.3
which states that the superpositionsx∗◦fandx∗◦gyield a solution of inequality2.1. Since, by assumption, the superpositionx∗◦g is unbounded, an appeal toTheorem 2.1shows that three results hold. Namely,ithe functionx∗◦fwithf0 0 solvesS,iithe function
x∗◦g withg0 0 solvesS, andiii, in particular, ifx∗◦gsatisfiesCλ, thenx∗◦fand
x∗◦g are solutions of the Wilson equationCλ
fg, and also ifx∗◦fsatisfiesCλ, thenx∗◦f andx∗◦gare solutions ofCλgf.
To put caseianother way, bearing the linear multiplicativity ofx∗ in mind, for all
x, y∈G, the differenceD:G×G → C,defined by
DSx, y:f
xy
2
2
−f x−y
2
2
−fxfy, DS
falls into the kernel ofx∗. Therefore, in view of the unrestricted choice ofx∗, we infer that
Since the algebraEhas been assumed to be semisimple, the last term of the above formula coincides with the singleton{0}, that is,
DSx, y0 ∀x, y∈G, 3.5
as claimed. The other cases also are the same.
Theorem 3.2. LetE, · be a semisimple commutative Banach space. Assume thatf, g :G → E
satisfy one of each inequalities f
xy±gx−y−λfxfy ≤ε, 3.6
fxy±gx−y−λgxgy ≤ε, ∀x, y∈G. 3.7
For an arbitrary linear multiplicative functionalx∗∈E∗,
iin case3.6, eitherx∗◦fis bounded orfsatisfiesS, iiin case3.7, eitherx∗◦gis bounded orgsatisfiesS.
Remark 3.3. By applying the same procedure as inRemark 2.9, we obtain the superstability for aforemensioned theorems on the Banach space, which are also in4,5,7–10,12,14.
Acknowledgments
The author would like to thank the referee’s valuable comment. This research was supported by Basic Science Research Program through the National Research Foundation of Korea
NRF funded by the Ministry of Education, Science and Technology Grant no. 2009-0077113.
References
1 J. A. Baker, J. Lawrence, and F. Zorzitto, “The stability of the equationfxy fxfy,”Proceedings of the American Mathematical Society, vol. 74, no. 2, pp. 242–246, 1979.
2 J. A. Baker, “The stability of the cosine equation,”Proceedings of the American Mathematical Society, vol. 80, no. 3, pp. 411–416, 1980.
3 P. W. Cholewa, “The stability of the sine equation,”Proceedings of the American Mathematical Society, vol. 88, no. 4, pp. 631–634, 1983.
4 R. Badora, “On the stability of the cosine functional equation,”Rocznik Naukowo-Dydaktyczny. Prace Matematyczne, no. 15, pp. 5–14, 1998.
5 R. Badora and R. Ger, “On some trigonometric functional inequalities,” inFunctional Equations–Results and Advances, vol. 3 ofAdvances in Mathematics (Dordrecht), pp. 3–15, Kluwer Academic, Dordrecht, The Netherlands, 2002.
6 G. L. Forti, “On an alternative functional equation related to the Cauchy equation,” Aequationes Mathematicae, vol. 24, no. 2-3, pp. 195–206, 1982.
7 P. G˘avruta, “On the stability of some functional equations,” inStability of Mappings of Hyers-Ulam Type, Th. M. Rassias and J. Tabor, Eds., Hadronic Press Collection of Original Articles, pp. 93–98, Hadronic Press, Palm Harbor, Fla, USA, 1994.
8 G. H. Kim, “The stability of d’Alembert and Jensen type functional equations,”Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 237–248, 2007.
10 G. H. Kim and S. S. Dragomir, “On the stability of generalized d’Alembert and Jensen functional equations,”International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 43185, 12 pages, 2006.
11 Pl. Kannappan and G. H. Kim, “On the stability of the generalized cosine functional equations,”
Annales Academiae Paedagogicae Cracoviensis. Studia Mathematica, vol. 1, pp. 49–58, 2001.
12 G. H. Kim, “On the stability of trigonometric functional equations,”Advances in Difference Equations, vol. 2007, Article ID 90405, 10 pages, 2007.
13 G. H. Kim and Y. H. Lee, “boundedness of approximate trigonometric functional equations,”Applied Mathematics Letters, vol. 331, pp. 439–443, 2009.
