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Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 973709, 14 pages doi:10.1155/2009/973709 Research Article A Functional Inequality in Restricted Domains of Banach Modules M B Moghimi,1 Abbas Najati,1 and Choonkil Park2 Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil 56199–11367, Iran Department of Mathematics, Hanyang University, Seoul 133-791, South Korea Correspondence should be addressed to Choonkil Park, baak@hanyang.ac.kr Received 28 April 2009; Revised August 2009; Accepted 16 August 2009 Recommended by Binggen Zhang We investigate the stability problem for the following functional inequality αf x y /2α βf y z /2β γf z x /2γ ≤ f x y z on restricted domains of Banach modules over a C∗ -algebra As an application we study the asymptotic behavior of a generalized additive mapping Copyright q 2009 M B Moghimi et al 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 Introduction and Preliminaries The following question concerning the stability of group homomorphisms was posed by Ulam : Under what conditions does there exist a group homomorphism near an approximate group homomorphism? Hyers considered the case of approximately additive mappings f : E → E , where E and E are Banach spaces and f satisfies Hyers inequality f x y −f x −f y ≤ε 1.1 for all x, y ∈ E In 1950, Aoki provided a generalization of the Hyers’ theorem for additive mappings and in 1978, Rassias generalized the Hyers’ theorem for linear mappings by allowing the Cauchy difference to be unbounded see also The result of Rassias’ theorem has been generalized by Forti 6, and Gavruta who permitted the Cauchy difference to be bounded by a general control function During the last three decades a number of papers Advances in Difference Equations have been published on the generalized Hyers-Ulam stability to a number of functional equations and mappings see 9–23 We also refer the readers to the books 24–28 Throughout this paper, let A be a unital C∗ -algebra with unitary group U A , unit e, and norm | · | Assume that X is a left A-module and Y is a left Banach A-module An additive mapping T : X → Y is called A-linear if T ax aT x for all a ∈ A and all x ∈ X In this paper, we investigate the stability problem for the following functional inequality: αf x y 2α βf y z 2β γf z x ≤ f x 2γ y 1.2 z on restricted domains of Banach modules over a C∗ -algebra, where α, β, γ are nonzero positive real numbers As an application we study the asymptotic behavior of a generalized additive mapping Solutions of the Functional Inequality 1.2 Theorem 2.1 Let X and M be left A-modules and let α, β, γ be nonzero real numbers If a mapping f : X → M with f 0 satisfies the functional inequality αf ax ay 2α βf ay az 2β γaf z x ≤ f ax 2γ ay az 2.1 for all x, y, z ∈ X and all a ∈ U A , then f is A-linear Proof Letting z −x − y in 2.1 , we get αf ax ay 2α βf − for all x, y ∈ X and all a ∈ U A Letting x αf ay 2α γaf − y 2γ 0, ax 2β γaf − resp., y resp., αf y 2γ 2.2 in 2.2 , we get ax 2α βf − ax 2β 2.3 for all x, y ∈ X and all a ∈ U A Hence f ay −γ/α af −α/γ y and it follows from 2.2 and 