The Stability of a Quadratic Functional Equation with the Fixed Point Alternative //// Hindawi Publishing Corporation Home Journals About Us About this Journal Submit a Manuscript Table of Contents Journal Menu Abstracting and Indexing Aims and Scope Annual Issues Article Processing Charges Articles in Press Author Guidelines Bibliographic Information Contact Information Editorial Board Editorial Workflow Free eTOC Alerts Reviewers Acknowledgment Subscription Information Open Special Issues Published Special Issues Special Issue Guidelines Abstract Full-Text PDF Full-Text HTML Linked References How to Cite this Article Abstract and Applied Analysis Volume 2009 (2009), Article ID 907167, 11 pages doi:10.1155/2009/907167 Research Article <h2>The Stability of a Quadratic Functional Equation with the Fixed Point Alternative</h2> Choonkil Park 1 and Ji-Hye Kim 2 1 Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, South Korea 2 Department of Mathematics, Hanyang University, Seoul 133-791, South Korea Received 15 September 2009; Accepted 1 December 2009 Academic Editor: W. A. Kirk Copyright © 2009 Choonkil Park and Ji-Hye Kim. 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. Abstract Lee, An and Park introduced the quadratic functional equation 𝑓 ( 2 𝑥 + 𝑦 ) + 𝑓 ( 2 𝑥 − 𝑦 ) = 8 𝑓 ( 𝑥 ) + 2 𝑓 ( 𝑦 ) and proved the stability of the quadratic functional equation in the spirit of Hyers, Ulam and Th. M. Rassias. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation in Banach spaces. 1. Introduction The stability problem of functional equations originated from a question of Ulam [ 1 ] concerning the stability of 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 Th. M. Rassias [ 4 ] for linear mappings by considering an unbounded Cauchy difference. Theorem 1.1 (Th. M. Rassias). Let 𝑓 ∶ 𝐸 → 𝐸  be a mapping from a normed vector space 𝐸 into a Banach space 𝐸  subject to the inequality  ‖ 𝑓 ( 𝑥 + 𝑦 ) − 𝑓 ( 𝑥 ) − 𝑓 ( 𝑦 ) ‖ ≤ 𝜖 ‖ 𝑥 ‖ 𝑝 + ‖ 𝑦 ‖ 𝑝  ( 1 . 