Sang-baek Lee*, Jae-hyeong Bae**, and Won-gil Park***

Transcription

1 JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 6, No. 4, November ON THE HYERS-ULAM STABILITY OF AN ADDITIVE FUNCTIONAL INEQUALITY Sang-baek Lee*, Jae-hyeong Bae**, and Won-gil Park*** Abstract. In this paper, we prove the generalized Hyers-Ulam stability of the additive functional inequality in Banach spaces. f y) + fy z) + fz) f) 1. Introduction and preliminaries In 1940, Ulam [6] suggested the stability problem of functional equations concerning the stability of group homomorphisms as follows: Let G, ) be a group and let H,, d) be a metric group with the metric d, ). Given ε > 0, does there eist a δ = δε) > 0 such that if a mapping f : G H satisfies the inequality d f y), f) fy) ) < δ for all, y G, then a homomorphism F : G H eits with d f), F ) ) < ε for all G? In 1941, Hyers [] gave a first partial) affirmative answer to the question of Ulam for Banach spaces as follows: If δ > 0 and if f : E F is a mapping between Banach spaces E and F satisfying f + y) f) fy) δ Received March 7, 013; Accepted October 11, Mathematics Subject Classification: Primary 39B8; Secondary 46B, 47J. Key words and phrases: additive functional inequality, Banach space. Correspondence should be addressed to Won-gil Park, wgpark@mokwon.ac.kr. This research was supported by Basic Science Research Program through the National Research Foundation of KoreaNRF) funded by the Ministry of Education ).

2 67 Sang-baek Lee, Jae-hyeong Bae, and Won-gil Park for all, y E, then there is a unique additive mapping A : E F such that f) A) δ for all, y E. We will recall a fundamental result in fied point theory for eplicit later use. Theorem 1.1. The alternative of fied point) [1, 5] Suppose we are given a complete generalized metric space X, d) and a strictly contractive mapping Λ : X X, with the Lipschitz constant L. Then, for each given element X, either dλ n, Λ n+1 ) = for all nonnegative integers n or there eists a positive integer n 0 such that a) dλ n, Λ n+1 ) < for all n n 0 ; b) The sequence Λ n ) is convergent to a fied point y of Λ; c) y is the unique fied point of Λ in the set Y = {y X dλ n 0, y) < }; d) dy, y ) 1 1 Ldy, Λy) for all y Y.. Hyers-Ulam stability in Banach spaces Throughout this paper, let X be a normed linear space and Y a Banach space. In 007, Park, Cho and Han [4] proved the Hyers-Ulam stability of the additive functional inequality f) + fy) + fz) f + y + z) in Banach spaces. In 011, Lee, Park and Shin [3] prove the Hyers-Ulam stability of the additive functional inequality f) + fy) + fz) f + y + z) in Banach spaces. In this paper, we prove the generalized Hyers-Ulam stability of the additive functional inequality in Banach spaces. f y) + fy z) + fz) f)

3 Stability of an additive functional inequality 673 Lemma.1. Let f : X Y be a mapping. Then it is additive if and only if it satisfies.1) f y) + fy z) + fz) f) for all, y, z X. Proof. If f is additive, then clearly f y) + fy z) + fz) = f) for all, y, z X. Assume that f satisfies.1). Letting = y = z = 0 in.1), we gain 3f0) f0) and so f0) = 0. Putting = z = 0 in.1), we get f y) + fy) f0) = 0 and so f y) = fy) for all y X. Letting = 0 and replacing z by z in.1), we have fy + z) + f y) + f z) f0) = 0 for all y, z X. Thus we obtain for all y, z X. fy + z) = fy) + fz) Theorem.. Let f : X Y be a mapping with f0) = 0. If there is a function ϕ : X 3 [0, ) satisfying.) f y) + fy z) + fz) f) + ϕ, y, z) and.3) ϕ, y, z) := j=0 1 j ϕ ) j, ) j y, ) j z ) < for all, y, z X, then there eists a unique additive mapping A : X Y such that.4) f) A) 1 ϕ0,, ) for all X. Proof. Replacing, y, z by 0, ) n, ) n, respectively, and dividing by n+1 in.), since f0) = 0, we get f ) n+1 ) ) n+1 f ) n ) ) n 1 n+1 ϕ 0, ) n, ) n )

4 674 Sang-baek Lee, Jae-hyeong Bae, and Won-gil Park for all X and all nonnegative integers n. From the above inequality, we have f ) n ) ) n f ) m ) n 1 ) m f ) j+1 ) ) j+1 f ) j ) ) j.5) j=m n 1 j=m 1 j+1 ϕ 0, ) j, ) j ) for all X and all nonnegative { integers } m, n with m < n. By the condition.3), the sequence f ) n ) ) is a Cauchy sequence for all n { } f ) n ) ) n X. Since Y is complete, the sequence X. So one can define a mapping A : X Y by f ) n ) A) := lim n ) n converges for all for all X. Taking m = 0 and letting n tend to in.5), we have the inequality.4). Replacing, y, z by ) n, ) n y, ) n z, respectively, and dividing by n in.), we obtain f ) n y) ) ) n + f ) n y z) ) ) n + f ) n z) ) ) n f ) n ) ) ) n + 1 n ϕ ) n, ) n y, ) n z ) for all, y, z X and all nonnegative integers n. Since.3) gives that lim n 1 n ϕ ) n, ) n y, ) n z ) = 0 for all, y, z X, letting n tend to in the above inequality, we see that A satisfies the inequality.1) and so it is additive by Lemma.1. Let A : X Y be another additive mapping satisfying.4). Since both A and A are additive, we have