2.3 that and f ax ay /2α − f ax/2α − f ay/2α for all x, y ∈ X and all rf x for all x ∈ X a ∈ U A Therefore f x y f x f y for all x, y ∈ X Hence f rx and all rational numbers r Now let a ∈ A a / and let m be an integer number with m > 4|a| Then by Theorem of 29 , there exist elements u1 , u2 , u3 ∈ U A such that 3/m a u1 u2 u3 Since f is Advances in Difference Equations −γ/α rbf −α/γ x for all x ∈ X, all rational numbers r and all additive and f rbx b ∈ U A , we have m f ax m m f u1 x mγ − u1 α f ax α u3 f − x γ u2 u2 x u3 x m f u1 x mγ α − af − x αm γ f u2 x f u3 x γ α − af − x α γ 2.4 for all x ∈ X Replacing −γ/α x instead of x in the above equation, we have γ f − ax α γ − af x α 2.5 for all x ∈ X Since a is an arbitrary nonzero element in A in the previous paragraph, one can replace −α/γ a instead of a in 2.5 Thus we have f ax af x for all x ∈ X and all a ∈ A a / So f : X → Y is A-linear The following theorem is another version of Theorem 2.1 on a restricted domain when α, β, γ > Theorem 2.2 Let X and M be left A-modules and let d, α, β, γ be nonzero positive real numbers Assume that a mapping f : X → M satisfies f 0 and the functional inequality 2.1 for all x, y, z ∈ X with x y z ≥ d and all a ∈ U A Then f is A-linear Proof Letting z −x − y with x ax αf for all a ∈ U A Let δ βx y ≥ d in 2.1 , we get ay 2α βf − ax 2β γaf − max{|β|−1 d, |γ|−1 d} and let x γy ≥ β , γ x y y 2γ 2.6 y ≥ δ Then ≥ β , γ δ ≥ d 2.7 Therefore replacing x and y by 2βx and 2γy in 2.6 , respectively, we get αf for all x, y ∈ X with x βax γay α βf −ax y ≥ δ and all a ∈ U A γaf −y 2.8 Advances in Difference Equations Similar to the proof of Theorem of 30 see also 31 , we prove that f satisfies 2.8 for all x, y ∈ X and all a ∈ U A Suppose x y < δ If x y 0, let z ∈ X with z δ, otherwise ⎧ ⎪ δ ⎪ ⎪ ⎪ ⎪ ⎨ z: x , x x if x ≥ y ; 2.9 ⎪ ⎪ ⎪ ⎪ δ ⎪ ⎩ y , y y if y ≥ x Since α, β, γ > 0, it is easy to verify that β−1 γ z β−1 γy βγ −1 x − z ≥ δ, x β−1 γ z β−1 γ z 2 2βγ −1 z ≥ δ, y ≥ δ, βγ −1 x − β−1 γ z β−1 γy 2.10 2βγ −1 z ≥ δ, z ≥ δ Therefore αf βax γay βf −ax α αf αf αf − αf − αf βax γay α βax γaz α β γ az α βax γaz α β γ az α γaf −y βf − β−1 γ az − β−1 γay βf −ax γay 2βγ −1 z − βγ −1 x γaf −z βf −2 βf −2 γay γaf β−1 γ az β−1 γ az βf − γaf γaf −y 2βγ −1 z − βγ −1 x β−1 γ az − β−1 γay γaf −z 2.11 Hence f satisfies 2.8 and we infer that f satisfies 2.2 for all x, y ∈ X and all a ∈ U A By Theorem 2.1, f is A-linear Advances in Difference Equations Generalized Hyers-Ulam Stability of 1.2 on a Restricted Domain In this section, we investigate the stability problem for A-linear mappings associated to the functional inequality 1.2 on a restricted domain For convenience, we use the following abbreviation for a given function f : X → Y and a ∈ U A : Da f x, y, z : αf ax ay 2α βf ay az 2β z γaf x 3.1 2γ for all x, y, z ∈ X Theorem 3.1 Let d, α, β, γ > 0, p ∈ 0, , and θ, ε ≥ be given Assume that a mapping f : X → Y satisfies the functional inequality ≤ f ax f Da f x, y, z ay az θ ε x p y p z p 3.2 for all x, y, z ∈ X with x y z ≥ d and all a ∈ U A Then there exist a unique A-linear mapping T : X → Y and a constant C > such that f x −T