1 ) for all 𝑥 , 𝑦 ∈ 𝐸 , where 𝜖 and 𝑝 are constants with 𝜖 > 0 and 𝑝 < 1 . Then the limit 𝐿 ( 𝑥 ) = l i m 𝑛 → ∞ 𝑓 ( 2 𝑛 𝑥 ) 2 𝑛 ( 1 . 2 ) exists for all 𝑥 ∈ 𝐸 , and 𝐿 ∶ 𝐸 → 𝐸  is the unique additive mapping which satisfies ‖ 𝑓 ( 𝑥 ) − 𝐿 ( 𝑥 ) ‖ ≤ 2 𝜖 2 − 2 𝑝 ‖ 𝑥 ‖ 𝑝 ( 1 . 3 ) for all 𝑥 ∈ 𝐸 . Also, if for each 𝑥 ∈ 𝐸 the function 𝑓 ( 𝑡 𝑥 ) is continuous in 𝑡 ∈ ℝ , then 𝐿 is ℝ -linear. The above inequality ( 1.1 ) has provided a lot of influence in the development of what is now known as a generalized Hyers-Ulam stability of functional equations. Beginning around the year 1980, the topic of approximate homomorphisms, or the stability of the equation of homomorphism, was studied by a number of mathematicians. Găvruta [ 5 ] generalized the Rassias’ result. Theorem 1.2 (see [ 6 – 8 ]). Let 𝑋 be a real normed linear space and 𝑌 a real complete normed linear space. Assume that 𝑓 ∶ 𝑋 → 𝑌 is an approximately additive mapping for which there exist constants 𝜃 ≥ 0 and 𝑝 ∈ ℝ − { 1 } such that 𝑓 satisfies inequality ‖ 𝑓 ( 𝑥 + 𝑦 ) − 𝑓 ( 𝑥 ) − 𝑓 ( 𝑦 ) ‖ ≤ 𝜃 ⋅ ‖ 𝑥 ‖ 𝑝 / 2 ⋅ ‖ 𝑦 ‖ 𝑝 / 2 ( 1 . 4 ) for all 𝑥 , 𝑦 ∈ 𝑋 . Then there exists a unique additive mapping 𝐿 ∶ 𝑋 → 𝑌 satisfying 𝜃 ‖ 𝑓 ( 𝑥 ) − 𝐿 ( 𝑥 ) ‖ ≤ | | 2 𝑝 | | − 2 ‖ 𝑥 ‖ 𝑝 ( 1 . 5 ) for all 𝑥 ∈ 𝑋 . If, in addition, 𝑓 ∶ 𝑋 → 𝑌 is a mapping such that the transformation 𝑡 → 𝑓 ( 𝑡 𝑥 ) is continuous in 𝑡 ∈ ℝ for each fixed 𝑥 ∈ 𝑋 , then 𝐿 is an ℝ -linear mapping. The functional equation 𝑓 ( 𝑥 + 𝑦 ) + 𝑓 ( 𝑥 − 𝑦 ) = 2 𝑓 ( 𝑥 ) + 2 𝑓 ( 𝑦 ) ( 1 . 6 ) is called a quadratic functional equation . In particular, every solution of the quadratic functional equation is said to be a quadratic function . A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [ 9 ] for mappings 𝑓 ∶ 𝑋 → 𝑌 , where 𝑋 is a normed space and 𝑌 is a Banach space. Cholewa [ 10 ] noticed that the theorem of Skof is still true if the relevant domain 