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On the stability of the pexiderized trigonometric functional equation
Applied Mathematics and Computation 203 (2008) 99–105
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On the stability of the pexiderized trigonometric functional equation Gwang Hui Kim Department of Mathematics, Kangnam University, Youngin, Gyeonggi 446-702, Republic of Korea
a r t i c l e
i n f o
Keywords: Stability Superstability Sine functional equation d’Alembert equation Trigonometric functional equation
a b s t r a c t The aim of this paper is to investigate the superstability problem of the cosine(d’Alembert), the Wilson, and the sine functional equation from the pexiderized trigonometric functional equation (Tgh) under the condition: jf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞj 6 uðxÞ or uðyÞ, and also will investigated the stability of the Jensen type functional equation (Jy) under the condition: jf ðx þ yÞ f ðx yÞ 2f ðyÞj 6 uðx; yÞ. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction Baker [4], and Bourgin [5] introduced the following: if f satisfies the inequality jE1 ðf Þ E2 ðf Þj 6 e, then either f is bounded or E1 ðf Þ ¼ E2 ðf Þ. This is frequently referred to as Superstability. The superstability of the cosine functional equation (also called the d’Alembert equation): f ðx þ yÞ þ f ðx yÞ ¼ 2f ðxÞf ðyÞ is investigated by Baker [3], which again is proved under a simple new technique by Gavruta [8]. The superstability of the sine functional equation: x þ y2 x y2 f ðxÞf ðyÞ ¼ f f 2 2
ðAÞ
ðSÞ
is investigated by Cholewa [6]. The d’Alembert functional equation (A) is generalized to the following functional equations: f ðx þ yÞ þ f ðx yÞ ¼ 2f ðxÞgðyÞ;
ðAfg Þ
f ðx þ yÞ þ f ðx yÞ ¼ 2gðxÞf ðyÞ;
ðAgf Þ
f ðx þ yÞ þ f ðx yÞ ¼ 2gðxÞgðyÞ:
ðAgg Þ
Eq. (Afg), introduced by Wilson, is sometimes referred to as the Wilson equation. The above functional equations have been investigated by Badora, Ger, Kannappan, Kim, Sinopoulos, etc. [1,2,10–13,16]. Due to the well known trigonometric identity ðsinða þ bÞ sinða bÞ ¼ 2 cos a sin bÞ, we will consider the following pexider type trigonometric functional equation: f ðx þ yÞ f ðx yÞ ¼ 2gðxÞhðyÞ; E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.04.011
ðTgh Þ
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G.H. Kim / Applied Mathematics and Computation 203 (2008) 99–105
which has special cases as follows: f ðx þ yÞ f ðx yÞ ¼ 2f ðxÞf ðyÞ;
ðTÞ
f ðx þ yÞ f ðx yÞ ¼ 2f ðxÞgðyÞ;
ðTfg Þ
f ðx þ yÞ f ðx yÞ ¼ 2gðxÞf ðyÞ;
ðTgf Þ
f ðx þ yÞ f ðx yÞ ¼ 2gðxÞgðyÞ:
ðTgg Þ
Let gðxÞ 1 in (Tgf), then we also obtain the Jensen type functional equation f ðx þ yÞ f ðx yÞ ¼ 2f ðyÞ:
ðJy Þ
Given mappings f ; g; h : G ! C, for above equations, we will denote their difference by an operator DT gh : G G ! C as DT gh ðx; yÞ :¼ f ðx þ yÞ f ðx yÞ 2gðxÞhðyÞ: The aim of this paper is to investigate the superstability problem of the cosine(d’Alembert), the Wilson, and the sine functional equation from the pexiderized trigonometric functional equation (Tgh) under the condition: jDT gh ðx; yÞj 6 uðxÞ or uðyÞ, and also will investigated the stability of the Jensen type functional equation (Jy) under the condition: jDJy ðx; yÞj 6 uðx; yÞ. As a consequence, we obtain the superstability of the cosine(d’Alembert), the Wilson, and the sine functional equation from the above trigonometric type Eqs. (T), (Tfg), (Tgf) and (Tgg) as corollaries, and also extend the obtained results to the Banach algebra. In this paper, let ðG; þÞ be an Abelian group, C the field of complex numbers, and R the field of real numbers. In particular, if there is not referring which is special, let ðG; þÞ be a uniquely 2-divisible group whenever the function is related to the sine functional equation (S), and it will be denoted by ‘‘under 2-divisible” for short. We may assume that f ; g and h are non-zero functions and e is a nonnegative real constant, and let w : G G ! R and u : G ! R be a mappings. 2. Stability on Eq. (Tgh) In this section, we investigate the superstability of the d’Alembert (A), the Wilson (Afg), the sine (S), and the mixed type functional equations (Agf), (Tfg) and (Tgf) from the approximately equation of (Tgh). Theorem 1. Suppose that f ; g; h : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞj 6 uðyÞ
ð2:1Þ
