Hindawi Publishing Corporation î e ScientiďŹc World Journal Volume 2014, Article ID 752146, 9 pages http://dx.doi.org/10.1155/2014/752146
Research Article Stability of a Quartic Functional Equation Abasalt Bodaghi Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran Correspondence should be addressed to Abasalt Bodaghi;
[email protected] Received 8 August 2013; Accepted 20 November 2013; Published 23 January 2014 Academic Editors: A. Ibeas and B. Meng Copyright Š 2014 Abasalt Bodaghi. 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. We obtain the general solution of the generalized quartic functional equation đ(đĽ + đđŚ) + đ(đĽ â đđŚ) = 2(7đ â 9)(đ â 1)đ(đĽ) + 2đ2 (đ2 â 1)đ(đŚ) â (đ â 1)2 đ(2đĽ) + đ2 {đ(đĽ + đŚ) + đ(đĽ â đŚ)} for a fixed positive integer đ. We prove the Hyers-Ulam stability for this quartic functional equation by the directed method and the fixed point method on real Banach spaces. We also investigate the Hyers-Ulam stability for the mentioned quartic functional equation in non-Archimedean spaces.
1. Introduction We say a functional equation F is stable if any function đ satisfying the equation F approximately is near to exact solution of F. Moreover, a functional equation F is hyperstable if any function đ satisfying the equation F approximately is a true solution of F. The study of stability problems for functional equations is related to a question of Ulam [1] concerning the stability of group homomorphisms, affirmatively answered for Banach spaces by Hyers [2]. Subsequently, the result of Hyers was generalized by a number of authors. For example, Bodaghi et al. investigated the Hyers-Ulam stability of Jordan âderivation pairs for the Cauchy additive functional equation and the Cauchy additive functional inequality in [3]. For some results on the stability of various functional equations, see also [4â9]. The oldest quartic functional equation was introduced by Rassias in [10] and then was employed by other authors. Rassias [10] investigated stability properties of the following quartic functional equation: đ (đĽ + 2đŚ) + đ (đĽ â 2đŚ) + 6đ (đĽ) = 4đ (đĽ + đŚ) + 4đ (đĽ â đŚ) + 24đ (đŚ) .
(1)
Since đ(2đĽ) = 16đ(đĽ), we get đ (đĽ + 2đŚ) + đ (đĽ â 2đŚ) = 10đ (đĽ) + 24đ (đŚ) â đ (2đĽ) + 4đ (đĽ + đŚ) + 4đ (đĽ â đŚ) .
(2)
In [11], Chung and Sahoo determined the general solution of (2) without assuming any regularity conditions on the unknown function. Indeed, they proved that the function đ : R â R is a solution of (2) if and only if đ(đĽ) = đ(đĽ, đĽ, đĽ, đĽ) where the function đ : R4 â R is symmetric and additive in each variable. The fact that every solution of (2) is even implies that it can be written as follows: đ (2đĽ + đŚ) + đ (2đĽ â đŚ) = 24đ (đĽ) â 6đ (đŚ) + 4đ (đĽ + đŚ) + 4đ (đĽ â đŚ) .
(3)
Lee et al. [12] obtained the general solution of (3) and proved the Hyers-Ulam stability of this equation. Also Park [13] investigated the stability problem of (3) in the orthogonality normed space. Lee and Chung [14] considered the following quartic functional equation, which is a generalization of (3): đ (đđĽ + đŚ) + đ (đđĽ â đŚ) = 2đ2 (đ2 â 1) đ (đĽ) â 2 (đ2 â 1) đ (đŚ)
(4)
+ đ2 đ (đĽ + đŚ) + đ2 đ (đĽ â đŚ)
for fixed integer đ with đ ≠ 0, Âą1. They obtained the general solution of (4) and proved its Hyers-Ulam stability. Bodaghi et al. [15] applied the fixed point alternative theorem (Theorem 8 of the current paper) to establish HyersUlam stability of (3). They also showed that the functional equation (3) can be hyperstable under some conditions. This
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method which is different from the âdirect method,â initiated by Hyers in 1941, had been applied by CËadariu and Radu for the first time in [16]. In other words, they employed this fixed point method to the investigation of the Cauchy functional equation [17] and of the quadratic functional equation [16] (for more applications of this method, see [18â20]). In this paper, we consider the following functional equation which is somewhat different from (2), (3), and (4): đ (đĽ + đđŚ) + đ (đĽ â đđŚ) = 2 (7đ â 9) (đ â 1) đ (đĽ) + 2đ2 (đ2 â 1) đ (đŚ) (5) â (đ â 1)2 đ (2đĽ) + đ2 {đ (đĽ + đŚ) + đ (đĽ â đŚ)} for a fixed positive integer đ. In case đ = 2, then (5) is the celebrated Jordan-von Neumann equation. Then we find out the general solution of (5). We also prove the Hyers-Ulam stability problem and the hyperstability for (5) by the directed method and the fixed point method.
2. General Solution of (5) To achieve our aim in this section, we need the following lemma. Lemma 1. Let X and Y be real vector spaces. If a function đ : X â Y satisfies the functional equation (5) for all integers đ ⼠3, then đ satisfies đ(đđĽ) = đ4 đ(đĽ) for all integers đ ⼠2. Proof. Letting đĽ = đŚ = 0 in (5), we get đ(0) = 0. Once more, by putting đŚ = 0 in (5), we obtain đ (2đĽ) = 24 đ (đĽ) .
