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Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2008, Article ID 210626, 13 pages doi:10.1155/2008/210626 ResearchArticleOntheStabilityofGeneralizedAdditiveFunctionalInequalitiesinBanach Spaces Jung Rye Lee, 1 Choonkil Park, 2 and Dong Yun Shin 3 1 Department of Mathematics, Daejin University, Kyeonggi 487-711, South Korea 2 Department of Mathematics, Hanyang University, Seoul 133-791, South Korea 3 Department of Mathematics, University of Seoul, Seoul 130-743, South Korea Correspondence should be addressed to Choonkil Park, baak@hanyang.ac.kr Received 18 February 2008; Accepted 2 May 2008 Recommended by Ram Verma We study the following generalizedadditivefunctional inequality afxbfycfz≤ fαx βy γz, associated with linear mappings inBanach spaces. Moreover, we prove the Hyers-Ulam-Rassias stabilityofthe above generalizedadditivefunctional inequality, associated with linear mappings inBanach spaces. Copyright q 2008 Jung Rye Lee 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. 1. Introduction and preliminaries Thestability problem offunctional equations originated from a question of Ulam 1 concerning thestabilityof group homomorphisms. Hyers 2 gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ theorem was generalized by Aoki 3 for additive mappings and by Rassias 4 for linear mappings by considering an unbounded Cauchy difference. A generalization ofthe Rassias theorem was obtained by G ˘ avrut¸a 5 by replacing the unbounded Cauchy difference by a general control function inthe spirit of Rassias’ approach. Rassias 6 during the 27th International Symposium onFunctional Equations asked the question whether such a theorem can also be proved for p ≥ 1. Gajda 7 following the same approach as in Rassias 4 gave an affirmative solution to this question for p>1. It was shown by Gajda 7 as well as by Rassias and ˇ Semrl 8 that one cannot prove Rassias’ theorem when p 1. The counterexamples of Gajda 7 as well as of Rassias and ˇ Semrl 8 have stimulated several mathematicians to create new definitions of approximately additive or approximately linear mappings cf. G ˘ avrut¸a 5,Jung9 who among others studied the Hyers-Ulam stabilityof 2 Journal ofInequalities and Applications functional equations. The paper of Rassias 4 had great influence onthe development of a generalization ofthe Hyers-Ulam stability concept. This new concept is known as Hyers-Ulam- Rassias stabilityoffunctional equations cf. the books of Czerwik 10, Hyers et al. 11.During the last two decades, a number of papers and research monographs have been published on various generalizations and applications ofthe Hyers-Ulam-Rassias stability to a number offunctional equations and mappings see 12–17. Gil ´ anyi 18 showed that if f satisfies thefunctional inequality 2fx2fy − fx − y ≤ fx y , 1.1 then f satisfies the quadratic functional equation 2fx2fyfx yfx − y , 1.2 see also 19. Fechner 20 and Gil ´ anyi 21 proved the Hyers-Ulam-Rassias stabilityofthefunctional inequality 1.1.Parketal.22 investigated the Jordan-von Neumann-type Cauchy- Jensen additive mappings and prove their stability, and Cho and Kim 23 proved the Hyers- Ulam-Rassias stabilityofthe Jordan-von Neumann-type Cauchy-Jensen additive mappings. The purpose of this paper is to investigate thegeneralizedadditivefunctional inequality inBanach spaces and the Hyers-Ulam-Rassias stabilityofgeneralizedadditivefunctionalinequalities associated with linear mappings inBanach spaces. Throughout this paper, we assume that X, Y are Banach spaces and that a, b, c, α, β, γ are nonzero complex numbers. 