for all x; y 2 G. If g fails to be bounded, then (i) h satisfies (S); (ii) In particular case g satisfies (A), g and h are solutions of Eq. (Tgh). Proof. (i) Let g be unbounded. Then we can choose a sequence fxn g in G such that as n ! 1:
0 6¼ jgðxn Þj ! 1;
ð2:2Þ
Taking x ¼ xn in (2.1) we obtain f ðxn þ yÞ f ðxn yÞ 6 uðyÞ ; hðyÞ j2gðx Þj 2gðx Þ n
n
that is hðyÞ ¼ lim acf ðxn þ yÞ f ðxn yÞ2gðxn Þ; n!1
x 2 G:
ð2:3Þ
Using (2.1), we have jf ððxn þ xÞ þ yÞ f ððxn þ xÞ yÞ 2gðxn þ xÞhðyÞ þ f ððxn xÞ þ yÞ f ððxn xÞ yÞ 2gðxn xÞhðyÞj 6 2uðyÞ; so that f ðxn þ ðx þ yÞÞ f ðxn ðx þ yÞÞ f ðxn þ ðx yÞÞ f ðxn ðx yÞÞ gðxn þ xÞ þ gðxn xÞ uðyÞ 2 hðyÞ 6 2gðx Þ 2gðx Þ 2gðx Þ jgðx Þj n
n
n
ð2:4Þ
n
for all x; y 2 G. Taking the limit as n ! 1 with the use of (2.2) and (2.3), we conclude that, for every x 2 G, there exists the limit k0 ðxÞ :¼ lim
n!1
gðxn þ xÞ þ gðxn xÞ ; gðxn Þ
ð2:5Þ
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where the function k0 : G ! C satisfies the equation hðx þ yÞ hðx yÞ ¼ k0 ðxÞhðyÞ
8x; y 2 G:
ð2:6Þ
From the definition of k0 , we get the equality k0 ð0Þ ¼ 2, which jointly with (2.6) implies that h has to be odd. Keeping this in mind, by means of (2.6), we infer the equality hðx þ yÞ2 hðx yÞ2 ¼ ½hðx þ yÞ þ hðx yÞ½hðx þ yÞ hðx yÞ ¼ ½hðx þ yÞ þ hðx yÞk0 ðxÞhðyÞ ¼ ½hð2x þ yÞ þ hð2x yÞhðyÞ ¼ ½hðy þ 2xÞ hðy 2xÞhðyÞ ¼ k0 ðyÞhð2xÞhðyÞ: Putting y ¼ x in (2.6) we get the equation hð2xÞ ¼ k0 ðxÞhðxÞ;
x 2 G:
This, in return, leads to the equation hðx þ yÞ2 hðx yÞ2 ¼ hð2xÞhð2yÞ valid for all x; y 2 G which, in the light of the unique 2-divisibility of G, states nothing else but (S). (ii) If g satisfies (A), the defined limit function k0 states nothing else but 2g, so (2.6) validates (Tgh), namely hðx þ yÞ hðx yÞ ¼ 2gðxÞhðyÞ. h Theorem 2. Suppose that f ; g; h : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞj 6 uðxÞ
8x; y 2 G:
ð2:7Þ
If h fails to be bounded, then (i) g satisfies (S) under one of the cases gð0Þ ¼ 0, f ðxÞ ¼ f ðxÞ; (ii) In particular case h satisfies (A) or (T), g and h are solutions of the Wilson equation (Agh ). Proof. Let h be unbounded solution of the inequality (2.7). Then, there exists a sequence fyn g in G such that 0 6¼ jhðyn Þj ! 1 as n ! 1. Taking y ¼ yn in the inequality (2.7), dividing both sides by j2hðyn Þj, and passing to the limit as n ! 1 we obtain that gðxÞ ¼ lim cf ðx þ yn Þ f ðx yn Þ2hðyn Þ; n!1
x 2 G:
ð2:8Þ
Replacing y by y þ yn and y yn in (2.7), we can, with an application of (2.8), state the existence of a limit function k1 ðyÞ :¼ lim
n!1
hðy þ yn Þ hðy yn Þ ; hðyn Þ
where the function k1 : G ! C satisfies the equation gðx þ yÞ þ gðx yÞ ¼ gðxÞk1 ðyÞ
8x; y 2 G:
ð2:9Þ
Applying the case gð0Þ ¼ 0 in (2.9), it implies that g is an odd function. Keeping this in mind, by means of (2.9), we infer the equality gðx þ yÞ2 gðx yÞ2 ¼ ½gðx þ yÞ þ gðx yÞ½gðx þ yÞ gðx yÞ ¼ gðxÞk1 ðyÞ½gðx þ yÞ gðx yÞ ¼ gðxÞ½gðx þ 2yÞ gðx 2yÞ ¼ gðxÞ½gð2y þ xÞ þ gð2y xÞ ¼ gð2yÞgðxÞk1 ðxÞ: Putting y ¼ x in (2.9) we get the equation gð2xÞ ¼ gðxÞk1 ðxÞ;
x 2 G:
This, in return, leads to the equation gðx þ yÞ2 gðx yÞ2 ¼ gð2xÞgð2yÞ valid for all x; y 2 G which, in the light of the unique 2-divisibility of G, states nothing else but (S). Next, in particular case f ðxÞ ¼ f ðxÞ, it is enough to show that gð0Þ ¼ 0. Suppose that this is not the case.
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Putting x ¼ 0 in (2.7), due to gð0Þ 6¼ 0 and f ðxÞ ¼ f ðxÞ, we obtain the inequality jhðyÞj 6
uð0Þ ; 2jgð0Þj
y 2 G:
This inequality means that h is globally bounded – a contradiction. Thus the claimed gð0Þ ¼ 0 holds. (ii) In the case h satisfies (T), the limit k1 states nothing else but 2h, so (2.9) validates ðAgh Þ, namely gðx þ yÞ þ gðx yÞ ¼ 2gðxÞhðyÞ. Finally, consider the case h satisfies (A). An obvious slight change in the steps of the proof applied after (2.3), with substitute y to y þ yn and y yn in (2.7), and an application of (2.8), implies the Wilson equation: gðx þ yÞ þ gðx yÞ ¼ 2gðxÞhðyÞ
8x; y 2 G:
By replacing h by f, g by f, h by g in Theorems 1 and 2, we obtain the following corollaries. Corollary 1. Suppose that f ; g : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2gðxÞf ðyÞj 6 uðyÞ
8x; y 2 G:
ð2:10Þ
If g fails to be bounded, then (i) f satisfies (S); (ii) In particular case g satisfies (A), f and g are solutions of Eq. (Tgf). Corollary 2. Suppose that f ; g : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2gðxÞf ðyÞj 6 uðxÞ
8x; y 2 G:
ð2:11Þ