(6)
Remark 2. It is shown in [21, Lemma 2.1] that a mapping đ : X â Y satisfies the functional equation (1) if and only if đ satisfies đ (đĽ + đđŚ) + đ (đĽ â đđŚ) = 2đ2 (đ2 â 1) đ (đŚ) â 2 (đ2 â 1) đ (đĽ) + đ2 {đ (đĽ + đŚ) + đ (đĽ â đŚ)} . There is a gap in its proof. In fact, in the proof, the author only showed that the functional equation (1) implies (9) but the converse is not proved. Theorem 3 resolved this problem. Indeed, we solve the equation of (5). Theorem 3. Let X and Y be real vector spaces. Then a mapping đ : X â Y satisfies the functional equation (2) if and only if it satisfies (5) where đ ⼠3. Therefore, every solution of the functional equation (5) is also a quartic mapping. Proof. Suppose that đ : X â Y satisfies the functional equation (2). Putting đĽ = đŚ = 0 in (2), we get đ(0) = 0. Let đŚ = 0 in (2) to get đ(2đĽ) = 16đ(đĽ) for all đĽ â X. Setting đĽ = 0 in (2) and using the fact that đ(đŚ) = 16đ(đŚ), we have đ(âđŚ) = đ(đŚ). Letting đŚ = đĽ in (2), we have đ(3đĽ) = 81đ(đĽ) for all đĽ â X. By induction, we obtain đ(đđĽ) = đ4 đ(đĽ) for all positive integers đ. Replacing đĽ by đĽ + đŚ and đĽ â đŚ in (2), respectively, we have đ (đĽ + 3đŚ) + đ (đĽ â 3đŚ) = 48đ (đĽ) + 144đ (đŚ) â 4đ (2đĽ) + 9 {đ (đĽ + đŚ) + đ (đĽ â đŚ)} . (10) In Similar way to the above, we get đ (đĽ + 4đŚ) + đ (đĽ â 4đŚ) = 114đ (đĽ) + 480đ (đŚ) â 9đ (2đĽ) + 16 {đ (đĽ + đŚ) + đ (đĽ â đŚ)} . (11)
In the case that đ = 3, by replacing đĽ, đŚ by 3đĽ, đĽ in (5), respectively, we have đ (6đĽ) = 48đ (3đĽ) + 144đ (đĽ) â 4đ (6đĽ) + 9 {đ (4đĽ) + đ (2đĽ)} .
(9)
Using the above method, we can deduce that đ (đĽ + đđŚ) + đ (đĽ â đđŚ)
(7)
= đđ đ (đĽ) + đđ đ (đŚ) â (đ â 1)2 đ (2đĽ)
4
The above equality and (6) imply that đ(3đĽ) = 3 đ(đĽ). Now, assume that, for every đ ⤠đ â 1, we have đ(đđĽ) = đ4 đ(đĽ). If đ = 2đ, then đ(đđĽ) = đ(2đđĽ) = 24 đ(đđĽ). Since đ ⤠đ â 1, we have đ(đđĽ) = đ4 đ(đĽ), and thus đ(đđĽ) = đ4 đ(đĽ). Let đ = 2đ+1. Then, by substituting đĽ, đŚ by đđĽ, đĽ in (5), respectively, we have đ (2đđĽ) = 2 (7đ â 9) (đ â 1) đ (đđĽ) + 2đ2 (đ2 â 1) đ (đĽ) â (đ â 1)2 đ (2đđĽ) (8) + đ2 {đ ((đ + 1) đĽ) + đ ((đ â 1) đĽ)} . Since đ = 2đ + 1, we have đ + 1 = 2(đ + 1) and đ = 2đ. Replacing these equalities in (8) and using (6), we get đ(đđĽ) = đ4 đ(đĽ). This completes the proof.
(12)
+ đ2 {đ (đĽ + đŚ) + đ (đĽ â đŚ)} for which đđ = âđđâ2 + 28đ2 â 120đ + 156, đ2 = 10,
đ3 = 48,
đđ = 2đđâ1 â đđâ2 + 24(đ â 1)2 , đ2 = 24,
(13)
đ3 = 144.
Solving the above recurrence equations is routine, and so we get đđ = 2 (7đ â 9) (đ â 1) ,
đđ = 2đ2 (đ2 â 1)
for all đĽ, đŚ â X and each positive integer đ ⼠2.
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Conversely, assume that đ : X â Y satisfies the functional equation for each đ ⼠đ, in particular, for đ = đ(đ â 1). Hence for each đĽ, đŚ â X, we have
2
= 2 (7đ â 9) (đ â 1) đ (đĽ) + 2đ (đ â 1) đ ((đ â 1) đŚ) â (đ â 1)2 đ (2đĽ) + đ2 {đ (đĽ + (đ â 1) đŚ)
Thus, if đ satisfies the functional equation (5) for all đ ⼠3, then it satisfies (5) for đ â 1. In particular, đ satisfies (2).
3. Hyers-Ulam Stability of (5)
+đ (đĽ â (đ â 1) đŚ)} . (15) By Lemma 1, we have đ((đ â 1)đŚ) = (đ â 1)4 đ(đŚ) and so đ (đĽ + đ (đ â 1) đŚ) + đ (đĽ â đ (đ â 1) đŚ) = 2 (7đ â 9) (đ â 1) đ (đĽ) + 2đ2 (đ2 â 1) (đ â 1)4 đ (đŚ) 2
+ (đ â 1)2 {đ (đĽ + đŚ) + đ (đĽ â đŚ)} . (19)
đ (đĽ + đ (đ â 1) đŚ) + đ (đĽ â đ (đ â 1) đŚ) 2
+ 2(đ â 1)2 ((đ â 1)2 â 1) đ (đŚ) â (đ â 2)2 đ (2đĽ)
2
â (đ â 1) đ (2đĽ) + đ {đ (đĽ + (đ â 1) đŚ)
Let đ be an integer with đ ⼠2. We use the abbreviation for the given mapping đ : X â Y as follows: Dđ đ (đĽ, đŚ) := đ (đĽ + đđŚ) + đ (đĽ â đđŚ) â 2 (7đ â 9) (đ â 1) đ (đĽ) â 2đ2 (đ2 â 1) đ (đŚ) + (đ â 1)2 đ (2đĽ) â đ2 {đ (đĽ + đŚ) + đ (đĽ â đŚ)} (đĽ, đŚ â X) . (20)
+đ (đĽ â (đ â 1) đŚ)} . (16) On the other hand, đ (đĽ + (đ2 â đ) đŚ) + đ (đĽ â (đ2 â đ) đŚ)
Theorem 4. Let đź be a real number and let đ : X â Y be a mapping for which there exists a function đ : X Ă X â [âđź, â) such that
= 2 (7 (đ2 â đ) â 9) ((đ2 â đ) â 1) đ (đĽ) 2
2
+ 2(đ2 â đ) ((đ2 â đ) â 1) đ (đŚ)
(17)
2
2
â ((đ â đ) â 1) đ (2đĽ) + (đ â đ) {đ (đĽ + đŚ) + đ (đĽ â đŚ)} . Using (16) and (17), we get đ2 {đ (đĽ + (đ â 1) đŚ) + đ (đĽ â (đ â 1) đŚ)} = 2 [(7 (đ2 â đ) â 9) ((đ2 â đ) â 1) â (7đ â 9) (đ â 1) ] đ (đĽ) 2
2
2
âđ2 (đ2 â 1) (đ â 1)4 ] đ (đŚ) 2
â [((đ2 â đ) â 1) â (đ â 1)2 ] đ (2đĽ) 2
2
+ đ (đ â 1) {đ (đĽ + đŚ) + đ (đĽ â đŚ)} . A calculation shows that đ (đĽ + (đ â 1) đŚ) + đ (đĽ â (đ â 1) đŚ) = 2 (7đ â 16) (đ â 2) đ (đĽ)
(21)
for all đĽ â X.