𝑋 is replaced by an Abelian group. Czerwik [ 11 ] proved the generalized Hyers-Ulam stability of the quadratic functional equation. Several functional equations have been investigated in [ 12 – 25 ]. Let 𝑋 be a set. A function 𝑑 ∶ 𝑋 × 𝑋 → [ 0 , ∞ ] is called a generalized metric on 𝑋 if 𝑑 satisfies (1) 𝑑 ( 𝑥 , 𝑦 ) = 0 if and only if 𝑥 = 𝑦 ; (2) 𝑑 ( 𝑥 , 𝑦 ) = 𝑑 ( 𝑦 , 𝑥 ) for all 𝑥 , 𝑦 ∈ 𝑋 ; (3) 𝑑 ( 𝑥 , 𝑧 ) ≤ 𝑑 ( 𝑥 , 𝑦 ) + 𝑑 ( 𝑦 , 𝑧 ) for all 𝑥 , 𝑦 , 𝑧 ∈ 𝑋 . We recall a fundamental result in fixed point theory. Theorem 1.3 (see [ 26 – 28 ]). Let ( 𝑋 , 𝑑 ) be a complete generalized metric space and let 𝐽 ∶ 𝑋 → 𝑋 be a strictly contractive mapping with Lipschitz constant 𝐿 < 1 . Then for each given element 𝑥 ∈ 𝑋 , either 𝑑  𝐽 𝑛 𝑥 , 𝐽 𝑛 + 1 𝑥  = ∞ ( 1 . 7 ) for all nonnegative integers 𝑛 or there exists a positive integer 𝑛 0 such that (1) 𝑑 ( 𝐽 𝑛 𝑥 , 𝐽 𝑛 + 1 𝑥 ) < ∞ , for all 𝑛 ≥ 𝑛 0 ; (2) the sequence { 𝐽 𝑛 𝑥 } converges to a fixed point 𝑦 ∗ of 𝐽 ; (3) 𝑦 ∗ is the unique fixed point of 𝐽 in the set 𝑌 = { 𝑦 ∈ 𝑋 ∣ 𝑑 ( 𝐽 𝑛 0 𝑥 , 𝑦 ) < ∞ } ; (4) 𝑑 ( 𝑦 , 𝑦 ∗ ) ≤ ( 1 / ( 1 − 𝐿 ) ) 𝑑 ( 𝑦 , 𝐽 𝑦 ) for all 𝑦 ∈ 𝑌 . Lee et al. [ 29 ] proved that a mapping 𝑓 ∶ 𝑋 → 𝑌 satisfies 𝑓 ( 2 𝑥 + 𝑦 ) + 𝑓 ( 2 𝑥 − 𝑦 ) = 8 𝑓 ( 𝑥 ) + 2 𝑓 ( 𝑦 ) ( 1 . 8 ) for all 𝑥 , 𝑦 ∈ 𝑋 if and only if the mapping 𝑓 ∶ 𝑋 → 𝑌 satisfies 𝑓 ( 𝑥 + 𝑦 ) + 𝑓 ( 𝑥 − 𝑦 ) = 2 𝑓 ( 𝑥 ) + 2 𝑓 ( 𝑦 ) ( 1 . 9 ) for all 𝑥 , 𝑦 ∈ 𝑋 . Using the fixed point method, Park [ 14 ] proved the generalized Hyers-Ulam stability of the quadratic functional equation 𝑓 ( 2 𝑥 + 𝑦 ) = 4 𝑓 ( 𝑥 ) + 𝑓 ( 𝑦 ) + 𝑓 ( 𝑥 + 𝑦 ) − 𝑓 ( 𝑥 − 𝑦 ) ( 1 . 1 0 ) in Banach spaces. In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation ( 1.8 ) in Banach spaces. Throughout this paper, assume that 𝑋 is a normed vector space with norm | | ⋅ | | and that 𝑌 is a Banach space with norm ‖ ⋅ ‖ . 2. Fixed Points and Generalized Hyers-Ulam Stability of a Quadratic Functional Equation For a given mapping 𝑓 ∶ 𝑋 → 𝑌 , we define 𝐶 𝑓 ( 𝑥 , 𝑦 ) ∶ = 𝑓 ( 2 𝑥 + 𝑦 ) + 𝑓 ( 2 𝑥 − 𝑦 ) − 8 𝑓 ( 𝑥 ) − 2 𝑓 ( 𝑦 ) ( 2 . 