5 Stability of an additive functional inequality 675 A) A ) = 1 n A ) n ) A ) n ) 1 A ) n n ) f ) n ) + f ) n ) A ) n ) ) 1 n ϕ 0, ) n, ) n ) = j=n 1 j ϕ 0, ) j, ) j ) which goes to zero as n for all X by.3). Therefore, A is a unique additive mapping satisfying.4), as desired. Corollary.3. Let θ [0, ) and p [0, 1) and let f : X Y be an odd mapping such that.6) f y) + fy z) + fz) f) + θ p + y p + z p ) for all, y, z X. Then there eists a unique Cauchy additive mapping A : X Y such that.7) for all X. f) A) θ p p Proof. In Theorem., take ϕ, y, z) := θ p + y p + z p ) for all, y, z X. Then we have the desired result. Theorem.4. Let f : X Y be a mapping with f0) = 0. If there is a function ϕ : X 3 [0, ) satisfying.) and ).8) ϕ, y, z) := j ϕ ) j, y ) j, z ) j < j=1 for all, y, z X, then there eists a unique additive mapping A : X Y such that.9) for all X. f) A) 1 ϕ0,, )

6 676 Sang-baek Lee, Jae-hyeong Bae, and Won-gil Park Proof. Replacing, y, z by 0, ), n ) n, respectively, and multiplying by n 1 in.), since f0) = 0, we have ) )n f ) n ) n 1 f ) n 1 ) n 1 ϕ 0, ) n, for all X and all n N. From the above inequality, we get ) ).10) )n f ) n ) m f ) m n ) ) )j f ) j ) j 1 f ) j 1 j=m+1 n j=m+1 j 1 ϕ 0, ) j, ) ) j ) ) n for all X and { all nonnegative )} integers m, n with m < n. From.8), the sequence ) f n ) is a Cauchy sequence for all X. n { )} Since Y is complete, the sequence ) f n ) converges for all n X. So one can define a mapping A : X Y by ) A) := lim n )n f ) n for all X. To prove that A satisfies.9), putting m = 0 and letting n in.10), we have ) f) A) j 1 ϕ 0, ) j, ) j = 1 ϕ0,, ) j=1 for all X. Replacing, y, z by by n in.), we obtain ) y )n f ) n )n f ) n ), n y ) n, y z + ) n f ) ) n + n ϕ z ) n, respectively, and multiplying ) n, ) ) z + ) n f ) ) n y ) n, z ) n for all, y, z X and all nonnegative integers n. Since.8) gives that ) lim n n ϕ ) n, y ) n, z ) n = 0

7 Stability of an additive functional inequality 677 for all, y, z X, if we let n in the above inequality, then we have A y) + Ay z) + Az) A) for all, y, z X. By Lemma.1, the mapping A is additive. The rest of the proof is similar to the corresponding part of the proof of Theorem.. Corollary.5. Let p > 1 and θ be non-negative real numbers and let f : X Y be an odd mapping such that.11) f y) + fy z) + fz) f) + θ p + y p + z p ) for all, y, z X. Then there eists a unique Cauchy additive mapping A : X Y such that.1) f) A) θ p p for all X. Proof. In Theorem.4, take ϕ, y, z) := θ p + y p + z p ) for all, y, z X. Then, we have the desired result. 3. Hyers-Ulam stability using fied point methods Now, using the fied point method, we investigate the Hyers-Ulam stability of the functional inequality.1) in Banach spaces. Theorem 3.1. Suppose that an odd mapping f : X Y satisfies the inequality 3.1) f y) + fy z) + fz) f) + φ, y, z) for all, y, z X, where φ : X 3 [0, ) is a function. If there eists L < 1 such that 3.) φ, y, z) 1 Lφ, y, z) for all, y, z X, then there eists a unique Cauchy additive mapping A : X Y satisfying 3.3) for all X. f) A) L φ0,, ) L