2
+ 2 [(đ â đ) ((đ â đ) â 1)
â 1 đĚ (đĽ, đŚ) := â 4đ đ (2đ đĽ, 2đ đŚ) < â, đ=0 2
óľŠ óľŠóľŠ óľŠóľŠDđ đ (đĽ, đŚ)óľŠóľŠóľŠY ⤠đź + đ (đĽ, đŚ) (22) for all đĽ, đŚ â X, where đ is an integer with đ ⼠2. Then there exists a unique quartic mapping T : X â Y such that 2 óľŠóľŠ óľŠóľŠóľŠ óľŠóľŠđ (đĽ) â T (đĽ) â 2đ (đ + 1) đ (0)óľŠóľŠóľŠ óľŠóľŠ óľŠóľŠ 15 (đ â 1) óľŠóľŠY óľŠóľŠ (23) đĚ (đĽ, 0) đź ⤠+ 15(đ â 1)2 16(đ â 1)2
2
2
Throughout this section, we assume that X is a normed real linear space with norm â â
âX and Y is a real Banach space with norm â â
âY . We are going to prove the stability of the quartic functional equation (5).
(18)
Proof. Putting đŚ = 0 in (22), we have óľŠóľŠ óľŠóľŠ 2 2 2 2 óľŠóľŠóľŠ(đ â 1) đ (2đĽ) â 16(đ â 1) đ (đĽ) â 2đ (đ â 1) đ (0)óľŠóľŠóľŠY ⤠đź + đ (đĽ, 0) (24) for all đĽ â X. Thus óľŠóľŠ óľŠóľŠ đ2 (đ + 1) óľŠóľŠ 1 óľŠóľŠ óľŠóľŠ đ (2đĽ) â đ (đĽ) â óľŠóľŠ đ (0) óľŠóľŠ 8 (đ â 1) óľŠóľŠóľŠ 16 óľŠY đ (đĽ, 0) đź ⤠+ 2 16(đ â 1) 16(đ â 1)2
(25)
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for all đĽ â X. Replacing đĽ by 2đĽ in (25) and continuing this method, we get óľŠóľŠ óľŠóľŠ óľŠóľŠ óľŠóľŠ đ (2đ đĽ) đ2 (đ + 1) đâ1 1 óľŠóľŠ óľŠóľŠ â đ â đ â (đĽ) (0) óľŠóľŠ óľŠóľŠ 24đ 4đ 8 (đ â 1) đ=0 2 óľŠóľŠY óľŠóľŠ đ
â¤
đź 1 đâ1 1 đâ1 đ (2 đĽ, 0) + . â 4đ â 16 đ=0 2 (đ â 1)2 16 đ=0 24đ (đ â 1)2
(26)
=
đź 1 â 1 + đ (2đ+đ đĽ, 0) â 15(đ â 1)2 24đ 16 đ=0 24(đ+đ)
=
1 â 1 đź + đ (2đ đĽ, 0) â 15(đ â 1)2 24đ 16 đ=0 24đ
for all đĽ â X. Taking đ â â in the preceding inequality, we immediately find the uniqueness of T. This completes the proof.