1 ) for all 𝑥 , 𝑦 ∈ 𝑋 . Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation 𝐶 𝑓 ( 𝑥 , 𝑦 ) = 0 . Theorem 2.1. Let 𝑓 ∶ 𝑋 → 𝑌 be a mapping for which there exists a function 𝜑 ∶ 𝑋 2 → [ 0 , ∞ ) with 𝑓 ( 0 ) = 0 such that ‖ 𝐷 𝑓 ( 𝑥 , 𝑦 ) ‖ ≤ 𝜑 ( 𝑥 , 𝑦 ) ( 2 . 2 ) for all 𝑥 , 𝑦 ∈ 𝑋 . If there exists an 𝐿 < 1 such that 𝜑 ( 𝑥 , 𝑦 ) ≤ 4 𝐿 𝜑 ( 𝑥 / 2 , 𝑦 / 2 ) for all 𝑥 , 𝑦 ∈ 𝑋 , then there exists a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 satisfying ( 1.8 ) and 1 ‖ 𝑓 ( 𝑥 ) − 𝑄 ( 𝑥 ) ‖ ≤ 8 − 8 𝐿 𝜑 ( 𝑥 , 0 ) ( 2 . 3 ) for all 𝑥 ∈ 𝑋 . Proof. Consider the set 𝑆 ∶ = { 𝑔 ∶ 𝑋 ⟶ 𝑌 } , ( 2 . 4 ) and introduce the generalized metric on 𝑆 : 𝑑  ( 𝑔 , ℎ ) = i n f 𝐾 ∈ ℝ +  ∶ ‖ 𝑔 ( 𝑥 ) − ℎ ( 𝑥 ) ‖ ≤ 𝐾 𝜑 ( 𝑥 , 0 ) , ∀ 𝑥 ∈ 𝑋 . ( 2 . 5 ) It is easy to show that ( 𝑆 , 𝑑 ) is complete. Now we consider the linear mapping 𝐽 ∶ 𝑆 → 𝑆 such that 1 𝐽 𝑔 ( 𝑥 ) ∶ = 4 𝑔 ( 2 𝑥 ) ( 2 . 6 ) for all 𝑥 ∈ 𝑋 . By [ 30 , Theorem 3 . 1 ], 𝑑 ( 𝐽 𝑔 , 𝐽 ℎ ) ≤ 𝐿 𝑑 ( 𝑔 , ℎ ) ( 2 . 7 ) for all 𝑔 , ℎ ∈ 𝑆 . Letting 𝑦 = 0 in ( 2.2 ), we get ‖ 2 𝑓 ( 2 𝑥 ) − 8 𝑓 ( 𝑥 ) ‖ ≤ 𝜑 ( 𝑥 , 0 ) ( 2 . 8 ) for all 𝑥 ∈ 𝑋 . So ‖ ‖ ‖ 1 𝑓 ( 𝑥 ) − 4 ‖ ‖ ‖ ≤ 1 𝑓 ( 2 𝑥 ) 8 𝜑 ( 𝑥 , 0 ) ( 2 . 9 ) for all 𝑥 ∈ 𝑋 . Hence 𝑑 ( 𝑓 , 𝐽 𝑓 ) ≤ 1 / 8 . By Theorem 1.3 , there exists a mapping 𝑄 ∶ 𝑋 → 𝑌 such that (1) 𝑄 is a fixed point of 𝐽 , that is, 𝑄 ( 2 𝑥 ) = 4 𝑄 ( 𝑥 ) ( 2 . 1 0 ) for all 𝑥 ∈ 𝑋 . The mapping 𝑄 is a unique fixed point of 𝐽 in the set 𝑀 = { 𝑔 ∈ 𝑆 ∶ 𝑑 ( 𝑓 , 𝑔 ) < ∞ } . ( 2 . 1 1 ) This implies that 𝑄 is a unique mapping satisfying ( 2.10 ) such that there exists 𝐾 ∈ ( 0 , ∞ ) satisfying ‖ 𝑓 ( 𝑥 ) − 𝑄 ( 𝑥 ) ‖ ≤ 𝐾 𝜑 ( 𝑥 , 0 ) ( 2 . 1 2 ) for all 𝑥 ∈ 𝑋 . (2) 𝑑 ( 𝐽 𝑛 𝑓 , 𝑄 ) → 0 as 𝑛 → ∞ . This implies the equality l i m 𝑛 → ∞ 𝑓 ( 2 𝑛 𝑥 ) 4 𝑛 = 𝑄 ( 𝑥 ) ( 2 . 1 3 ) for all 𝑥 ∈ 𝑋 . (3) 𝑑 ( 𝑓 , 𝑄 ) ≤ ( 1 / ( 1 − 𝐿 ) ) 𝑑 ( 𝑓 , 𝐽 𝑓 ) , which implies the inequality 1 𝑑 ( 𝑓 , 𝑄 ) ≤ . 