2. Generalizedadditivefunctionalinequalities Consider a mapping f : X→Y satisfying the following functional inequality: afxbfycfz ≤ fαx βy γz 2.1 for all x, y, z ∈ X. We investigate thegeneralizedadditivefunctional inequality inBanach spaces. We will use that for an additive mapping f,wehavefm/nxm/nf x for any positive integers n, m and all x ∈ X and so frxrfx for any rational number r and all x ∈ X. Theorem 2.1. Let f : X→Y be a nonzero mapping satisfying f00 and 2.1. Then the following hold: a f is additive; b if α/β, β/γ are rational numbers, then a/α b/β c/γ; c if α is a rational number, then |a|≤|α|. Proof. a Letting y −α/βx, z 0in2.1,wegetafxbf− α/βx0. Letting y 0,z −α/γx in 2.1,wegetafxcf−α/γx0. Letting x 0,yα/βx, z −α/γx in 2.1,wegetbfα/βxcf−α/γx0. Jung Rye Lee et al. 3 Thus, we get f−α/βx−fα/βx and so f−x−fx,bfxafβ/αx,and b a f α β x c b f β γ x a c f γ α x fx2.2 for all x ∈ X. Onthe other hand, letting z −αx βy/γ −α/γx β/αy in 2.1,weget afxbfycf − α γ x β α y 0. 2.3 The facts that cf − α γ x β α y c − a c f x β α y −af x β α y 2.4 and bfyafβ/αy give that f x β α y fxf β α y 2.5 and so fx yfxfy for all x, y ∈ X, which implies that f is additive. b Since f is additive by a and since α/β and β/γ are rational numbers, the facts that b/afα/βxfx and c/bfβ/γxfx give that b a · α β fx c b · β γ fxfx2.6 for all x ∈ X. Since f is nonzero, we conclude that a/α b/β c/γ. c Letting y z 0in2.1, since α is a rational number, we get afx ≤ fαx αfx 2.7 for all x ∈ X. Since f is nonzero, we conclude that |a|≤|α|, as desired. As an application of Theorem 2.1, if we consider a mapping f : X→Y satisfying fxfyfz ≤ fx 2y 3z 2.8 for all x, y, z ∈ X, then we conclude that f ≡ 0. Actually, for a mapping f : X→Y satisfying f00and afxbfycfz ≤ fαx βy γz 2.9 for all x, y, z ∈ X,whenα/β , β/γ are rational numbers, the above theorem says that f ≡ 0 unless a/α b/β c/γ. Here, we consider functionalinequalities similar to 2.1. 4 Journal ofInequalities and Applications Remark 2.2. Let f : X→Y be a mapping with f00. If f satisfies afxbfycfz ≤ fαx βy 2.10 for all x, y, z ∈ X, then by letting x y 0, we get cfz0 for all z ∈ X and so f ≡ 0. And if f satisfies afxbfy ≤ fαx βy γz 2.11 for all x, y, z ∈ X, then by letting y 0,z −αx/γ,wegetafx0 for all x ∈ X and so f ≡ 0. In order to generalize the inequality 2.1, inthe following corollaries, we assume that a k ’s and α k ’s, k 1, 2, ,n n ≥ 3 are nonzero complex numbers. Corollary 2.3. Let f : X→Y be a nonzero mapping satisfying f00 and n k1 a k f x k ≤ f n k1 α k x k 2.12 for all x k ∈ X. Then the following hold: a f is additive; b if α j /α i is a rational number, then a i /α i a j /α j ; c if α i is a rational number, then |a i |≤|α i |. Proof. a Let x k 0in2.12 except for three x k ’s. Then by the same reasoning as inthe proof of Theorem 2.1, it is proved and so we omit the details. b Letting x i x, x j y, by the same reasoning as inthe corresponding part ofthe proof of Theorem 2.1, we can prove it. c Letting x k 0 for all k with k / i, 2.12 gives that a i f x i ≤ f α i x i α i f x i . 