óľŠóľŠ óľŠ đâ1 2 óľŠóľŠ đ (2đ đĽ) đ (2đ đĽ) óľŠóľŠóľŠ óľŠóľŠ óľŠóľŠ ⤠đ (đ + 1) â 1 óľŠóľŠóľŠđ (0)óľŠóľŠóľŠ óľŠóľŠ 4đ â óľŠ óľŠY 8 (đ â 1) đ=đ 24đ óľŠ óľŠóľŠ 2 24đ óľŠóľŠóľŠ óľŠ óľŠY đź 1 đâ1 â 4đ 16 đ=đ 2 (đ â 1)2
đĚ (đĽ) đź 1 [ + ] 2 4đ 2 15(đ â 1) 16(đ â 1)2
(30)
On the other hand, we can use induction to obtain
+
â¤
Corollary 5. Let đź, đ˝, đž, đ, and đ be nonnegative real numbers such that đ > 0 and đ, đ < 4. Suppose that đ : X â Y is a mapping fulfilling (27)
óľŠ óľŠ óľŠđ óľŠóľŠ đ óľŠóľŠDđ đ (đĽ, đŚ)óľŠóľŠóľŠY ⤠đź + đ˝âđĽâX + đžóľŠóľŠóľŠđŚóľŠóľŠóľŠX
(31)
for all đĽ, đŚ â X, where đ is an integer with đ ⼠2. Then there exists a unique quartic mapping T : X â Y such that
đ 1 đâ1 đ (2 đĽ, 0) + â 4đ 16 đ=đ 2 (đ â 1)2
for all đĽ â X, and đ > đ ⼠0. Thus the sequence {đ(2đ đĽ)/24đ } is Cauchy by (21) and (27). Completeness of Y allows us to assume that there exists a map T so that
2đź đź óľŠ óľŠóľŠ â óľŠóľŠđ (đĽ) â T (đĽ)óľŠóľŠóľŠY ⤠2 4 2 2đ + 13đ â 30đ + 15 15(đ â 1) +
đ˝ âđĽâđX (đ â 1) (24 â 2đ ) 2
(32) đ
lim
đââ
đ (2 đĽ) = T (đĽ) . 24đ
(28)
Taking the limit as đ â â in (26) and applying (28), we can see that inequality (23) holds. Now, we replace đĽ, đŚ by 2đ đĽ, 2đ đŚ, respectively, in (22); then đ (2đ đĽ, 2đ đŚ) 1 óľŠóľŠ 1 đ đ óľŠ . óľŠóľŠDđ đ (2 đĽ, 2 đŚ)óľŠóľŠóľŠY ⤠4đ đź + 4đ 2 2 24đ
(29)
for all đĽ â X and all đĽ â X \ {0} if đ < 0. Proof. Setting đ(đĽ, đŚ) = đ˝âđĽâđX + đžâđŚâđ X in Theorem 12, we have óľŠóľŠ óľŠóľŠ 2đ2 (đ + 1) óľŠóľŠ óľŠóľŠ óľŠóľŠđ (đĽ) â T (đĽ) â óľŠóľŠ đ (0) óľŠóľŠ óľŠóľŠ 15 â 1) (đ óľŠ óľŠY (33) đ˝ đź đ ⤠+ âđĽâX . 15(đ â 1)2 (đ â 1)2 (24 â 2đ ) It follows from (31) that
Letting the limit as đ â â, we obtain Dđ T(đĽ, đŚ) = 0 for all positive integers đ ⼠2 and all đĽ, đŚ â X. Hence, by Theorem 3, it indicates that T : X â Y is a quartic mapping. Now, let Tó¸ : X â Y be another quartic mapping satisfying (23). Then we have
By these statements we can get the result.
óľŠ óľŠóľŠ óľŠóľŠT (đĽ) â Tó¸ (đĽ)óľŠóľŠóľŠ óľŠY óľŠ
We have the following result which is analogous to Theorem 12 for the quartic functional equation (5). We include its proof.
1 óľŠóľŠ óľŠ óľŠóľŠT (2đ đĽ) â Tó¸ (2đ đĽ)óľŠóľŠóľŠ óľŠY 24đ óľŠ óľŠóľŠ óľŠ 1 óľŠóľŠ 2đ2 (đ + 1) óľŠ đ (0)óľŠóľŠóľŠóľŠ ⤠4đ (óľŠóľŠóľŠóľŠT (2đ đĽ) â đ (2đ đĽ) + 2 15 (đ â 1) óľŠóľŠY óľŠóľŠ =
óľŠóľŠ óľŠóľŠ 2đ2 (đ + 1) óľŠ óľŠ đ (0)óľŠóľŠóľŠóľŠ ) +óľŠóľŠóľŠóľŠđ (2đ đĽ) â Tó¸ (2đ đĽ) â 15 (đ â 1) óľŠóľŠ óľŠóľŠY
đź óľŠ óľŠóľŠ . óľŠóľŠđ (0)óľŠóľŠóľŠY ⤠2đ4 + 13đ2 â 30đ + 15
(34)
Theorem 6. Suppose that đ : X â Y is a mapping for which there exists a function đ : X Ă X â [0, â) such that â đĽ đŚ đĚ (đĽ, đŚ) := â 24đ đ ( đ , đ ) < â, 2 2 đ=1
(35)
óľŠ óľŠóľŠ óľŠóľŠDđ đ (đĽ, đŚ)óľŠóľŠóľŠY ⤠đ (đĽ, đŚ)
(36)
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for all đĽ, đŚ â X, where đ is an integer with đ ⼠2. Then there exists a unique quartic mapping T : X â Y such that đĚ (đĽ, 0) óľŠ óľŠóľŠ óľŠóľŠđ (đĽ) â T (đĽ)óľŠóľŠóľŠY ⤠(đ â 1)2
(37)
for all đĽ â X. Proof. It follows from (35) that đ(0, 0) = 0. Thus from (36) we have đ(0) = 0. Putting đŚ = 0 in (36), we get óľŠ óľŠóľŠ óľŠóľŠ(đ â 1)2 đ (2đĽ) â 16(đ â 1)2 đ (đĽ)óľŠóľŠóľŠ ⤠đ (đĽ, 0) óľŠY óľŠ
(38)
for all đĽ â X. If we replace đĽ by đĽ/2 in the above inequality and divide both sides by (đ â 1)2 , we have óľŠóľŠ óľŠ óľŠóľŠđ (đĽ) â 16đ ( đĽ )óľŠóľŠóľŠ ⤠đ (đĽ/2, 0) . óľŠóľŠ óľŠ 2 óľŠóľŠY (đ â 1)2 óľŠ
(39)
Using triangular inequality and proceeding this way, we obtain đ óľŠ óľŠóľŠ 1 đĽ óľŠóľŠđ (đĽ) â 24đ đ ( đĽ )óľŠóľŠóľŠ ⤠24đ đ ( đ , 0) â óľŠ óľŠóľŠ 2 đ óľŠ 2 óľŠY (đ â 1) đ=1 óľŠ 2 4đ
(40) đ
for all đĽ â X. If we show that the sequence {2 đ(đĽ/2 )} is Cauchy, then it will be convergent by the completeness of Y. For this, if we replace đĽ by đĽ/2đ in (40) and then multiply both sides by 24đ , then we get óľŠóľŠ 4(đ+đ) đĽ đĽ óľŠóľŠ óľŠóľŠ2 đ ( đ ) â 24đ đ ( đ )óľŠóľŠóľŠóľŠ óľŠóľŠ 2 óľŠ 2 óľŠY â¤
đ 1 đĽ â 24(đ+đ) đ ( đ+đ , 0) 2 2 â 1) (đ đ=1
=
đ+đ 1 đĽ 24đ đ ( đ , 0) â 2 2 (đ â 1) đ=đ+1
đââ
đĽ ). 2đ
(41)
(42)
Corollary 7. Let đ˝, đž, đ, and đ be nonnegative real numbers such that đ, đ > 4. Suppose that đ : X â Y is a mapping fulfilling (43)
for all đĽ, đŚ â X, where đ is an integer with đ ⼠2. Then there exists a unique quartic mapping T : X â Y such that óľŠ óľŠóľŠ óľŠóľŠđ (đĽ) â T (đĽ)óľŠóľŠóľŠY ⤠for all đĽ â X.