Then either f is bounded or g satisfies (A). Proof. Replacing h by f in Theorem 2, the limit equation (2.8) transfer to gðxÞ ¼ lim
n!1
f ðx þ yn Þ f ðx yn Þ : 2f ðyn Þ
ð2:12Þ
An obvious slight change in the steps of the proof applied in (2.8) of Theorem 2, with an application of (2.12), gives us the required result. h Corollary 3. Suppose that f ; h : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2f ðxÞhðyÞj 6 uðyÞ
8x; y 2 G:
If f fails to be bounded, then (i) h satisfies (S); (ii) In particular case f satisfies (A), f and h are solutions of hðx þ yÞ hðx yÞ ¼ 2f ðxÞhðyÞ. Corollary 4. Suppose that f ; h : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2f ðxÞhðyÞj 6 uðxÞ
8x; y 2 G:
If h fails to be bounded, then (i) f satisfies (S) under one of the cases f ð0Þ ¼ 0, f ðxÞ ¼ f ðxÞ; (ii) In particular case h satisfies (A) or (T), f and h are solutions of ðAfh Þ : f ðx þ yÞ þ f ðx yÞ ¼ 2f ðxÞhðyÞ. Corollary 5. Suppose that f ; g : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2gðxÞgðyÞj 6 uðxÞ or uðyÞ
8x; y 2 G:
Then either g is bounded or g satisfies (S) under 2-divisible. Corollary 6. Suppose that a non-zero function f : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2f ðxÞf ðyÞj 6 uðxÞ or uðyÞ
8x; y 2 G:
Then f is bounded. Proof. Assume that f is not bounded. Then, by applying g ¼ f in Corollaries 1 and 2, f satisfies simultaneously (A) and (T). This forces that f is a zero function. But we know that there exists the cosine function which satisfies (A) as a non-zero. Hence we arrive the result by a contradiction. h
G.H. Kim / Applied Mathematics and Computation 203 (2008) 99–105
103
By applying uðxÞ ¼ uðyÞ ¼ e in Theorems 1 and 2, we obtain the following corollaries. Corollary 7. Suppose that f ; g; h : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞj 6 e for all x; y 2 G. If g fails to be bounded, then (i) h satisfies (S); (ii) In particular case g satisfies (A), g and h are solutions of the equation (Tgh). Corollary 8. Suppose that f ; g; h : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞj 6 e
8x; y 2 G:
If h fails to be bounded, then (i) g satisfies (S) under one of the cases gð0Þ ¼ 0, f ðxÞ ¼ f ðxÞ; (ii) In particular case h satisfies (A) or (T), g and h are solutions of the Wilson equation (Agh ). Remark 1. By applying uðxÞ ¼ uðyÞ ¼ e in Corollaries 1–6, we can obtain the same types as them. For the sake of brevity, we will omit them to write.
3. Extension to the Banach algebra The obtained results of Section 2 can be extended to the Banach algebra. For simplify, we only will represent one of them (16 results), and the applications to the other theorems and corollaries will be omitted. Theorem 3. Let ðE; k kÞ be a semisimple commutative Banach algebra. Assume that f ; g; h : G ! E satisfy the inequality kf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞk 6 uðyÞ
8x; y 2 G:
ð3:1Þ
For an arbitrary linear multiplicative functional x 2 E , if the superposition x g fails to be bounded, then (i) h satisfies (S); (ii) In particular case x g satisfies (A), g and h are solutions of equation (Tgh). Proof. (i) Assume that (3.1) holds, and fix arbitrarily a linear multiplicative functional x 2 E . As is well known, we have kx k ¼ 1 whence, for every x; y 2 G, we have uðyÞ P kf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞk ¼ sup y ðf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞÞj ky k¼1
P jx ðf ðx þ yÞÞ x ðf ðx yÞÞ 2x ðgðxÞÞx ðhðyÞÞj; which states that the superpositions x f , x g and x h yield a solution of inequality (2.11). Since, by assumption, the superposition x g is unbounded, an appeal to Theorem 2 shows that the function x h solves Eq. (S). In other words, bearing the linear multiplicativity of x in mind, for all x; y 2 G, the difference DSh ðx; yÞ :¼ 2 2 hðxÞhðyÞ h xþy þ h xy falls into the kernel of x . Therefore, in view of the unrestricted choice of x , we infer that 2 2 DSh ðx; yÞ 2 \fker x : x is a multiplicative member of E g for all x; y 2 G. Since the algebra E has been assumed to be semisimple, the last term of the above formula coincides with the singleton f0g, i.e. x þ y2 x y2 hðxÞhðyÞ h þh ¼ 0 for all x; y 2 G; 2 2 as claimed. (ii) Under the assumption that the superposition x g satisfies (A), we know from Theorem 2 that the superpositions x h and x g are solutions of the equation x ðhðx þ yÞÞ x ðhðx yÞÞ ¼ 2x ðgðxÞÞx ðhðyÞÞ: Namely hðx þ yÞ hðx yÞ 2gðxÞhðyÞ 2 \fker x : x is a multiplicative member of E g: The other argument is similar.