2.13 Since f is nonzero, we conclude that |a i |≤|α i |, as desired. Inthe above corollary, similar to Remark 2.2, we notice that if a mapping f satisfies f00and p k1 a k f x k ≤ f q k1 α k x k 2.14 for some p, q ∈{1, 2, ,n} with p / q and all x k ∈ X,thenf ≡ 0. Corollary 2.4. For an invertible 3 × 3 matrix a ij of complex numbers, let f : X→Y be a nonzero mapping satisfying f00 and af a 11 x a 12 y a 13 z bf a 21 x a 22 y a 23 z cf a 31 x a 32 y a 33 z ≤ f αa 11 βa 21 γa 31 x αa 12 βa 22 γa 32 y αa 13 βa 23 γa 33 z 2.15 Jung Rye Lee et al. 5 for all x, y, z ∈ X. Then the following hold: a f is additive; b if α/β, β/γ are rational numbers, then a/α b/β c/γ; c if α is a rational number, then |a| |α|. Proof. If we let s a 11 x a 12 y a 13 z, t a 21 x a 22 y a 23 z, u a 31 x a 32 y a 33 z, then since a matrix a ij is invertible and αa 11 βa 21 γa 31 x αa 12 βa 22 γa 32 y αa 13 βa 23 γa 33 z αs βt γu, 2.16 inequality 2.15 is equivalent to afsbftcfu ≤ fαs βt γu 2.17 for all s, t, u ∈ X. Thus by applying Theorem 2.1, our proofs are clear. By the same reasoning as in Remark 2.2, we obtain the following result. Remark 2.5. For an invertible 3 × 3 matrix a ij of complex numbers, let f : X→Y be a mapping with f00. If f satisfies af a 11 x a 12 y a 13 z bf a 21 x a 22 y a 23 z cf a 31 x a 32 y a 33 z ≤ f αa 11 βa 21 x αa 12 βa 22 y αa 13 βa 23 z 2.18 or af a 11 x a 12 y a 13 z bf a 21 x a 22 y a 23 z ≤ f αa 11 βa 21 γa 31 x αa 12 βa 22 γa 32 y αa 13 βa 23 γa 33 z 2.19 for all x, y, z ∈ X,thenf ≡ 0. Now we investigate linearity of a mapping f : X→Y . The following is a well-known and useful lemma. Lemma 2.6. Let f : X→Y be an additive mapping satisfying lim t∈R,t→0 ftx0 for all x ∈ X.Then f is an R-linear mapping. Theorem 2.7. Let f : X→Y be a nonzero mapping satisfying 2.1 and lim t∈R,t→0 ftx0 for all x ∈ X. Then the following hold: a f is R-linear; b if α/β, β/γ are real numbers, then a/α b/β c/γ. 6 Journal ofInequalities and Applications Proof. a For a mapping f satisfying lim t∈R,t→0 ftx0 for all x ∈ X,ifweletx 0, then we get f00. Since f satisfies 2.1,froma in Theorem 2.1 and Lemma 2.6 we conclude that f is R-linear. b Since f is R-linear by a and α/β, β/γ are real numbers, by the same reasoning as inthe proof of Theorem 2.1b, we can prove it. 3. StabilityofgeneralizedadditivefunctionalinequalitiesIn this section, we study the Hyers-Ulam-Rassias stabilityofgeneralizedadditivefunctionalinequalitiesinBanach spaces. First of all, we introduce α-additivity of a mapping and investigate its properties. Definition 3.1. For a mapping f : X→Y , we say that f is α-additive if fx αyfxαfy3.1 for all x, y ∈ X. Proposition 3.2. If a mapping f : X→Y is α-additive, then f is additive and 1/α-additive. Proof. Let f : X→Y be an α-additive mapping. Letting x y 0in3.1,wegetf00. Letting x 0in3.1,wegetfαyαfy for all y ∈ X. Moreover, letting x 0 and replacing y by y/α in 3.1,wegetfy/α1/αfy for all y ∈ X. Hence we obtain fx yf x α· y α fxαf y α fxfy3.2 for all x, y ∈ X and so f is additive. Onthe other hand, we have f x 1 α y f 1 α y αx 1 α fy αxfx 1 α fy3.3 for all x, y ∈ X and so f is 1/α-additive. Remark 3.3. If a mapping f : X→Y is α-additive and β-additive, then we have fx αβyfxαfβyfxαβfy3.4 for all x, y ∈ X, which implies that f is αβ-additive. Inthe following lemma, we give conditions for a mapping f : X→Y to be C-linear. Lemma 3.4. Let f : X→Y be an α-additive mapping satisfying lim t∈R,t→0 ftx0 for all x ∈ X.Ifα is not a real number, then f is a C-linear mapping. Proof. Let f be an α-additive mapping satisfying lim t∈R,t→0 ftx0 for all x ∈ X. Since f is additive, by Lemma 2.6, f is R-linear. When α is not real, if we let α a bi for some real numbers a, b b / 0, then since f is additive and R-linear, we have a bifxf a bix faxfbixafxbfix3.5 and so fixifx for all x ∈ X, which implies that f is