đ˝ âđĽâđX (đ â 1) (2đ â 24 ) 2
Theorem 8 (the fixed point alternative theorem). Let (Î, đ) be a complete generalized metric space and let J : Î â Î be a mapping with Lipschitz constant đż < 1. Then, for each element đź â Î, either đ(Jđ đź, Jđ+1 đź) = â for all đ ⼠0 or there exists a natural number đ0 such that (i) đ(Jđ đź, Jđ+1 đź) < â for all đ ⼠đ0 ;
(ii) the sequence {Jđ đź} is convergent to a fixed point đ˝â of J; (iii) đ˝â is the unique fixed point of J in the set Î 1 = {đ˝ â Î : đ(Tđ0 đź, đ˝) < â}; (iv) đ(đ˝, đ˝â ) ⤠(1/(1 â đż))đ(đ˝, Jđ˝) for all đ˝ â Î 1 . Theorem 9. Let đ : X â Y be a mapping with đ(0) = 0 and let đ : X Ă X â [0, â) be a function such that óľŠóľŠ óľŠ óľŠóľŠDđ đ (đĽ, đŚ)óľŠóľŠóľŠY ⤠đ (đĽ, đŚ)
(44)
(45)
for all đĽ, đŚ â X, where đ is an integer with đ ⼠2. If there exists a constant đ â (0, 1), such that (46)
for all đĽ, đŚ â X, then there exists a unique quartic mapping T : X â Y such that óľŠ óľŠóľŠ óľŠóľŠđ (đĽ) â T (đĽ)óľŠóľŠóľŠY â¤
Now, in a similar way to the proof of Theorem 12, we can complete the rest of the proof.
óľŠ óľŠ óľŠđ óľŠóľŠ đ óľŠóľŠDđ đ (đĽ, đŚ)óľŠóľŠóľŠY ⤠đ˝âđĽâX + đžóľŠóľŠóľŠđŚóľŠóľŠóľŠX
We are going to investigate the hyperstability of the given quartic functional equation (5) by using the fixed point method. First, we bring the next theorem which was proved in [22]. This result plays a fundamental role to achieve our goal.
đ (2đĽ, 2đŚ) ⤠16đđ (đĽ, đŚ)
for all đĽ â X, and đ > đ > 0. Thus the mentioned sequence is convergent to the mapping T; that is, T (đĽ) := lim 24đ đ (
Proof. First, we note that if we put đĽ = đŚ = 0 in (43), we have đ(0) = 0. Taking đ(đĽ, đŚ) = đ˝âđĽâđX + đžâđŚâđ X in Theorem 14, we can obtain the desired result.
1 đ (đĽ, 0) 16(đ â 1)2 (1 â đ)
(47)
for all đĽ â X. Proof. We wish to make the conditions of Theorem 8. We consider the set Î = {đ : X ół¨â Y | đ (0) = 0}
(48)
and define the mapping D on Î Ă Î as follows: óľŠ óľŠ D (đ, â) := inf {đś â (0, â) : óľŠóľŠóľŠđ (đĽ) â â (đĽ)óľŠóľŠóľŠY ⤠đśđ (đĽ, 0) , (âđĽ â X) } , (49) if there exists such constant đś, and D(đ, â) = â, otherwise. In a similar way to the proof of [15, Theorem 2.2], we can show that D is a generalized metric on Î and the metric space (Î, D) is complete. Here, we define the mapping J : Î â Î by Jâ (đĽ) =
1 â (2đĽ) , 16
(đĽ â X) .
(50)
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If đ, â â Î such that D(đ, â) < đś, by definitions of D and J, we have 1 1 óľŠóľŠ óľŠóľŠóľŠ 1 óľŠóľŠ đ (2đĽ) â â (2đĽ)óľŠóľŠóľŠ ⤠đśđ (2đĽ, 0) (51) óľŠ óľŠóľŠ 16 16 óľŠY 16 for all đĽ â X. Using (46), we get óľŠóľŠ 1 óľŠ óľŠóľŠ đ (2đĽ) â 1 â (2đĽ)óľŠóľŠóľŠ ⤠đśđđ (đĽ, 0) óľŠóľŠ óľŠóľŠ 16 óľŠ 16 óľŠY
(52)
for all đĽ â X. The above inequality shows that D(Jđ, Jâ) ⤠đD(đ, â) for all đ, â â Î. Hence, J is a strictly contractive mapping on Î with a Lipschitz constant đ. We now show that D(Jđ, đ) < â. Putting đŚ = 0 in (45), we obtain óľŠ óľŠóľŠ 1 óľŠóľŠ đ (2đĽ) â đ (đĽ)óľŠóľŠóľŠ ⤠đ (đĽ, 0) óľŠóľŠ óľŠóľŠ óľŠY 16(đ â 1)2 óľŠ 16
(53)
for all đĽ â X. We conclude from the last inequality that D (Jđ, đ) â¤
1 . 16(đ â 1)2
(54)
Theorem 8 shows that D(Jđ đ, Jđ+1 đ) < â for all đ ⼠0, and thus in this theorem we have đ0 = 0. Consequently, the parts (iii) and (iv) of Theorem 8 hold on the whole Î. Hence there exists a unique mapping T : X â Y such that T is a fixed point of J and that Jđ đ â T as đ â â. Thus đ (2đ đĽ) = T (đĽ) đ â â 24đ
(55)
lim
for all đĽ â X, and so đ (đ, T) â¤
1 1 . (56) đ (Jđ, đ) ⤠1âđ 16(đ â 1)2 (1 â đ)
The above inequalities show that (47) is true for all đĽ â X. Now, it follows from (46) that đ (2đ đĽ, 2đ đĽ) = 0. đââ 24đ
(57)
lim
đ
đ
Substituting đĽ and đŚ by 2 đĽ and 2 đŚ, respectively, in (45), we get đ
đ
đ (2 đĽ, 2 đŚ) 1 óľŠóľŠ đ đ óľŠ . óľŠóľŠDđ đ (2 đĽ, 2 đŚ)óľŠóľŠóľŠY ⤠4đ 2 24đ
(58)
Taking the limit as đ â â, we obtain Dđ T(đĽ, đŚ) = 0 for all integers đ ⼠2 and all đĽ, đŚ â X. It follows from Theorem 3 that T : X â Y is a quartic mapping which is unique. Corollary 10. Let đź, đ˝, and đ be nonnegative real numbers with đ, đ < 4 and let đ : X â Y be a mapping such that óľŠóľŠ óľŠ óľŠ óľŠđ đ óľŠóľŠDđ đ (đĽ, đŚ)óľŠóľŠóľŠY ⤠đźâđĽâX + đ˝óľŠóľŠóľŠđŚóľŠóľŠóľŠX (59) for all đĽ, đŚ â X. Then there exists a unique quartic mapping T : X â Y satisfying đź óľŠ óľŠóľŠ âđĽâđX óľŠóľŠđ (đĽ) â T (đĽ)óľŠóľŠóľŠY ⤠16(đ â 1)2 (24 â 2đ ) for all đĽ â X.
(60)
Proof. Note that inequality (59) implies that đ(0) = 0. If we put đ(đĽ, đŚ) = đźâđĽâđX + đ˝âđŚâđX in Theorem 9, we obtain the desired result. In the next result, we prove the hyperstability of quartic functional equations under some conditions. Corollary 11. Let đ, đ , and đź be nonnegative real numbers with 0 < đ + đ ≠ 4 and let đ : X â Y be a mapping such that óľŠ óľŠóľŠ đ óľŠ óľŠđ óľŠóľŠDđ đ (đĽ, đŚ)óľŠóľŠóľŠY ⤠đźâđĽâX óľŠóľŠóľŠđŚóľŠóľŠóľŠX
(61)
for all đĽ, đŚ â X. Then đ is a quartic mapping on X. Proof. Putting đĽ = đŚ = 0 in (61), we get đ(0) = 0. Again, if we put đŚ = 0 in (61), then we have đ(2đĽ) = 16đ(đĽ) for all đĽ â X. It is easy to check that đ(2đ đĽ) = 24đ đ(đĽ), and so đ(đĽ) = đ(2đ đĽ)/24đ for all đĽ â X and đ â N. Now, it follows from Theorem 9 that đ is a quartic mapping when đ(đĽ, đŚ) = đźâđĽâđX âđŚâđ X .
4. Stability of (5) in Non-Archimedean Spaces We recall some basic facts concerning non-Archimedean spaces and some preliminary results. By a non-Archimedean field we mean a field K equipped with a function (valuation) | â
| from K into [0, â) such that |đ| = 0 if and only if đ = 0, |đđ | = |đ||đ |, and |đ + đ | ⤠max{|đ|, |đ |} for all đ, đ â K. Clearly |1| = | â 1| = 1 and |đ| ⤠1 for all đ â N. Let X be a vector space over a scalar field K with a nonArchimedean nontrivial valuation | â
|. A function â â
â : X â R is a non-Archimedean norm (valuation) if it satisfies the following conditions: (i) âđĽâ = 0 if and only if đĽ = 0; (ii) âđđĽâ = |đ|âđĽâ, (đĽ â X, đ â K); (iii) the strong triangle inequality (ultrametric); namely, óľŠ óľŠ óľŠ óľŠóľŠ óľŠóľŠđĽ + đŚóľŠóľŠóľŠ ⤠max {âđĽâ , óľŠóľŠóľŠđŚóľŠóľŠóľŠ} ,
(đĽ, đŚ â X) .