x ≤C 24 × 2p αp−1 ε x − 2p p 3.3 for all x ∈ X −x − y with x Proof Let z ax ay 2α y ≥ d Then 3.2 implies that ax 2β y 2γ βf − γaf − p ≤ f θ ε x ≤ f αf θ 2ε x y p y p x p y p 3.4 Thus αf ax ay α βf − for all x, y ∈ X with x x y ≥ δ Then βx αf βax γay α ax β γaf − y γ ≤ f θ 2p ε x p y p 3.5 y ≥ d and all a ∈ U A Let δ max{β−1 d, γ −1 d} and let γy ≥ d Therefore it follows from 3.5 that βf −ax γaf −y ≤ f θ 2p ε βx p γy p 3.6 Advances in Difference Equations for all x, y ∈ X with x y ≥ δ and all a ∈ U A For the case x y < δ, let z be an element of X which is defined in the proof of Theorem 2.2 It is clear that z ≤ 2δ Using 2.11 and 3.6 , we get αf βax ≤ γay βax αf γay αf αf αf f βax γaz β−1 γ az − β−1 γay βf −ax α β γ az α βax γaz γay γ az α β−1 γ az βf − 4p εδp 2β θ β−1 γ az βf −2 γay γaf 2βγ −1 z − βγ −1 x γaf −z βf −2 α β γaf −y βf − α αf ≤5 βf −ax α γ p γaf −y γaf 2βγ −1 z − βγ −1 x β−1 γ az − β−1 γay 2p β γ p γp γaf −z × 2p ε βx p γy p 3.7 for all x, y ∈ X with x αf βax y < δ and all a ∈ U A Hence γay γaf −y ≤K 4p εδp 2β γ βf −ax α × 2p ε βx p γy p 3.8 for all x, y ∈ X and all a ∈ U A , where K: Letting x and y θ f p 2p β γ p γp 3.9 in 3.8 , respectively, we get αf γay α βf αf βax α βf −ax γaf −y ≤K × 2p ε γy p , 3.10 γaf ≤K × 2p ε βx p for all x, y ∈ X and all a ∈ U A It follows from 3.8 and 3.10 that f x y −f x −f y ≤ α−1 β γ f 3K 12 × 2p ε αx p αy p 3.11 Advances in Difference Equations for all x, y ∈ X By the results of Hyers and Rassias , there exists a unique additive mapping T : X → Y given by T x limn → ∞ 2−n f 2n x such that ≤ α−1 β f x −T x γ 24 × 2p αp−1 ε x − 2p 3K f p 3.12 for all x ∈ X It follows from the definition of T and 3.2 that T 0 and Da T x, y, z ≤ T ax ay az for all x, y, z ∈ X with x y z ≥ d and all a ∈ U A Hence T is A-linear by Theorem 2.2 We apply the result of Theorem 3.1 to study the asymptotic behavior of a generalized additive mapping An asymptotic property of additive mappings has been proved by Skof 32 see also 30, 33 Corollary 3.2 Let α, β, γ be nonzero positive real numbers Assume that a mapping f : X → Y with f 0 satisfies Da f x, y, z − f ax ay −→ az as x y z −→ ∞ 3.13 for all a ∈ U A , then f is A-linear Proof It follows from 3.13 that there exists a sequence {δn }, monotonically decreasing to zero, such that Da f x, y, z − f ax for all x, y, z ∈ X with x y ay az ≤ δn 3.14 z ≥ n and all a ∈ U A Therefore Da f x, y, z ≤ f ax ay δn az 3.15 for all x, y, z ∈ X with x y z ≥ n and all a ∈ U A Applying 3.15 and Theorem 3.1, we obtain a sequence {Tn : X → Y} of unique A-linear mappings satisfying f x − Tn x ≤ 15α−1 δn 3.16 for all x ∈ X Since the sequence {δn } is monotonically decreasing, we conclude f x − Tm x ≤ 15α−1 δm ≤ 15α−1 δn for all x ∈ X and all m ≥ n The uniqueness of Tn implies Tm n → ∞ in 3.16 , we obtain that f is A-linear 3.17 Tn for all m ≥ n Hence letting The following theorem is another version of Theorem 3.1 for the case p > 8 Advances in Difference Equations Theorem 3.3 Let p > 1, d > 0, ε ≥ be given and let α, β, γ be nonzero real numbers Assume that a mapping f : X → Y with f 0 satisfies the functional inequality ≤ f ax Da f x, y, z