C-linear. Jung Rye Lee et al. 7 Now we are ready to investigate the Hyers-Ulam-Rassias stabilityofgeneralizedadditivefunctional inequality associated with a linear mapping. Here, we give a lemma for our main result. Lemma 3.5. Let f : X→Y be a mapping. If there exists a function ψ : X→0, ∞ satisfying fαx − αfx ≤ ψx, 3.6 ∞ j0 ψ α j x |α| j < ∞ 3.7 for all x ∈ X, then there exists a unique mapping L : X→Y satisfying LαxαLx and fx − Lx ≤ 1 |α| ∞ j0 ψ α j x |α| j 3.8 for all x ∈ X. If, in addition, f is additive, then L is α-additive. Note that this lemma is a special case ofthe results of 24. Proof. Replacing x by α j x in 3.6,wegetfα j1 x − αfα j x≤ψα j x. Dividing by |α| j1 inthe above inequality, we get f α j1 x α j1 − f α j x α j ≤ ψ α j x |α| j1 3.9 for all x ∈ X. From the above inequality, we have f α n1 x α n1 − f α q x α q ≤ n jq f α j1 x α j1 − f α j x α j ≤ n jq 1 |α| ψ α j x |α| j 3.10 for all x ∈ X and all nonnegative integers q, n with q<n.Thusby3.7, the sequence {fα n x/α n } is Cauchy for all x ∈ X. Since Y is complete, the sequence {fα n x/α n } converges for all x ∈ X. So we can define a mapping L : X→Y by Lx : lim n→∞ f α n x α n 3.11 for all x ∈ X. In order to prove that L satisfies 3.8,ifweputq 0andletn→∞ inthe above inequality, then we obtain fx − Lx ≤ ∞ j0 1 |α| ψ α j x |α| j 3.12 for all x ∈ X. 8 Journal ofInequalities and Applications Onthe other hand, Lαx lim n→∞ f α n αx α n αlim n→∞ f α n1 x α n1 αLx3.13 for all x ∈ X, as desired. Now to prove the uniqueness of L,letL : X→Y be another mapping satisfying L αx αL x and 3.8.Thenwehave Lx − L x 1 |α| n L α n x − L α n x ≤ 1 |α| n L α n x − f α n x L α n x − f α n x ≤ 2 |α| n · 1 |α| ∞ j0 ψ α j α n x |α| j 2 |α| ∞ jn ψ α j x |α| j 3.14 whichgoestozeroasn→∞ for all x ∈ X by 3.7. Consequently, L is a unique desired mapping. In addition, when f is additive, L is also additive and so the fact of LαxαLx for all x ∈ X gives that L is α-additive. According to Theorem 2.1, the inequality 2.1 can be reduced as the following additivefunctional inequality αfxβfyγfz ≤ fαx βy γz 3.15 for all x, y, z ∈ X. Inthe following theorem, we prove the Hyers-Ulam-Rassias stabilityofthe above additivefunctional inequality. Theorem 3.6. Let ξ −α/β and let f : X→Y be a mapping satisfying lim t∈R,t→0 ftx0 for all x ∈ X.Ifthereexistsafunctionϕ : X 3 →0, ∞ satisfying αfxβfyγfz ≤ fαx βy γz ϕx, y, z, 3.16 ∞ j0 ϕ ξ j x, ξ j y, ξ j z |ξ| j < ∞, 3.17 lim t∈R,t→0 ∞ j0 ϕ ξ j tx, ξ j1 tx, 0 |ξ| j 0 3.18 for all x, y, z ∈ X, then there exists a unique R-linear and ξ-additive mapping L : X→Y satisfying fx − Lx ≤ 1 |α| ∞ j0 ϕ ξ j x, ξ j1 x, 0 |ξ| j 3.19 for all x ∈ X. If, in addition, ξ is not a real number, then L is a C-linear mapping. Jung Rye Lee et al. 9 Proof. Replacing y −α/βx, z 0in3.16, since αfxβf − α β x ≤ ϕ x, − α β x, 0 , 3.20 we get fξx − ξfx ≤ 1 |β| ϕx, ξx, 03.21 for all x ∈ X.Ifwereplaceψx in Lemma 3.5 by 1/|β|ϕx, ξx, 0,thenby3.17 and Lemma 3.5, there exists a unique mapping L : X→Y satisfying LξxξLx for all x ∈ X and 3.19.Infact,Lx : lim n→∞ fξ n x/ξ n for all x ∈ X. Moreover, by lim t∈R,t→0 ftx0 for all x ∈ X and 3.18,weget lim t∈R,t→0 Ltx − ftx ≤ lim t∈R,t→0 1 |α| ∞ j0 ϕ ξ j tx, ξ j1 tx, 0 |ξ| j 0 3.22 and so lim t∈R,t→0 Ltx0 for all x ∈ X. Since 3.16 and 3.17 give αLxβLyγLz lim n→∞ αf ξ n x βf ξ n y γf ξ n z ξ n ≤ lim n→∞ f ξ n αx βy γz ξ n lim n→∞ ϕ ξ n x, ξ n y, ξ n z |ξ| n Lαx βy γz 0 Lαx βy γz , 3.23 we conclude that by Theorem 2.1 and Lemma 2.6, a mapping L is R-linear and ξ-additive. When ξ is not a real number, by Lemma 3.4, a mapping L is C-linear. Inthe above theorem, we remark that when ξ is −γ/β or −α/γ, we obtain the same result as in Theorem 3.6. As an application of Theorem 3.6, we obtain the following stability. Corollary 3.7. Let f : X→Y be a mapping satisfying lim t∈R,t→0 ftx0 for all x ∈ X and ξ −α/β. When |α| > |β| and 0 <p<1,or|α| < |β| and p>1,ifthereexistsaθ ≥ 0 satisfying αfxβfyγfz ≤ fαx βy γz θ x p y p z p 3.24 for all x, y, z ∈ X, then there exists a unique R-linear and ξ-additive mapping L : X→Y satisfying fx − Lx ≤ θ |α| p |β| p |α||β| |β| p−1 −|α| p−1 x p 3.25 for all x ∈ X. 10 Journal ofInequalities and Applications Proof. If we define ϕx, y, z : θx p y p z p ,thenϕ satisfies the conditions of 3.17 and 3.18. Thanks to Theorem 3.6, it is proved. Before closing this section, we establish another stabilityofgeneralizedadditivefunctional inequalities. Lemma 3.8. Let f : X→Y be a mapping. If there exists a function ψ : X→0, ∞ satisfying 3.6 and ∞ j1 |α| j ψ x α j < ∞ 3.26 for all x ∈ X, then there exists a unique mapping L : X→Y satisfying LαxαLx and fx − Lx ≤ 1 |α| ∞ j1 |α| j ψ x α j 3.27 for all x ∈ X. If, in addition, f is additive, then L is α-additive. Note that this lemma is a special case ofthe results of 24. Proof. Replacing x by x/α j in 3.6,wegetfx/α j−1 − αfx/α j ≤ψx/α j . Multiplying by |α| j−1 inthe above inequality, we get α j−1 f x α j−1 − α j f x α j ≤|α| j−1 ψ x α j 3.28 for all x ∈ X. From the above inequality, we have α n f x α n − α q−1 f x α q−1 ≤ n jq α j f x α j − α j−1 f x α j−1 ≤ n jq 1 |α| |α| j ψ x α j 3.29 for all x ∈ X and all nonnegative integers q, n with q<n.Thusby3.26 the sequence {α n fx/α n } is Cauchy for all x ∈ X. Since Y is complete, the sequence {α n fx/α n } converges for all x ∈ X. So we can define a mapping L : X→Y by Lx : lim n→∞ α n f x α n 3.30 for all x ∈ X. In order to prove that L satisfies 3.27,ifweputq 1andletn→∞ inthe above inequality, then we obtain fx − Lx ≤ 1 |α| ∞ j1 |α| j ϕ x α j 1 |α| ∞ j1 |α| j ψ x α j 3.31 for all x ∈ X. 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Nonlinear Differential Equations and Their Applications, Birkh¨ user, Boston, Mass, USA, 1998 a 12 D G Bourgin, “Classes of transformations and bordering transformations,” Bulletin ofthe American Mathematical Society, vol 57, pp 223–237, 1951 13 Th M Rassias, The problem of S M Ulam for approximately multiplicative mappings,” Journal of Mathematical Analysis and Applications, vol 246, no 2, pp 352–378,... ≤ lim 0 3.38 12 Journal ofInequalities and Applications and so limt∈R, t→0 L tx αL x βL y 0 for all x ∈ X It follows from 3.16 and 3.34 that γL z lim ξ n αf n→∞ x ξn βy ξn αx ξn ≤ lim ξ n f n→∞ y ξn βf L αx βy L αx βy γz ξn γz z ξn γf x y z , , ξn ξn ξn lim |ξ|n ϕ n→∞ 3.39 γz 0 for all x, y, z ∈ X The rest ofthe proof is the same as inthe corresponding part of the proof of Theorem 3.6 Corollary 3.10 . it. 3. Stability of generalized additive functional inequalities In this section, we study the Hyers-Ulam-Rassias stability of generalized additive functional inequalities in Banach spaces. First of. x k ’s. Then by the same reasoning as in the proof of Theorem 2.1, it is proved and so we omit the details. b Letting x i x, x j y, by the same reasoning as in the corresponding part of the proof. studied the Hyers-Ulam stability of 2 Journal of Inequalities and Applications functional equations. The paper of Rassias 4 had great in uence on the development of a generalization of the Hyers-Ulam