(62)
Then (X, â â
â) is called a non-Archimedean space. Due to the fact that óľŠ óľŠ óľŠóľŠ óľŠ óľŠóľŠđĽđ â đĽđ óľŠóľŠóľŠ ⤠max {óľŠóľŠóľŠóľŠđĽđ+1 â đĽđ óľŠóľŠóľŠóľŠ ; đ ⤠đ ⤠đ â 1} , (đ ⼠đ) (63) a sequence {đĽđ } is Cauchy if and only if {đĽđ+1 â đĽđ } converges to zero in a non-Archimedean normed space X. By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent. In [23], Hensel discovered the đ-adic numbers as a number theoretical analogue of power series in complex analysis. The most interesting example of non-Archimedean spaces is đ-adic numbers. A key property of đ-adic numbers is that they do not satisfy the Archimedean axiom: for all đĽ, đŚ > 0, there exists an integer đ such that đĽ < đđŚ. Let đ be a prime number. For any nonzero rational number đĽ = đđ (đ/đ) in which đ and đ are coprime to the prime number đ. Consider the đ-adic absolute value
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|đĽ|đ = đđ on Q. It is easy to check that | â
| is a nonArchimedean norm on Q. The completion of Q with respect to | â
| which is denoted by Qđ is said to be the đ-adic number field. One should remember that if đ > 3, then |2đ | = 1 for all integers đ. In [24], the stability of some functional equations in non-Archimedean normed spaces is investigated (see also [25]). Here and subsequently, we assume that X is a normed space and Y is a complete non-Archimedean space unless otherwise stated explicitly. In the upcoming theorem, we prove the stability of the functional equation (5). Theorem 12. Let đ : X Ă X â [0, â) such that lim
1
đ â â |16|đ
đ (2đ đĽ, 2đ đŚ) = 0
(64)
for all đĽ, đŚ â X. Suppose that đ : X â Y is a mapping satisfying the equality óľŠ óľŠóľŠ óľŠóľŠDđ đ (đĽ, đŚ)óľŠóľŠóľŠ ⤠đ (đĽ, đŚ)
(65)
for all đĽ, đŚ â X, where đ is an integer with đ ⼠2. Then there exists a unique quartic mapping đ : X â Y such that 1 óľŠ óľŠóľŠ óľŠóľŠđ (đĽ) â đ (đĽ)óľŠóľŠóľŠ ⤠óľ¨óľ¨ óľ¨ đĚ (đĽ) óľ¨óľ¨16(đ â 1)2 óľ¨óľ¨óľ¨ óľ¨ óľ¨
(66)
For each đĽ â X and non-negative integers đ, we have óľŠóľŠ đ (2đ đĽ) óľŠóľŠóľŠóľŠ óľŠóľŠ óľŠóľŠ óľŠóľŠđ (đĽ) â óľŠóľŠ 16đ óľŠóľŠóľŠ óľŠ óľŠóľŠđâ1 óľŠ óľŠóľŠ đ (2đ đĽ) đ (2đ+1 đĽ) óľŠóľŠóľŠóľŠ óľŠóľŠ = óľŠóľŠóľŠóľŠ â ( â ) óľŠóľŠ 16đ 16đ+1 óľŠóľŠđ=0 óľŠóľŠ óľŠ óľŠóľŠ óľŠ óľŠóľŠ đ (2đ đĽ) đ (2đ+1 đĽ) óľŠóľŠóľŠ óľŠ óľŠóľŠ : 0 ⤠đ < đ} óľŠ â¤ max {óľŠóľŠ â óľŠ 16đ+1 óľŠóľŠóľŠ óľŠóľŠ 16đ óľŠ óľŠ đ (2đ đĽ, 0) 1 ⤠óľ¨óľ¨ max { : 0 ⤠đ < đ} . óľ¨ đ óľ¨óľ¨16(đ â 1)2 óľ¨óľ¨óľ¨ |16| óľ¨ óľ¨ Taking đ â â in (71) and applying (70), we can see that the inequality (66) holds when đ ⼠2. It follows from (64), (65), and (70) that, for all đĽ, đŚ â X, 1 óľŠóľŠ óľŠóľŠ óľŠ đ đ óľŠ óľŠóľŠDđ đ (đĽ, đŚ)óľŠóľŠóľŠ = đlim óľŠD đ (2 đĽ, 2 đŚ)óľŠóľŠóľŠ â â |16|đ óľŠ đ 1 ⤠lim đ (2đ đĽ, 2đ đŚ) = 0. đ â â |16|đ
óľŠ óľŠóľŠ óľŠóľŠđ (đĽ) â đó¸ (đĽ)óľŠóľŠóľŠ óľŠ óľŠ = lim
Proof. Putting đŚ = 0 in (65), we get
⤠lim
for all đĽ â X. Thus we have óľŠ óľŠóľŠ óľŠóľŠđ (2đĽ) â 16đ (đĽ)óľŠóľŠóľŠ â¤
1 đ (đĽ, 0) |đ â 1|2
(68)
for all đĽ â X. Replacing đĽ by 2đ đĽ in (68) and then dividing both sides by |16|đ+1 , we have óľŠ óľŠóľŠ 1 1 1 đ+1 đ óľŠ óľŠ óľŠóľŠ đ (2đ đĽ, 0) óľŠóľŠ đ+1 đ (2 đĽ) â đ đ (2 đĽ)óľŠóľŠóľŠ ⤠16 óľŠ |đ â 1|2 |16|đ+1 óľŠ 16 (69) for all đĽ â X and all nonnegative integers đ. Thus the sequence {đ(2đ đĽ)/16đ } is Cauchy by (64) and (69). Due to the completeness of Y as a non-Archimedean space, there exists a mapping đ so that đ (2đ đĽ) = đ (đĽ) . đ â â 16đ lim
1
đ â â |16|đ
1
đ â â |16|đ
(67)
(70)
(72)
Hence, the mapping đ satisfies (5). Now, let đó¸ : X â Y be another quartic mapping satisfying (66). Then we have
Ě = sup{đ(2đ đĽ, 0)/|16|đ : đ â N ⪠{0}}. for all đĽ â X where đ(đĽ)
óľŠóľŠ óľŠ óľŠóľŠ(đ â 1)2 đ (2đĽ) â 16(đ â 1)2 đ (đĽ)óľŠóľŠóľŠ ⤠đ (đĽ, 0) óľŠ óľŠ
(71)
óľŠ óľŠóľŠ óľŠóľŠđ (2đ đĽ) â đó¸ (2đ đĽ)óľŠóľŠóľŠ óľŠ óľŠ óľŠ óľŠ óľŠ óľŠ max {óľŠóľŠóľŠóľŠđ (2đ đĽ) â đ (2đ đĽ)óľŠóľŠóľŠóľŠ , óľŠóľŠóľŠóľŠđ (2đ đĽ) â đó¸ (2đ đĽ)óľŠóľŠóľŠóľŠ}
đ (2đ đĽ, 0) 1 ⤠óľ¨óľ¨ lim lim max { : đ ⤠đ < đ + đ} óľ¨ óľ¨óľ¨16(đ â 1)2 óľ¨óľ¨óľ¨ đ â â đ â â |16|đ óľ¨ óľ¨ đ (2đ đĽ, 0) 1 lim sup { : đ ⤠đ < â} = 0 = óľ¨óľ¨ óľ¨ óľ¨óľ¨16(đ â 1)2 óľ¨óľ¨óľ¨ đ â â |16|đ óľ¨ óľ¨
(73)
for all đĽ â X. This shows the uniqueness of đ. Corollary 13. Let đź > 0, X be a non-Archimedean space and let Î : [0, â) â [0, â) be a function satisfying Î(|đ|đ ) ⤠Î(|đ|)Î(đ ) for all đ, đ â [0, â) for which Î(|2|) < |16|. Suppose that đ : X â Y is a mapping satisfying the inequality óľŠ óľŠ óľŠ óľŠóľŠ óľŠóľŠDđ đ (đĽ, đŚ)óľŠóľŠóľŠ ⤠đź (Î (âđĽâ) + Î (óľŠóľŠóľŠđŚóľŠóľŠóľŠ))
(74)
for all đĽ, đŚ â X, where đ is an integer with đ ⼠2. Then there exists a unique quartic mapping đ : X â Y such that đźÎ (âđĽâ) óľŠ óľŠóľŠ óľŠóľŠđ (đĽ) â đ (đĽ)óľŠóľŠóľŠ ⤠óľ¨óľ¨ óľ¨ óľ¨óľ¨16(đ â 1)2 óľ¨óľ¨óľ¨ óľ¨ óľ¨ for all đĽ â X.