8 − 8 𝐿 ( 2 . 1 4 ) This implies that the inequality ( 2.3 ) holds. It follows from ( 2.2 ) and ( 2.13 ) that ‖ 𝐶 𝑄 ( 𝑥 , 𝑦 ) ‖ = l i m 𝑛 → ∞ 1 4 𝑛 ‖ 𝐶 𝑓 ( 2 𝑛 𝑥 , 2 𝑛 𝑦 ) ‖ ≤ l i m 𝑛 → ∞ 1 4 𝑛 𝜑 ( 2 𝑛 𝑥 , 2 𝑛 𝑦 ) ≤ l i m 𝑛 → ∞ 𝐿 𝑛 𝜑 ( 𝑥 , 𝑦 ) = 0 ( 2 . 1 5 ) for all 𝑥 , 𝑦 ∈ 𝑋 . So 𝐶 𝑄 ( 𝑥 , 𝑦 ) = 0 for all 𝑥 , 𝑦 ∈ 𝑋 . By [ 29 , Proposition 2 . 1 ], the mapping 𝑄 ∶ 𝑋 → 𝑌 is quadratic, as desired. Corollary 2.2. Let 0 < 𝑝 < 2 and 𝜃 be positive real numbers, and let 𝑓 ∶ 𝑋 → 𝑌 be a mapping such that  ‖ 𝐶 𝑓 ( 𝑥 , 𝑦 ) ‖ ≤ 𝜃 ‖ 𝑥 ‖ 𝑝 + ‖ 𝑦 ‖ 𝑝  ( 2 . 1 6 ) for all 𝑥 , 𝑦 ∈ 𝑋 . Then there exists a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 satisfying ( 1.8 ) and 𝜃 ‖ 𝑓 ( 𝑥 ) − 𝑄 ( 𝑥 ) ‖ ≤ 8 − 2 𝑝 + 1 ‖ 𝑥 ‖ 𝑝 ( 2 . 1 7 ) for all 𝑥 ∈ 𝑋 . Proof. The proof follows from Theorem 2.1 by taking 𝜑  ( 𝑥 , 𝑦 ) ∶ = 𝜃 ‖ 𝑥 ‖ 𝑝 + ‖ 𝑦 ‖ 𝑝  ( 2 . 1 8 ) for all 𝑥 , 𝑦 ∈ 𝑋 . Then 𝐿 = 2 𝑝 − 2 , and we get the desired result. Theorem 2.3. Let 𝑓 ∶ 𝑋 → 𝑌 be a mapping for which there exists a function 𝜑 ∶ 𝑋 2 → [ 0 , ∞ ) satisfying ( 2.2 ) and 𝑓 ( 0 ) = 0 . If there exists an 𝐿 < 1 such that 𝜑 ( 𝑥 , 𝑦 ) ≤ ( 𝐿 / 4 ) 𝜑 ( 2 𝑥 , 2 𝑦 ) for all 𝑥 , 𝑦 ∈ 𝑋 , then there exists a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 satisfying ( 1.8 ) and 𝐿 ‖ 𝑓 ( 𝑥 ) − 𝑄 ( 𝑥 ) ‖ ≤ 𝜑 8 − 8 𝐿 ( 𝑥 , 0 ) ( 2 . 1 9 ) for all 𝑥 ∈ 𝑋 . Proof. We consider the linear mapping 𝐽 ∶ 𝑆 → 𝑆 such that  𝑥 𝐽 𝑔 ( 𝑥 ) ∶ = 4 𝑔 2  ( 2 . 2 0 ) for all 𝑥 ∈ 𝑋 . It follows from ( 2.8 ) that ‖ ‖ ‖  𝑥 𝑓 ( 𝑥 ) − 4 𝑓 2  ‖ ‖ ‖ ≤ 1 2 𝜑  𝑥 2  ≤ 𝐿 , 0 8 𝜑 ( 𝑥 , 0 ) ( 2 . 2 1 ) for all 𝑥 ∈ 𝑋 . Hence 𝑑 ( 𝑓 , 𝐽 𝑓 ) ≤ 𝐿 / 8 . By Theorem 1.3 , there exists a mapping 𝑄 ∶ 𝑋 → 𝑌 such that (1) 𝑄 is a fixed point of 𝐽 , that is, 𝑄 ( 2 𝑥 ) = 4 𝑄 ( 𝑥 ) ( 2 . 2 2 ) for all 𝑥 ∈ 𝑋 . The mapping 𝑄 is a unique fixed point of 𝐽 in the set 𝑀 = { 𝑔 ∈ 𝑆 ∶ 𝑑 ( 𝑓 , 𝑔 ) < ∞ } . ( 2 . 2 3 ) This implies that 𝑄 is a unique mapping satisfying ( 2.22 ) such that there exists 𝐾 ∈ ( 0 , ∞ ) satisfying ‖ 𝑓 ( 𝑥 ) − 𝑄 ( 𝑥 ) ‖ ≤ 𝐾 𝜑 ( 𝑥 , 0 ) ( 2 . 