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4. Solution and stability of the functional equation (Jy) In this section, let E be a Banach space. We prove the stability in the sense of Gavruta [9] for the functional Eq. (Jy). As a consequence, we obtain some corollaries, which are the Hyers–Ulam stability and the other type stability [15,14]. Theorem 4. Suppose that f : G ! E satisfies the inequality kf ðx þ yÞ f ðx yÞ 2f ðyÞk 6 wðx; yÞ; ð4:1Þ P1 1 k1 k1 where wðP 0Þ satisfies Wðx; yÞ :¼ k¼1 2k wð2 x; 2 yÞ < 1 for all x; y 2 G.Then there exists an unique additive mapping A : G ! E as a solution of (Jy) such that AðxÞ ¼ lim acf ð2n xÞ2n
ð4:2Þ
n!1
and kf ðxÞ þ f ð0Þ AðxÞk 6 Wðx; xÞ
8x 2 G:
ð4:3Þ
Proof. Putting y ¼ x in (4.1) we have kf ð2xÞ f ð0Þ 2f ðxÞk 6 wðx; xÞ
ð4:4Þ
for all x 2 G. Let FðxÞ :¼ f ðxÞ þ f ð0Þ for all x 2 G. Then Fð0Þ ¼ 2f ð0Þ and jjFð2xÞ 2FðxÞjj 6 wðx; xÞ
ð4:5Þ
n
for all x 2 G. Replacing x by 2 x in (4.5) and dividing its result by 2 Fð2n xÞ Fð2nþ1 xÞ 1 wð2n x; 2n xÞ 6 2n 2nþ1 2nþ1
nþ1
we get ð4:6Þ
for all x 2 E and all nonnegative integers n. Using (4.6) and the triangle inequality we have n X Fð2m xÞ Fð2n xÞ 1 6 wð2k1 x; 2k1 xÞ n 2m k 2 k¼mþ1 2
ð4:7Þ
n
for all x 2 G and all nonnegative integers m and n with m < n. This shows that fFð22n xÞg is a Cauchy sequence for all x 2 G because the right side of (4.7) converges to zero by the assumption of u when m ! 1. Consequently, we can define a mapping A : G ! E by AðxÞ :¼ lim
n!1
Fð2n xÞ 2n
ð4:8Þ
for all x 2 G. Putting m ¼ 0 in (4.7) and taking the limit as n ! 1, we obtain (4.3). Also, we get (4.2) from the definition of F in (4.8), and kAðx þ yÞ Aðx yÞ 2AðyÞk 6 lim
n!1
1 1 ½kf ð2n x þ 2n yÞ f ð2n x 2n yÞ 2f ð2n yÞk 6 lim 2 nþ1 wð2n x; 2n yÞ ¼ 0 n!1 2n 2
for all x; y 2 G, which satisfies Eq. (Jy). It also follows that A is additive mapping with Að0Þ ¼ 0. Now, let A0 : G ! E be another mapping satisfying (4.3). Then we have kAðxÞ A0 ðxÞk ¼ 2n kAð2n xÞ A0 ð2n xÞk 6 2n ðkAð2n xÞ f ð2n xÞ f ð0Þk þ kA0 ð2n xÞ f ð2n xÞ f ð0ÞkÞ 6 2
1 X 1 k k¼nþ1 2
wð2k1 x;2k1 xÞ
for all x 2 G and all positive integers n. Taking the limit in the above inequality as n ! 1, we can conclude that AðxÞ ¼ A0 ðxÞ for all x 2 G. This proves the uniqueness of A. Corollary 9. Suppose that f : G ! E satisfies the inequality kf ðx þ yÞ f ðx yÞ 2f ðyÞk 6 eðkxkp þ kykp Þ for all x; y 2 G and for some fixed p 2 R such that 0 6 p < 1. Then there exists an unique additive mapping A : G ! E as a solution of (Jy), which satisfies (4.2) and kf ðxÞ þ f ð0Þ AðxÞk 6
2ekxkp 2 2p
8x 2 G:
Corollary 10. Suppose that f : G ! E satisfies the inequality kf ðx þ yÞ f ðx yÞ 2f ðyÞk 6 ekxkq kykr for all x; y 2 G and for some fixed q; r 2 R such that p :¼ q þ r 2 R and 0 6 p < 1.
G.H. Kim / Applied Mathematics and Computation 203 (2008) 99–105
105
Then there exists an unique additive mapping A : G ! E as a solution of (Jy), which satisfies (4.2) and kf ðxÞ þ f ð0Þ AðxÞk 6
ekxkp 2 2p
8x 2 G:
ð4:9Þ
Remark 2. Corollaries 9 and 10 can be expanded with case p > 1 by Gajda method [7], and can be expanded with case p 2 ð1; 0. Putting wðx; yÞ ¼ uðxÞ or wðx; yÞ ¼ uðyÞ in (4.1), it implies (4.10) and (3.1). Putting y ¼ x in them, then it is similar with the inequality (4.4) of Theorem 4. Hence the remainder of the proof runs along Theorem 4. Theorem 5. Suppose that f : G ! E satisfies the inequality kf ðx þ yÞ f ðx yÞ 2f ðyÞk 6 uðxÞ; P k1 1 where u satisfies UðxÞ :¼ 1 xÞ < 1 for all x; y 2 G. k¼1 2k uð2 Then there exists an unique additive mapping A : G ! E as a solution of (Jy), which satisfies (4.2) and kf ðxÞ þ f ð0Þ AðxÞk 6 UðxÞ
8x 2 G:
ð4:10Þ
ð4:11Þ
Theorem 6. Suppose that f : G ! E satisfies the inequality kf ðx þ yÞ f ðx yÞ 2f ðyÞk 6 uðyÞ; P k1 1 yÞ < 1 for all x; y 2 G. where u satisfies UðyÞ :¼ 1 k¼1 2k uð2 Then there exists an unique additive mapping A : G ! E as a solution of (Jy), which satisfies (4.2) and (4.11). Corollary 11. Suppose that f : G ! E satisfies the inequality kf ðx þ yÞ f ðx yÞ 2f ðyÞk 6 ekxkp or ekykp for all x; y 2 G and for some fixed p 2 R such that 0 6 p < 1. Then there exists an unique additive mapping A : G ! E as a solution of (Jy), which satisfies (4.2) and (4.9). Theorem 7. Suppose that f : G ! E satisfies the inequality kf ðx þ yÞ f ðx yÞ 2f ðyÞk 6 e: Then there exists an unique additive mapping A : G ! E as a solution of (Jy), which satisfies (4.2) and kf ðxÞ þ f ð0Þ AðxÞk 6 e
8x 2 G:
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R. Badora, On the stability of cosine functional equation, Rocznik Naukowo-Dydak. Prace Mat. 15 (1998) 1–14. R. Badora, R. Ger, On some trigonometric functional inequalities, Functional Equations – Results and Advances, 2002, pp. 3–15. J.A. Baker, The stability of the cosine equation, Proc. Am. Math. Soc. 80 (1980) 411–416. J.A. Baker, J. Lawrence, F. Zorzitto, The stability of the equation f ðx þ yÞ ¼ f ðxÞf ðyÞ, Proc. Am. Math. Soc. 74 (1979) 242–246. D.G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J. 16 (1949) 385–397. P.W. Cholewa, The stability of the sine equation, Proc. Am. Math. Soc. 88 (1983) 631–634. Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. 14 (1991) 431–434. P. Gavruta, On the stability of some functional equations, in: Th.M. Rassias, J. Tabor (Eds.), Stability of Mappings of Hyers–Ulam type, Hadronic Press, Palm Harbor, FL, 1994, pp. 93–98. MR 95 k:39027. P. Gavruta, A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994) 431–436. Pl. Kannappan, The functional equation f ðxyÞ þ f ðxy1 Þ ¼ 2f ðxÞf ðyÞ for groups, Proc. Am. Math. Soc. 19 (1968) 69–74. Pl. Kannappan, G.H. Kim, On the stability of the generalized cosine functional equations, Ann. Acad. Pedagog. Crac. Stud. Math. 1 (2001) 49–58. G.H. Kim, The stability of the d’Alembert and Jensen type functional equations, J. Math. Anal. Appl. 325 (2007) 237–248. G.H. Kim, A stability of the generalized sine functional equations, J. Math. Anal. Appl. 331 (2007) 886–894. J.M. Rassias, On approximation of approximately linear mapping by linear mappings, J. Funct. Anal. 46 (1982) 126–130. Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc. 72 (2) (1978) 297–300. P. Sinopoulos, Functional equations on semigroups, Aequationes Math. 59 (2000) 255–261.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
On the stability of the pexiderized trigonometric functional equation Gwang Hui Kim Department of Mathematics, Kangnam University, Youngin, Gyeonggi 446-702, Republic of Korea
a r t i c l e
i n f o
Keywords: Stability Superstability Sine functional equation d’Alembert equation Trigonometric functional equation
a b s t r a c t The aim of this paper is to investigate the superstability problem of the cosine(d’Alembert), the Wilson, and the sine functional equation from the pexiderized trigonometric functional equation (Tgh) under the condition: jf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞj 6 uðxÞ or uðyÞ, and also will investigated the stability of the Jensen type functional equation (Jy) under the condition: jf ðx þ yÞ f ðx yÞ 2f ðyÞj 6 uðx; yÞ. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction Baker [4], and Bourgin [5] introduced the following: if f satisfies the inequality jE1 ðf Þ E2 ðf Þj 6 e, then either f is bounded or E1 ðf Þ ¼ E2 ðf Þ. This is frequently referred to as Superstability. The superstability of the cosine functional equation (also called the d’Alembert equation): f ðx þ yÞ þ f ðx yÞ ¼ 2f ðxÞf ðyÞ is investigated by Baker [3], which again is proved under a simple new technique by Gavruta [8]. The superstability of the sine functional equation: x þ y2 x y2 f ðxÞf ðyÞ ¼ f f 2 2
ðAÞ
ðSÞ
is investigated by Cholewa [6]. The d’Alembert functional equation (A) is generalized to the following functional equations: f ðx þ yÞ þ f ðx yÞ ¼ 2f ðxÞgðyÞ;
ðAfg Þ
f ðx þ yÞ þ f ðx yÞ ¼ 2gðxÞf ðyÞ;
ðAgf Þ
f ðx þ yÞ þ f ðx yÞ ¼ 2gðxÞgðyÞ:
ðAgg Þ
Eq. (Afg), introduced by Wilson, is sometimes referred to as the Wilson equation. The above functional equations have been investigated by Badora, Ger, Kannappan, Kim, Sinopoulos, etc. [1,2,10–13,16]. Due to the well known trigonometric identity ðsinða þ bÞ sinða bÞ ¼ 2 cos a sin bÞ, we will consider the following pexider type trigonometric functional equation: f ðx þ yÞ f ðx yÞ ¼ 2gðxÞhðyÞ; E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.04.011
ðTgh Þ
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G.H. Kim / Applied Mathematics and Computation 203 (2008) 99–105
which has special cases as follows: f ðx þ yÞ f ðx yÞ ¼ 2f ðxÞf ðyÞ;
ðTÞ
f ðx þ yÞ f ðx yÞ ¼ 2f ðxÞgðyÞ;
ðTfg Þ
f ðx þ yÞ f ðx yÞ ¼ 2gðxÞf ðyÞ;
ðTgf Þ
f ðx þ yÞ f ðx yÞ ¼ 2gðxÞgðyÞ:
ðTgg Þ
Let gðxÞ 1 in (Tgf), then we also obtain the Jensen type functional equation f ðx þ yÞ f ðx yÞ ¼ 2f ðyÞ:
ðJy Þ