8 678 Sang-baek Lee, Jae-hyeong Bae, and Won-gil Park Proof. Consider a set S := {g g : X Y} and introduce a generalized metric d on S as follows: where dg, h) = d φ g, h) := inf S φ g, h), S φ g, h) := {C 0, ) : g) h) Cφ0,, ) for all X } for all g, h S. Now we show that S, d) is complete. Let {h n } be a Cauchy sequence in S, d). Then, for any ε > 0 there eists an integer N ε > 0 such that dh m, h n ) < ε for all m, n N ε. Since dh m, h n ) = inf S φ h m, h n ) < ε for all m, n N ε, there eists C 0, ε) such that 3.4) h m ) h n ) Cφ0,, ) εφ0,, ) for all m, n N ε and all X. So {h n )} is a Cauchy sequence in Y for each X. Since Y is complete, {h n )} converges for each X. Thus a mapping h : X Y can be defined by 3.5) h) := lim n h n) for all X. Letting n in 3.4), we have m N ε h m ) h) εφ0,, ) ε S φ h m, h) dh m, h) = inf S φ h m, h) ε for all X. This means that the Cauchy sequence {h n } converges to h in S, d). Hence S, d) is complete. Define a mapping Λ : S S by ) 3.6) Λh) := h for all X. We claim that Λ is strictly contractive on S. For any given g, h S, let C gh [0, ] be an arbitrary constant with dg, h) C gh. Then dg, h) C gh g) h) C gh φ0,, ) for all X ) ) g h Cgh φ 0,, ) for all X ) ) g h LCgh φ0,, ) for all X, that is, dλg, Λh) LC gh. Hence we see that dλg, Λh) Ldg, h) for any g, h S. Therefore Λ is strictly contractive mapping on S with the

9 Stability of an additive functional inequality 679 Lipschitz constant L 0, 1). Putting = 0, y = and z = in 3.1), we have 3.7) f) f) φ0,, ) for all X. It follows from 3.7) that ) f) f φ 0,, ) 3.8) L φ0,, ) for all X. Thus df, Λf) L. Therefore, it follows from Theorem 1.1 that the sequence {Λ n f} converges to a fied point A of Λ, i.e., A : X Y, A) = lim Λf)) = lim n n n f n ) and A) = A) for all X. Also A is the unique fied point of Λ in the set S = {g S df, g) < } and da, f) 1 L dλf, f) 1 L L, i.e., the inequality 3.3) holds for all X. It follows from the definition of A and 3.1) that A y) + Ay z) + Az) A) for all, y, z X. By Lemma.1, the mapping A : X Y is a Cauchy additive mapping. Therefore, there eists a unique Cauchy additive mapping A : X Y satisfying 3.3). Corollary 3.. Let p > 1 and θ be non-negative real numbers and let f : X Y be an odd mapping such that 3.9) f y) + fy z) + fz) f) + θ p + y p + z p ) for all, y, z X. Then there eists a unique Cauchy additive mapping A : X Y such that 3.10) f) A) p + 1 p θ p for all X. Proof. In Theorem 3.1, take φ, y, z) := θ p + y p + z p ) for all, y, z X. Then, we can choose L = 1 p and we have the desired result.

10 680 Sang-baek Lee, Jae-hyeong Bae, and Won-gil Park Theorem 3.3. Suppose that an odd mapping f : X Y satisfies the inequality 3.11) f y) + fy z) + fz) f) + φ, y, z) for all, y, z X, where φ : X 3 [0, ) is a function. If there eists L < 1 such that 3.1) φ, y, z) Lφ, y, z ) for all, y, z X, then there eists a unique Cauchy additive mapping A : X Y satisfying 3.13) for all X. f) A) 1 φ0,, ) L Proof. Consider the complete generalized metric space S, d) given in the proof of Theorem 3.1. Now we consider the linear mapping Λ : S S given by Λh) = 1 h) for all X. For any given g, h S, let C gh [0, ] be an arbitrary constant with dg, h) C gh. Hence we obtain dλg, Λh) Ldg, h) for all g, h S. It follows from 3.7) that df, Λf) 1. The rest of the proof is similar to the corresponding part of the proof of Theorem 3.1. Corollary 3.4. Let θ [0, ) and p [0, 1) and let f : X Y be an odd mapping such that 3.14) f y) + fy z) + fz) f) + θ p + y p + z p ) for all, y, z X. Then there eists a unique Cauchy additive mapping A : X Y such that 3.15) for all X. f) A) 1 + p p θ p Proof. In Theorem 3.3, take φ, y, z) := θ p + y p + z p ) for all, y, z X. Then we can choose L = p 1 and we have the desired result.

11 Stability of an additive functional inequality 681 References [1] J. B. Diaz and B. Margolis, A fied point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc ), [] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci ), -4. [3] J. Lee, C. Park, and D. Shin, Stability of an additive functional inequality in proper CQ -algebras, Bull. Korean Math. Soc ), [4] C. Park, Y. S. Cho, and M. H. Han, Functional inequalities associated with Jordan -von Neumann-type additive functional equations, J. Inequal. Appl ), Article ID 4180, 13 pages. [5] I. A. Rus, Principles and Applications of Fied Point Theory, in Romanian), Editura Dacia, Cluj-Napoca, [6] S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ., New York, * Department of Mathematics Chungnam National University Daejeon , Republic of Korea mcsquarelsb@hanmail.net ** Humanitas College Kyung Hee University Yongin , Republic of Korea jhbae@khu.ac.kr *** Department of Mathematics Education Mokwon University Daejeon 30-79, Republic of Korea wgpark@mokwon.ac.kr