2 4 ) for all 𝑥 ∈ 𝑋 . (2) 𝑑 ( 𝐽 𝑛 𝑓 , 𝑄 ) → 0 as 𝑛 → ∞ . This implies the equality l i m 𝑛 → ∞ 4 𝑛 𝑓  𝑥 2 𝑛  = 𝑄 ( 𝑥 ) ( 2 . 2 5 ) for all 𝑥 ∈ 𝑋 . (3) 𝑑 ( 𝑓 , 𝑄 ) ≤ ( 1 / ( 1 − 𝐿 ) ) 𝑑 ( 𝑓 , 𝐽 𝑓 ) , which implies the inequality 𝐿 𝑑 ( 𝑓 , 𝑄 ) ≤ , 8 − 8 𝐿 ( 2 . 2 6 ) which implies that the inequality ( 2.19 ) holds. The rest of the proof is similar to the proof of Theorem 2.1 . Corollary 2.4. Let 𝑝 > 2 and 𝜃 be positive real numbers, and let 𝑓 ∶ 𝑋 → 𝑌 be a mapping satisfying ( 2.16 ). Then there exists a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 satisfying ( 1.8 ) and 𝜃 ‖ 𝑓 ( 𝑥 ) − 𝑄 ( 𝑥 ) ‖ ≤ 2 𝑝 + 1 − 8 ‖ 𝑥 ‖ 𝑝 ( 2 . 2 7 ) for all 𝑥 ∈ 𝑋 . Proof. The proof follows from Theorem 2.3 by taking 𝜑  ( 𝑥 , 𝑦 ) ∶ = 𝜃 ‖ 𝑥 ‖ 𝑝 + ‖ 𝑦 ‖ 𝑝  ( 2 . 2 8 ) for all 𝑥 , 𝑦 ∈ 𝑋 . Then 𝐿 = 2 2 − 𝑝 and, we get the desired result. Theorem 2.5. Let 𝑓 ∶ 𝑋 → 𝑌 be a mapping for which there exists a function 𝜑 ∶ 𝑋 2 → [ 0 , ∞ ) satisfying ( 2.2 ). If there exists an 𝐿 < 1 such that 𝜑 ( 𝑥 , 𝑦 ) ≤ 9 𝐿 𝜑 ( 𝑥 / 3 , 𝑦 / 3 ) for all 𝑥 , 𝑦 ∈ 𝑋 , then there exists a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 satisfying ( 1.8 ) and 1 ‖ 𝑓 ( 𝑥 ) − 𝑄 ( 𝑥 ) ‖ ≤ 9 − 9 𝐿 𝜑 ( 𝑥 , 𝑥 ) ( 2 . 2 9 ) for all 𝑥 ∈ 𝑋 . Proof. Consider the set 𝑆 ∶ = { 𝑔 ∶ 𝑋 ⟶ 𝑌 } , ( 2 . 3 0 ) and introduce the generalized metric on 𝑆 : 𝑑  ( 𝑔 , ℎ ) = i n f 𝐾 ∈ ℝ +  ∶ ‖ 𝑔 ( 𝑥 ) − ℎ ( 𝑥 ) ‖ ≤ 𝐾 𝜑 ( 𝑥 , 𝑥 ) , ∀ 𝑥 ∈ 𝑋 . ( 2 . 3 1 ) It is easy to show that ( 𝑆 , 𝑑 ) is complete. Now we consider the linear mapping 𝐽 ∶ 𝑆 → 𝑆 such that 1 𝐽 𝑔 ( 𝑥 ) ∶ = 9 𝑔 ( 3 𝑥 ) ( 2 . 3 2 ) for all 𝑥 ∈ 𝑋 . By [ 30 , Theorem 3 . 1 ], 𝑑 ( 𝐽 𝑔 , 𝐽 ℎ ) ≤ 𝐿 𝑑 ( 𝑔 , ℎ ) ( 2 . 3 3 ) for all 𝑔 , ℎ ∈ 𝑆 . Letting 𝑦 = 𝑥 in ( 2.2 ), we get ‖ 𝑓 ( 3 𝑥 ) − 9 𝑓 ( 𝑥 ) ‖ ≤ 𝜑 ( 𝑥 , 𝑥 ) ( 2 . 3 4 ) for all 𝑥 ∈ 𝑋 . So ‖ ‖ ‖ 1 𝑓 ( 𝑥 ) − 9 ‖ ‖ ‖ ≤ 1 𝑓 ( 3 𝑥 ) 9 𝜑 ( 𝑥 , 𝑥 ) ( 2 . 