Given mappings f ; g; h : G ! C, for above equations, we will denote their difference by an operator DT gh : G G ! C as DT gh ðx; yÞ :¼ f ðx þ yÞ f ðx yÞ 2gðxÞhðyÞ: The aim of this paper is to investigate the superstability problem of the cosine(d’Alembert), the Wilson, and the sine functional equation from the pexiderized trigonometric functional equation (Tgh) under the condition: jDT gh ðx; yÞj 6 uðxÞ or uðyÞ, and also will investigated the stability of the Jensen type functional equation (Jy) under the condition: jDJy ðx; yÞj 6 uðx; yÞ. As a consequence, we obtain the superstability of the cosine(d’Alembert), the Wilson, and the sine functional equation from the above trigonometric type Eqs. (T), (Tfg), (Tgf) and (Tgg) as corollaries, and also extend the obtained results to the Banach algebra. In this paper, let ðG; þÞ be an Abelian group, C the field of complex numbers, and R the field of real numbers. In particular, if there is not referring which is special, let ðG; þÞ be a uniquely 2-divisible group whenever the function is related to the sine functional equation (S), and it will be denoted by ‘‘under 2-divisible” for short. We may assume that f ; g and h are non-zero functions and e is a nonnegative real constant, and let w : G G ! R and u : G ! R be a mappings. 2. Stability on Eq. (Tgh) In this section, we investigate the superstability of the d’Alembert (A), the Wilson (Afg), the sine (S), and the mixed type functional equations (Agf), (Tfg) and (Tgf) from the approximately equation of (Tgh). Theorem 1. Suppose that f ; g; h : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞj 6 uðyÞ
ð2:1Þ
for all x; y 2 G. If g fails to be bounded, then (i) h satisfies (S); (ii) In particular case g satisfies (A), g and h are solutions of Eq. (Tgh). Proof. (i) Let g be unbounded. Then we can choose a sequence fxn g in G such that as n ! 1:
0 6¼ jgðxn Þj ! 1;
ð2:2Þ
Taking x ¼ xn in (2.1) we obtain f ðxn þ yÞ f ðxn yÞ 6 uðyÞ ; hðyÞ j2gðx Þj 2gðx Þ n
n
that is hðyÞ ¼ lim acf ðxn þ yÞ f ðxn yÞ2gðxn Þ; n!1
x 2 G:
ð2:3Þ
Using (2.1), we have jf ððxn þ xÞ þ yÞ f ððxn þ xÞ yÞ 2gðxn þ xÞhðyÞ þ f ððxn xÞ þ yÞ f ððxn xÞ yÞ 2gðxn xÞhðyÞj 6 2uðyÞ; so that f ðxn þ ðx þ yÞÞ f ðxn ðx þ yÞÞ f ðxn þ ðx yÞÞ f ðxn ðx yÞÞ gðxn þ xÞ þ gðxn xÞ uðyÞ 2 hðyÞ 6 2gðx Þ 2gðx Þ 2gðx Þ jgðx Þj n
n
n
ð2:4Þ
n
for all x; y 2 G. Taking the limit as n ! 1 with the use of (2.2) and (2.3), we conclude that, for every x 2 G, there exists the limit k0 ðxÞ :¼ lim
n!1
gðxn þ xÞ þ gðxn xÞ ; gðxn Þ
ð2:5Þ
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G.H. Kim / Applied Mathematics and Computation 203 (2008) 99–105
where the function k0 : G ! C satisfies the equation hðx þ yÞ hðx yÞ ¼ k0 ðxÞhðyÞ
8x; y 2 G:
ð2:6Þ
From the definition of k0 , we get the equality k0 ð0Þ ¼ 2, which jointly with (2.6) implies that h has to be odd. Keeping this in mind, by means of (2.6), we infer the equality hðx þ yÞ2 hðx yÞ2 ¼ ½hðx þ yÞ þ hðx yÞ½hðx þ yÞ hðx yÞ ¼ ½hðx þ yÞ þ hðx yÞk0 ðxÞhðyÞ ¼ ½hð2x þ yÞ þ hð2x yÞhðyÞ ¼ ½hðy þ 2xÞ hðy 2xÞhðyÞ ¼ k0 ðyÞhð2xÞhðyÞ: Putting y ¼ x in (2.6) we get the equation hð2xÞ ¼ k0 ðxÞhðxÞ;
x 2 G:
This, in return, leads to the equation hðx þ yÞ2 hðx yÞ2 ¼ hð2xÞhð2yÞ valid for all x; y 2 G which, in the light of the unique 2-divisibility of G, states nothing else but (S). (ii) If g satisfies (A), the defined limit function k0 states nothing else but 2g, so (2.6) validates (Tgh), namely hðx þ yÞ hðx yÞ ¼ 2gðxÞhðyÞ. h Theorem 2. Suppose that f ; g; h : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞj 6 uðxÞ
8x; y 2 G:
ð2:7Þ
If h fails to be bounded, then (i) g satisfies (S) under one of the cases gð0Þ ¼ 0, f ðxÞ ¼ f ðxÞ; (ii) In particular case h satisfies (A) or (T), g and h are solutions of the Wilson equation (Agh ). Proof. Let h be unbounded solution of the inequality (2.7). Then, there exists a sequence fyn g in G such that 0 6¼ jhðyn Þj ! 1 as n ! 1. Taking y ¼ yn in the inequality (2.7), dividing both sides by j2hðyn Þj, and passing to the limit as n ! 1 we obtain that gðxÞ ¼ lim cf ðx þ yn Þ f ðx yn Þ2hðyn Þ; n!1
x 2 G:
ð2:8Þ
Replacing y by y þ yn and y yn in (2.7), we can, with an application of (2.8), state the existence of a limit function k1 ðyÞ :¼ lim
n!1
hðy þ yn Þ hðy yn Þ ; hðyn Þ
where the function k1 : G ! C satisfies the equation gðx þ yÞ þ gðx yÞ ¼ gðxÞk1 ðyÞ
8x; y 2 G:
ð2:9Þ
Applying the case gð0Þ ¼ 0 in (2.9), it implies that g is an odd function. Keeping this in mind, by means of (2.9), we infer the equality gðx þ yÞ2 gðx yÞ2 ¼ ½gðx þ yÞ þ gðx yÞ½gðx þ yÞ gðx yÞ ¼ gðxÞk1 ðyÞ½gðx þ yÞ gðx yÞ ¼ gðxÞ½gðx þ 2yÞ gðx 2yÞ ¼ gðxÞ½gð2y þ xÞ þ gð2y xÞ ¼ gð2yÞgðxÞk1 ðxÞ: Putting y ¼ x in (2.9) we get the equation gð2xÞ ¼ gðxÞk1 ðxÞ;
x 2 G:
This, in return, leads to the equation gðx þ yÞ2 gðx yÞ2 ¼ gð2xÞgð2yÞ valid for all x; y 2 G which, in the light of the unique 2-divisibility of G, states nothing else but (S). Next, in particular case f ðxÞ ¼ f ðxÞ, it is enough to show that gð0Þ ¼ 0. Suppose that this is not the case.