for all x, y, z ∈ X with x y mapping φ : X → Y such that p ε x p y z p 3.18 ≤ 2p × 2p |α|p−1 ε x 2p − p 3.19 limn → ∞ 2n f 2−n x for all x ∈ X with x ≤ d/8|α| and φ x −x − y in 3.18 , we get Proof Letting z ax ay 2α βf − for all x, y ∈ X with x αf az z ≤ d and all a ∈ U A Then there exists a unique A-linear φ x −f x αf ay ax ay for all x, y ∈ X with x γaf − y 2γ ≤ε x p y p x y x p y 3.20 y ≤ d/2 and all a ∈ U A Hence βf − α ax 2β ax β γaf − y γ ≤ 2p ε x p p y p 3.21 y ≤ d/4 and all a ∈ U A It follows from 3.21 that αf ax α ay αf α βf − ax β y γaf − γ ≤ 2p ε x p , 3.22 ≤2 p ε y p for all x, y ∈ X with x , y ≤ d/4 and all a ∈ U A Adding 3.21 to 3.22 , we get αf ax ay α − αf ay ax − αf α α ≤ 2p ε x p p y x p y 3.23 for all x, y ∈ X with x , y ≤ d/8 and all a ∈ U A Therefore f x y −f x −f y ≤ 2p |α|p−1 ε x p y p x y p for all x, y ∈ X with x , y ≤ d/8|α| Let x ∈ X with x ≤ d/8|α| We may put y 3.24 to obtain f 2x − 2f x ≤ 2p × 2p |α|p−1 ε x p 3.24 x in 3.25 Advances in Difference Equations We can replace x by x/2n in 3.25 for all nonnegative integers n Then using a similar argument given in , we have 2n f 2−n−1 x − 2n f 2−n x ≤ 2p × 2p n |α|p−1 ε x p 3.26 Hence we have the following inequality: 2n f 2−n−1 x − 2m f 2−m x ≤ n 2k f 2−k−1 x − 2k f 2−k x k m ≤ p−1 |α| p n ε k m 2p 3.27 k x p for all x ∈ X with x ≤ d/8|α| and all integers n ≥ m ≥ Since Y is complete, 3.27 shows that the limit T x limn → ∞ 2n f 2−n x exists for all x ∈ X with x ≤ d/8|α| Letting m and n → ∞ in 3.27 , we obtain that T satisfies inequality 3.19 for all x ∈ X with x ≤ d/8|α| It follows from the definition of T and 3.24 that T x for all x, y ∈ X with x , y , x y T x T y 3.28 y ≤ d/8|α| Hence T x T x 3.29 for all x ∈ X with x ≤ d/8|α| We extend the additivity of T to the whole space X by using an extension method of Skof 34 Let δ : d/8|α| and x ∈ X be given with x > δ Let k k x be the smallest integer such that 2k−1 δ < x ≤ 2k δ We define the mapping φ : X → Y by ⎧ ⎪T x , ⎪ ⎪ ⎨ φ x : if x ≤ δ, ⎪ ⎪ ⎪ ⎩ k T 2−k x , 3.30 if x > δ Let x ∈ X be given with x > δ and let k k x be the smallest integer such that 2k−1 δ < x ≤ 2k δ Then k − is the smallest integer satisfying 2k−2 δ < x/2 ≤ 2k−1 δ If k 1, we have φ x/2 T x/2 and φ x 2T x/2 Therefore φ x/2 1/2 φ x For the case k > 1, it follows from the definition of φ that φ x 2k−1 T 2− k−1 x k · T 2−k x φ x 3.31 10 Advances in Difference Equations From the definition of φ and 3.29 , we get that φ x/2 1/2 φ x holds true for all x ∈ X Let x ∈ X and let k be an integer such that x ≤ 2k δ Then φ x 2k φ 2−k x 2k T 2−k x lim 2n k f 2− n k n→∞ lim 2n f 2−n x x n→∞ 3.32 It remains to prove that φ is A-linear Let x, y ∈ X and let n be a positive integer such that 1/2 φ x for all x ∈ X and T satisfies 3.28 , we have x , y , x y ≤ 2n δ Since φ x/2 φ x y 2n φ x y 2n 2n T x y x 2n 2n T 2n x φ n n T y 2n 3.33 y φ n φ x φ y Hence φ is additive Since φ x limn → ∞ 2n f 2−n x for all x ∈ X, we have from 3.22 that αφ ay/α γaφ y/γ for all y ∈ X and all a ∈ U A Letting a e, we get αφ y/α γφ y/γ Therefore φ ay aφ y for all y ∈ X and all a ∈ U A This proves that φ is A-linear Also, φ satisfies