3 5 ) for all 𝑥 ∈ 𝑋 . Hence 𝑑 ( 𝑓 , 𝐽 𝑓 ) ≤ 1 / 9 . By Theorem 1.3 , there exists a mapping 𝑄 ∶ 𝑋 → 𝑌 such that (1) 𝑄 is a fixed point of 𝐽 , that is, 𝑄 ( 3 𝑥 ) = 9 𝑄 ( 𝑥 ) ( 2 . 3 6 ) for all 𝑥 ∈ 𝑋 . The mapping 𝑄 is a unique fixed point of 𝐽 in the set 𝑀 = { 𝑔 ∈ 𝑆 ∶ 𝑑 ( 𝑓 , 𝑔 ) < ∞ } . ( 2 . 3 7 ) This implies that 𝑄 is a unique mapping satisfying ( 2.36 ) such that there exists 𝐾 ∈ ( 0 , ∞ ) satisfying ‖ 𝑓 ( 𝑥 ) − 𝑄 ( 𝑥 ) ‖ ≤ 𝐾 𝜑 ( 𝑥 , 𝑥 ) ( 2 . 3 8 ) for all 𝑥 ∈ 𝑋 . (2) 𝑑 ( 𝐽 𝑛 𝑓 , 𝑄 ) → 0 as 𝑛 → ∞ . This implies the equality l i m 𝑛 → ∞ 𝑓 ( 3 𝑛 𝑥 ) 9 𝑛 = 𝑄 ( 𝑥 ) ( 2 . 3 9 ) for all 𝑥 ∈ 𝑋 . (3) 𝑑 ( 𝑓 , 𝑄 ) ≤ ( 1 / ( 1 − 𝐿 ) ) 𝑑 ( 𝑓 , 𝐽 𝑓 ) , which implies the inequality 1 𝑑 ( 𝑓 , 𝑄 ) ≤ . 9 − 9 𝐿 ( 2 . 4 0 ) This implies that the inequality ( 2.29 ) holds. It follows from ( 2.2 ) and ( 2.39 ) that ‖ 𝐶 𝑄 ( 𝑥 , 𝑦 ) ‖ = l i m 𝑛 → ∞ 1 9 𝑛 ‖ 𝐶 𝑓 ( 3 𝑛 𝑥 , 3 𝑛 𝑦 ) ‖ ≤ l i m 𝑛 → ∞ 1 9 𝑛 𝜑 ( 3 𝑛 𝑥 , 3 𝑛 𝑦 ) ≤ l i m 𝑛 → ∞ 𝐿 𝑛 𝜑 ( 𝑥 , 𝑦 ) = 0 ( 2 . 4 1 ) for all 𝑥 , 𝑦 ∈ 𝑋 . So 𝐶 𝑄 ( 𝑥 , 𝑦 ) = 0 for all 𝑥 , 𝑦 ∈ 𝑋 . By [ 29 , Proposition 2 . 1 ], the mapping 𝑄 ∶ 𝑋 → 𝑌 is quadratic, as desired. Corollary 2.6. Let 0 < 𝑝 < 2 and 𝜃 be positive real numbers, and let 𝑓 ∶ 𝑋 → 𝑌 be a mapping satisfying ( 2.16 ). Then there exists a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 satisfying ( 1.8 ) and ‖ 𝑓 ( 𝑥 ) − 𝑄 ( 𝑥 ) ‖ ≤ 2 𝜃 9 − 3 𝑝 ‖ 𝑥 ‖ 𝑝 ( 2 . 4 2 ) for all 𝑥 ∈ 𝑋 . Proof. The proof follows from Theorem 2.5 by taking 𝜑  ( 𝑥 , 𝑦 ) ∶ = 𝜃 ‖ 𝑥 ‖ 𝑝 + ‖ 𝑦 ‖ 𝑝  ( 2 . 4 3 ) for all 𝑥 , 𝑦 ∈ 𝑋 . Then 𝐿 = 3 𝑝 − 2 and, we get the desired result. Corollary 2.7. Let 0 < 𝑝 < 1 and 𝜃 be positive real numbers, and let 𝑓 ∶ 𝑋 → 𝑌 be a mapping such that ‖ 𝐷 𝑓 ( 𝑥 , 𝑦 ) ‖ ≤ 𝜃 ⋅ ‖ 𝑥 ‖ 𝑝 ⋅ ‖ 𝑦 ‖ 𝑝 ( 2 . 4 4 ) for all 𝑥 , 𝑦 ∈ 𝑋 . Then there exists a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 satisfying ( 1.8 ) and 𝜃 ‖ 𝑓 ( 𝑥 ) − 𝑄 ( 𝑥 ) ‖ ≤ 9 − 9 𝑝 ‖ 𝑥 ‖ 2 𝑝 ( 2 . 4 5 ) for all 𝑥 ∈ 𝑋 . Proof. The proof follows from Theorem 2.5 by taking 𝜑 ( 𝑥 , 𝑦 ) ∶ = 𝜃 ⋅ ‖ 𝑥 ‖ 𝑝 ⋅ ‖ 𝑦 ‖ 𝑝 ( 2 . 