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Putting x ¼ 0 in (2.7), due to gð0Þ 6¼ 0 and f ðxÞ ¼ f ðxÞ, we obtain the inequality jhðyÞj 6
uð0Þ ; 2jgð0Þj
y 2 G:
This inequality means that h is globally bounded – a contradiction. Thus the claimed gð0Þ ¼ 0 holds. (ii) In the case h satisfies (T), the limit k1 states nothing else but 2h, so (2.9) validates ðAgh Þ, namely gðx þ yÞ þ gðx yÞ ¼ 2gðxÞhðyÞ. Finally, consider the case h satisfies (A). An obvious slight change in the steps of the proof applied after (2.3), with substitute y to y þ yn and y yn in (2.7), and an application of (2.8), implies the Wilson equation: gðx þ yÞ þ gðx yÞ ¼ 2gðxÞhðyÞ
8x; y 2 G:
By replacing h by f, g by f, h by g in Theorems 1 and 2, we obtain the following corollaries. Corollary 1. Suppose that f ; g : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2gðxÞf ðyÞj 6 uðyÞ
8x; y 2 G:
ð2:10Þ
If g fails to be bounded, then (i) f satisfies (S); (ii) In particular case g satisfies (A), f and g are solutions of Eq. (Tgf). Corollary 2. Suppose that f ; g : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2gðxÞf ðyÞj 6 uðxÞ
8x; y 2 G:
ð2:11Þ
Then either f is bounded or g satisfies (A). Proof. Replacing h by f in Theorem 2, the limit equation (2.8) transfer to gðxÞ ¼ lim
n!1
f ðx þ yn Þ f ðx yn Þ : 2f ðyn Þ
ð2:12Þ
An obvious slight change in the steps of the proof applied in (2.8) of Theorem 2, with an application of (2.12), gives us the required result. h Corollary 3. Suppose that f ; h : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2f ðxÞhðyÞj 6 uðyÞ
8x; y 2 G:
If f fails to be bounded, then (i) h satisfies (S); (ii) In particular case f satisfies (A), f and h are solutions of hðx þ yÞ hðx yÞ ¼ 2f ðxÞhðyÞ. Corollary 4. Suppose that f ; h : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2f ðxÞhðyÞj 6 uðxÞ
8x; y 2 G:
If h fails to be bounded, then (i) f satisfies (S) under one of the cases f ð0Þ ¼ 0, f ðxÞ ¼ f ðxÞ; (ii) In particular case h satisfies (A) or (T), f and h are solutions of ðAfh Þ : f ðx þ yÞ þ f ðx yÞ ¼ 2f ðxÞhðyÞ. Corollary 5. Suppose that f ; g : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2gðxÞgðyÞj 6 uðxÞ or uðyÞ
8x; y 2 G:
Then either g is bounded or g satisfies (S) under 2-divisible. Corollary 6. Suppose that a non-zero function f : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2f ðxÞf ðyÞj 6 uðxÞ or uðyÞ
8x; y 2 G:
Then f is bounded. Proof. Assume that f is not bounded. Then, by applying g ¼ f in Corollaries 1 and 2, f satisfies simultaneously (A) and (T). This forces that f is a zero function. But we know that there exists the cosine function which satisfies (A) as a non-zero. Hence we arrive the result by a contradiction. h
G.H. Kim / Applied Mathematics and Computation 203 (2008) 99–105
103
By applying uðxÞ ¼ uðyÞ ¼ e in Theorems 1 and 2, we obtain the following corollaries. Corollary 7. Suppose that f ; g; h : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞj 6 e for all x; y 2 G. If g fails to be bounded, then (i) h satisfies (S); (ii) In particular case g satisfies (A), g and h are solutions of the equation (Tgh). Corollary 8. Suppose that f ; g; h : G ! C satisfy the inequality jf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞj 6 e
8x; y 2 G:
If h fails to be bounded, then (i) g satisfies (S) under one of the cases gð0Þ ¼ 0, f ðxÞ ¼ f ðxÞ; (ii) In particular case h satisfies (A) or (T), g and h are solutions of the Wilson equation (Agh ). Remark 1. By applying uðxÞ ¼ uðyÞ ¼ e in Corollaries 1–6, we can obtain the same types as them. For the sake of brevity, we will omit them to write.
3. Extension to the Banach algebra The obtained results of Section 2 can be extended to the Banach algebra. For simplify, we only will represent one of them (16 results), and the applications to the other theorems and corollaries will be omitted. Theorem 3. Let ðE; k kÞ be a semisimple commutative Banach algebra. Assume that f ; g; h : G ! E satisfy the inequality kf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞk 6 uðyÞ
8x; y 2 G:
ð3:1Þ
For an arbitrary linear multiplicative functional x 2 E , if the superposition x g fails to be bounded, then (i) h satisfies (S); (ii) In particular case x g satisfies (A), g and h are solutions of equation (Tgh). Proof. (i) Assume that (3.1) holds, and fix arbitrarily a linear multiplicative functional x 2 E . As is well known, we have kx k ¼ 1 whence, for every x; y 2 G, we have uðyÞ P kf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞk ¼ sup y ðf ðx þ yÞ f ðx yÞ 2gðxÞhðyÞÞj ky k¼1
P jx ðf ðx þ yÞÞ x ðf ðx yÞÞ 2x ðgðxÞÞx ðhðyÞÞj; which states that the superpositions x f , x g and x h yield a solution of inequality (2.11). Since, by assumption, the superposition x g is unbounded, an appeal to Theorem 2 shows that the function x h solves Eq. (S). In other words, bearing the linear multiplicativity of x in mind, for all x; y 2 G, the difference DSh ðx; yÞ :¼ 2 2 hðxÞhðyÞ h xþy þ h xy falls into the kernel of x . Therefore, in view of the unrestricted choice of x , we infer that 2 2 DSh ðx; yÞ 2 \fker x : x is a multiplicative member of E g for all x; y 2 G. Since the algebra E has been assumed to be semisimple, the last term of the above formula coincides with the singleton f0g, i.e. x þ y2 x y2 hðxÞhðyÞ h þh ¼ 0 for all x; y 2 G; 2 2 as claimed. (ii) Under the assumption that the superposition x g satisfies (A), we know from Theorem 2 that the superpositions x h and x g are solutions of the equation x ðhðx þ yÞÞ x ðhðx yÞÞ ¼ 2x ðgðxÞÞx ðhðyÞÞ: Namely hðx þ yÞ hðx yÞ 2gðxÞhðyÞ 2 \fker x : x is a multiplicative member of E g: The other argument is similar.