(75)
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Proof. Defining đ : XĂX â [0, â) by đ(đĽ, đŚ) = đź(Î(âđĽâ)+ Î(âđŚâ)), we have 1 Î (|2|) đ đ đ đ đĽ, 2 đŚ) ⤠lim (2 ( ) đ (đĽ, đŚ) = 0 (76) đ â â |16|đ đââ |16| lim
for all đĽ, đŚ â X. We also have đĚ (đĽ) = sup {
đ (2đ đĽ, 0) |16|đ
: 0 ⤠đ < â}
(77)
= đ (đĽ, 0) = đź (Î (âđĽâ))
for all đĽ â X and non-negative integers đ. Since the right hand side of inequality (84) goes to 0 as đ â â, by applying (83), we deduce inequality (80). Now, in a similar way to the proof of Theorem 12, we can complete the rest of the proof. Corollary 15. Let đź > 0, X be a non-Archimedean space and let Î : [0, â) â [0, â) be a function satisfying Î(|đ|đ ) ⤠Î(|đ|)Î(đ ) for all đ, đ â [0, â) for which Î(|2|â1 ) < |16|â1 . Suppose that đ : X â Y is a mapping satisfying the inequality óľŠ óľŠ óľŠ óľŠóľŠ óľŠóľŠDđ đ (đĽ, đŚ)óľŠóľŠóľŠ ⤠đź (Î (âđĽâ) + Î (óľŠóľŠóľŠđŚóľŠóľŠóľŠ))
for all đĽ â X. Now, Theorem 12 implies the desired result. We have the following result which is analogous to Theorem 12 for the functional equation (5).
for all đĽ, đŚ â X, where đ is an integer with đ ⼠2. Then there exists a unique quartic mapping đ : X â Y such that óľŠ đźÎ (âđĽ/2â) óľŠóľŠ óľŠóľŠđ (đĽ) â đ (đĽ)óľŠóľŠóľŠ ⤠|đ â 1|2
Theorem 14. Let đ : X Ă X â [0, â) such that đĽ đŚ lim |16|đ đ ( đ , đ ) = 0 đââ 2 2
(78)
for all đĽ, đŚ â X. Suppose that đ : X â Y is a mapping satisfying the inequality óľŠ óľŠóľŠ óľŠóľŠDđ đ (đĽ, đŚ)óľŠóľŠóľŠY ⤠đ (đĽ, đŚ) (79) for all đĽ, đŚ â X, where đ is an integer with đ ⼠2. Then there exists a unique quartic mapping đ : X â Y such that 1 đĚ (đĽ) â |đ 1|2
óľŠ óľŠóľŠ óľŠóľŠđ (đĽ) â Q (đĽ)óľŠóľŠóľŠ â¤
(80)
Ě for all đĽ â X where đ(đĽ) = sup{|16|đ đ(đĽ/2đ+1 , 0) : đ â N ⪠{0}}. Proof. In a similar way to the proof of Theorem 12, we have óľŠ óľŠóľŠ óľŠóľŠđ (2đĽ) â 16đ (đĽ)óľŠóľŠóľŠ â¤
1 đ (đĽ, 0) |đ â 1|2
(81)
for all đĽ â X. If we replace đĽ by đĽ/2đ+1 in the above inequality and multiply both sides of (81) to |16|đ , we get đ óľŠ óľŠóľŠ đ óľŠóľŠ16 đ ( đĽ ) â 16đ+1 đ ( đĽ )óľŠóľŠóľŠ ⤠|16| đ ( đĽ , 0) óľŠ óľŠóľŠ 2đ 2đ+1 óľŠóľŠ |đ â 1|2 2đ+1 óľŠ (82)
for all đĽ â X and all non-negative integers đ. Thus, we conclude from (78) and (82) that the sequence {2đ đ(đĽ/2đ )} is Cauchy. Since the non-Archimedean space Y is complete, this sequence converges in Y to the mapping đ. Indeed, đ (đĽ) = lim 16đ đ ( đââ
đĽ ), 2đ
(đĽ â X) .
Using induction and (81), one can show that đĽ óľŠóľŠ óľŠóľŠóľŠ óľŠóľŠđ (đĽ) â 16đ đ ( đ )óľŠóľŠóľŠ óľŠóľŠ 2 óľŠóľŠ â¤
1 đĽ max {|16|đ đ ( đ+1 , 0) : 0 ⤠đ < đ} 2 |đ â 1|2
(83)
(84)
(85)
(86)
for all đĽ â X. Proof. The proof is a direct consequence of Theorem 14 and similar to the proof of Corollary 13.
Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.
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