inequality 3.19 for all x ∈ X with x ≤ d/8|α|, by the definition of φ For the case p counterexample we use the Gajda’s example 35 to give the following Example 3.4 Let φ : C → C be defined by φ x : ⎧ ⎨x, for |x| < 1, ⎩1, for |x| ≥ 3.34 Consider the function f : C → C by the formula ∞ f x : n φ 2n x 2n 3.35 It is clear that f is continuous, bounded by on C and f x y −f x −f y ≤ |x| y 3.36 for all x, y ∈ C see 35 It follows from 3.36 that the following inequality: f x y z −f x −f y −f z ≤ 12 |x| y |z| 3.37 holds for all x, y, z ∈ C First we show that f λx − λf x ≤2 |λ| |x| 3.38 Advances in Difference Equations 11 for all x, λ ∈ C If f satisfies 3.38 for all |λ| ≥ 1, then f satisfies 3.38 for all λ ∈ C To see this, Then |f λ−1 x −λ−1 f x | ≤ let < |λ| < the result is obvious when λ all x ∈ C Replacing x by λx, we get that |f λx − λf x | ≤ 2|λ|2 |λ|−1 |x| for all x ∈ C Hence we may assume that |λ| ≥ If λx or |λx| ≥ 1, then ≤2 f λx − λf x |λ| ≤ 2|λ| |λ|−1 |x| for 21 |λ| |x| |λ| |x| ≤ |λ| |x| 3.39 Now suppose that < |λx| < Then there exists an integer k ≥ such that 1 ≤ |λx| < k 2k 3.40 2k |x|, 2k |λx| ∈ −1, 3.41 2m |x|, 2m |λx| ∈ −1, 3.42 Therefore Hence for all m 0, 1, , k From the definition of f and 3.40 , we have ∞ f λx − λf x n k ≤ 1 φ 2n λx − λφ 2n x 2n ∞ |λ| n k 2n |λ| 2k 3.43 ≤ 2|λ| |λ| |x| ≤ 2 |λ| |x| Therefore f satisfies 3.38 Now we prove that Dμ f x, y, z − f μx ≤ 16 |α|−1 |α| μy μz β −1 β for all x, y, z ∈ C and all μ ∈ T1 : {λ ∈ C : |λ| Dμ f x, y, z : αf μx μy 2α γ −1 γ |x| y |z| 3.44 1}, where βf μy μz 2β γμf z x 2γ 3.45 12 Advances in Difference Equations It follows from 3.37 and 3.38 that Dμ f x, y, z − f μx ≤ αf μx γμf f ≤ ≤ 16 μy 2α z μy |α|−1 |α| |α| z μf μz f x β −1 β y μy μz 2β βf x μy f μy − μf 2γ |α|−1 μz μx −f x μx μy β z x μz −1 γ −1 2 β μx − f μx μz μz −f μx μy −f y γ μy γ 10 z μz |x| −1 γ |x z| |z| y 3.46 for all x, y, z ∈ C and all μ ∈ T1 Thus f satisfies inequality 3.18 for p a linear functional such that f x −T x ≤ M|x| Let T : C → C be 3.47 for all x ∈ C, where M is a positive constant Then there exists a constant c ∈ C such that T x cx for all rational numbers x So we have f x ≤ M |c| |x| 3.48 for all rational numbers x Let m ∈ N with m > M |c| If x0 ∈ 0, 2−m for all n 0, 1, , m − So f x0 ≥ m−1 n φ 2n x0 2n mx0 > M |c| x0 , ∩ Q, then 2n x0 ∈ 0, 3.49 which contradicts 3.48 Acknowledgments The third author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology NRF-2009-0070788 The authors would like to thank the referees 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Aoki, “On the stability of the linear transformation in Banach spaces,” Journal of the Mathematical Society of Japan, vol 2, pp 64–66, 1950 Th M Rassias, “On the stability of the linear mapping... Hyers-Ulam Stability of 1.2 on a Restricted Domain In this section, we investigate the stability problem for A- linear mappings associated to the functional inequality 1.2 on a restricted domain For... stability of functional equations in several variables,” Aequationes Mathematicae, vol 50, no 1-2, pp 143–190, 1995 P Gavruta, ? ?A generalization of the Hyers-Ulam-Rassias stability of approximately