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4. Solution and stability of the functional equation (Jy) In this section, let E be a Banach space. We prove the stability in the sense of Gavruta [9] for the functional Eq. (Jy). As a consequence, we obtain some corollaries, which are the Hyers–Ulam stability and the other type stability [15,14]. Theorem 4. Suppose that f : G ! E satisfies the inequality kf ðx þ yÞ f ðx yÞ 2f ðyÞk 6 wðx; yÞ; ð4:1Þ P1 1 k1 k1 where wðP 0Þ satisfies Wðx; yÞ :¼ k¼1 2k wð2 x; 2 yÞ < 1 for all x; y 2 G.Then there exists an unique additive mapping A : G ! E as a solution of (Jy) such that AðxÞ ¼ lim acf ð2n xÞ2n
ð4:2Þ
n!1
and kf ðxÞ þ f ð0Þ AðxÞk 6 Wðx; xÞ
8x 2 G:
ð4:3Þ
Proof. Putting y ¼ x in (4.1) we have kf ð2xÞ f ð0Þ 2f ðxÞk 6 wðx; xÞ
ð4:4Þ
for all x 2 G. Let FðxÞ :¼ f ðxÞ þ f ð0Þ for all x 2 G. Then Fð0Þ ¼ 2f ð0Þ and jjFð2xÞ 2FðxÞjj 6 wðx; xÞ
ð4:5Þ
n
for all x 2 G. Replacing x by 2 x in (4.5) and dividing its result by 2 Fð2n xÞ Fð2nþ1 xÞ 1 wð2n x; 2n xÞ 6 2n 2nþ1 2nþ1
nþ1
we get ð4:6Þ
for all x 2 E and all nonnegative integers n. Using (4.6) and the triangle inequality we have n X Fð2m xÞ Fð2n xÞ 1 6 wð2k1 x; 2k1 xÞ n 2m k 2 k¼mþ1 2
ð4:7Þ
n
for all x 2 G and all nonnegative integers m and n with m < n. This shows that fFð22n xÞg is a Cauchy sequence for all x 2 G because the right side of (4.7) converges to zero by the assumption of u when m ! 1. Consequently, we can define a mapping A : G ! E by AðxÞ :¼ lim
n!1
Fð2n xÞ 2n
ð4:8Þ
for all x 2 G. Putting m ¼ 0 in (4.7) and taking the limit as n ! 1, we obtain (4.3). Also, we get (4.2) from the definition of F in (4.8), and kAðx þ yÞ Aðx yÞ 2AðyÞk 6 lim
n!1
1 1 ½kf ð2n x þ 2n yÞ f ð2n x 2n yÞ 2f ð2n yÞk 6 lim 2 nþ1 wð2n x; 2n yÞ ¼ 0 n!1 2n 2
for all x; y 2 G, which satisfies Eq. (Jy). It also follows that A is additive mapping with Að0Þ ¼ 0. Now, let A0 : G ! E be another mapping satisfying (4.3). Then we have kAðxÞ A0 ðxÞk ¼ 2n kAð2n xÞ A0 ð2n xÞk 6 2n ðkAð2n xÞ f ð2n xÞ f ð0Þk þ kA0 ð2n xÞ f ð2n xÞ f ð0ÞkÞ 6 2
1 X 1 k k¼nþ1 2
wð2k1 x;2k1 xÞ
for all x 2 G and all positive integers n. Taking the limit in the above inequality as n ! 1, we can conclude that AðxÞ ¼ A0 ðxÞ for all x 2 G. This proves the uniqueness of A. Corollary 9. Suppose that f : G ! E satisfies the inequality kf ðx þ yÞ f ðx yÞ 2f ðyÞk 6 eðkxkp þ kykp Þ for all x; y 2 G and for some fixed p 2 R such that 0 6 p < 1. Then there exists an unique additive mapping A : G ! E as a solution of (Jy), which satisfies (4.2) and kf ðxÞ þ f ð0Þ AðxÞk 6
2ekxkp 2 2p
8x 2 G:
Corollary 10. Suppose that f : G ! E satisfies the inequality kf ðx þ yÞ f ðx yÞ 2f ðyÞk 6 ekxkq kykr for all x; y 2 G and for some fixed q; r 2 R such that p :¼ q þ r 2 R and 0 6 p < 1.
G.H. Kim / Applied Mathematics and Computation 203 (2008) 99–105
105
Then there exists an unique additive mapping A : G ! E as a solution of (Jy), which satisfies (4.2) and kf ðxÞ þ f ð0Þ AðxÞk 6
ekxkp 2 2p
8x 2 G:
ð4:9Þ
Remark 2. Corollaries 9 and 10 can be expanded with case p > 1 by Gajda method [7], and can be expanded with case p 2 ð1; 0. Putting wðx; yÞ ¼ uðxÞ or wðx; yÞ ¼ uðyÞ in (4.1), it implies (4.10) and (3.1). Putting y ¼ x in them, then it is similar with the inequality (4.4) of Theorem 4. Hence the remainder of the proof runs along Theorem 4. Theorem 5. Suppose that f : G ! E satisfies the inequality kf ðx þ yÞ f ðx yÞ 2f ðyÞk 6 uðxÞ; P k1 1 where u satisfies UðxÞ :¼ 1 xÞ < 1 for all x; y 2 G. k¼1 2k uð2 Then there exists an unique additive mapping A : G ! E as a solution of (Jy), which satisfies (4.2) and kf ðxÞ þ f ð0Þ AðxÞk 6 UðxÞ
8x 2 G:
ð4:10Þ
ð4:11Þ
Theorem 6. Suppose that f : G ! E satisfies the inequality kf ðx þ yÞ f ðx yÞ 2f ðyÞk 6 uðyÞ; P k1 1 yÞ < 1 for all x; y 2 G. where u satisfies UðyÞ :¼ 1 k¼1 2k uð2 Then there exists an unique additive mapping A : G ! E as a solution of (Jy), which satisfies (4.2) and (4.11). Corollary 11. Suppose that f : G ! E satisfies the inequality kf ðx þ yÞ f ðx yÞ 2f ðyÞk 6 ekxkp or ekykp for all x; y 2 G and for some fixed p 2 R such that 0 6 p < 1. Then there exists an unique additive mapping A : G ! E as a solution of (Jy), which satisfies (4.2) and (4.9). Theorem 7. Suppose that f : G ! E satisfies the inequality kf ðx þ yÞ f ðx yÞ 2f ðyÞk 6 e: Then there exists an unique additive mapping A : G ! E as a solution of (Jy), which satisfies (4.2) and kf ðxÞ þ f ð0Þ AðxÞk 6 e
8x 2 G:
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