4 6 ) for all 𝑥 , 𝑦 ∈ 𝑋 . Then 𝐿 = 9 𝑝 − 1 and, we get the desired result. Theorem 2.8. Let 𝑓 ∶ 𝑋 → 𝑌 be a mapping for which there exists a function 𝜑 ∶ 𝑋 2 → [ 0 , ∞ ) satisfying ( 2.2 ). If there exists an 𝐿 < 1 such that 𝜑 ( 𝑥 , 𝑦 ) ≤ ( 𝐿 / 9 ) 𝜑 ( 3 𝑥 , 3 𝑦 ) for all 𝑥 , 𝑦 ∈ 𝑋 , then there exists a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 satisfying ( 1.8 ) and 𝐿 ‖ 𝑓 ( 𝑥 ) − 𝑄 ( 𝑥 ) ‖ ≤ 𝜑 9 − 9 𝐿 ( 𝑥 , 𝑥 ) ( 2 . 4 7 ) for all 𝑥 ∈ 𝑋 . Proof. We consider the linear mapping 𝐽 ∶ 𝑆 → 𝑆 such that  𝑥 𝐽 𝑔 ( 𝑥 ) ∶ = 9 𝑔 3  ( 2 . 4 8 ) for all 𝑥 ∈ 𝑋 . The rest of the proof is similar to the proof of Theorem 2.1 . Corollary 2.9. Let 𝑝 > 2 and 𝜃 be positive real numbers, and let 𝑓 ∶ 𝑋 → 𝑌 be a mapping satisfying ( 2.16 ). Then there exists a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 satisfying ( 1.8 ) and ‖ 𝑓 ( 𝑥 ) − 𝑄 ( 𝑥 ) ‖ ≤ 2 𝜃 3 𝑝 − 9 ‖ 𝑥 ‖ 𝑝 ( 2 . 4 9 ) for all 𝑥 ∈ 𝑋 . Proof. The proof follows from Theorem 2.8 by taking 𝜑  ( 𝑥 , 𝑦 ) ∶ = 𝜃 ‖ 𝑥 ‖ 𝑝 + ‖ 𝑦 ‖ 𝑝  ( 2 . 5 0 ) for all 𝑥 , 𝑦 ∈ 𝑋 . Then 𝐿 = 3 2 − 𝑝 , and we get the desired result. Corollary 2.10. Let 𝑝 > 1 and 𝜃 be positive real numbers, and let 𝑓 ∶ 𝑋 → 𝑌 be a mapping satisfying ( 2.44 ). Then there exists a unique quadratic mapping 𝑄 ∶ 𝑋 → 𝑌 satisfying ( 1.8 ) and 𝜃 ‖ 𝑓 ( 𝑥 ) − 𝑄 ( 𝑥 ) ‖ ≤ 9 𝑝 − 9 ‖ 𝑥 ‖ 2 𝑝 ( 2 . 5 1 ) for all 𝑥 ∈ 𝑋 . Proof. The proof follows from Theorem 2.8 by taking 𝜑 ( 𝑥 , 𝑦 ) ∶ = 𝜃 ⋅ ‖ 𝑥 ‖ 𝑝 ⋅ ‖ 𝑦 ‖ 𝑝 ( 2 . 5 2 ) for all 𝑥 , 𝑦 ∈ 𝑋 . Then 𝐿 = 9 1 − 𝑝 , and we get the desired result. Acknowledgment The first author was supported by Hanyang University in 2009. <h4>References</h4> S. M. Ulam, Problems in Modern Mathematics , John Wiley & Sons, New York, NY, USA, 1960. D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America , vol. 27, pp. 222–224, 1941. T. 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. 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