Correspondence should be addressed to Abasalt Bodaghi;

Function Spaces, Article ID 532463, 5 pages http://dx.doi.org/0.55/204/532463 Research Article Approximation on the Quadratic Reciprocal Functional Equation Abasalt Bodaghi and Sang Og Kim 2 Department of Mathematics, Garmsar Branch, Islamic Azad University, Garmsar, Iran 2 Department of Mathematics, Hallym University, Chuncheon 200-702, Republic of Korea Correspondence should be addressed to Abasalt Bodaghi; abasalt.bodaghi@gmail.com Received 9 November 203; Accepted 9 January 204; Published 20 February 204 Academic Editor: Bing Xu Copyright 204 A. Bodaghi and S. O. 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. The quadratic reciprocal functional equation is introduced. The Ulam stability problem for an ε-quadratic reciprocal mapping f:x Ybetween nonzero real numbers is solved. The Găvruţa stability for the quadratic reciprocal functional equations is established as well.. Introduction In [], Ulam proposed the well-known Ulam stability problem and one year later, the problem for linear mappings was solved by Hyers [2]. Bourgin [3]alsostudiedtheUlamproblem for additive mappings. Gruber [4] claimed that thiskind of stability problem is of particular interest in the probability theory and in the case of functional equations of different types. The result of Hyers was generalized for approximately additive mappings by Aoki [5] and for approximately linear mappings, by considering the unbounded Cauchy differences by Rassias [6]. A further generalization was obtained by Găvruţa [7] by replacing the Cauchy differences with a control function φ satisfying a very simple condition of convergence. Skof [8]was the first authorto solve the Ulam problem for quadratic mappings on Banach algebras. Cholewa [9] demonstrated that the theorem of Skof is still true if relevant domain is replaced with an abelian group (see also [0 4]). Ravi and Senthil Kumar [5] studied the Hyers-Ulam stability for the reciprocal functional equation f(x+y)= f (x) f(y) f (x) +f(y), () where f:x Yis a mapping in the space of nonzero real numbers. It is easy to check that the reciprocal function f(x) = /x is a solution of the functional equation (). Other results regarding the stability of various forms of the reciprocal functional equation can be found in [6 22]. In this paper, we study the Ulam-Găvruţa-Rassias stability for a new 2-dimensional quadratic reciprocal functional mapping f:x Ysatisfying the Rassias quadratic reciprocal functional equation f(2x+y)+f(2x y)= 2f (x) f(y)[4f(y)+f(x)]. It is easily verified that the quadratic reciprocal function f(x) = /x 2 is a solution of the functional equation (2). As some corollaries, we investigate the pertinent stability of the Rassias equation (2)controlledbythe sum,product,andthe mixed product-sum of powers of norms. 2. ε-stability of (2) Throughout this paper, we denote the space of nonzero real numbers by R. Definition. Amappingf:R R is called Rassias quadratic reciprocal, if the Rassias quadratic reciprocal functional equation (2)holdsforallx, y R. Discussion on the above Definition and (2). We firstly note that, in the above definition, the equalities x = y/2 and (2)

2 Function Spaces x= y/2can not occur because 2x yand 2x+ydo not belong to R. On the other hand, if 4f(y) = f(x), weconsider (2) which is equivalent to (f(2x+y)+f(2x y)) =2f(x) f (y) [4f (y) + f (x)]. Since f(y) =0,wehavef(x) =0.If4f(y) + f(x) =0,then f(2x+y)+f(2x y) is not defined. This is a contradiction. So, 4f(y) + f(x) = 0. Hence,itfollowsthatf(x) = f(y) = 4f(y)+f(x) = 0.However,inthecase4f(y)+f(x) = 0,there is no problem in the definition of (2). In the following theorem, we obtain an approximation for approximate quadratic reciprocal mappings on nonzero real numbers. Theorem 2. Let f:r R be a mapping for which there exists a constant ε (independent of x and y) such that the functional inequality (3) 8 9 ε (4) holds for all x, y R.Thenthelimit Q (x) = lim n 3 2n f(x ) (5) 3n exists for all x R, n Nand Q:R R is the unique mapping satisfying the Rassias quadratic reciprocal functional equation (2),suchthat f (x) Q(x) ε (6) Moreover,thefunctionalidentity Q (x) = 3 2n Q(x ) (7) 3n holds for all x R and n N. Proof. Putting y=xin (4), we get f (3x) 3 2 f (x) 8ε 9 Thuswehave f (x) 3 2 f(x 3 ) 8ε (9) 9 Substitutingx by x/3 in (9) and then dividing both sides by 3 2,weobtain (8) 3 2 f(x 3 ) 3 4 f(x 3 2 ) 8ε (0) 3 4 It follows from (9)and(0)that f (x) 3 4 f(x 3 2 ) 8ε 9 ( + 3 2 ) () The above process can be repeated to obtain f (x) 3 2n f(x 3 n ) 8ε 9 ( + 3 2 + 3 4 + + 3 ) 2(n ) (2) for all x R and all n N. In order to prove the convergence of the sequence {(/3 2n )f(x/3 n )},wehaveifn>k>0,then by (2) 3 2n f(x ) 3n 3 f(x 2k 3 ) k = 3 2k 3 ( x 2(n k) 3 n ) f(x 3 ) k = y f( 3 2k 32(n k) 3 ) f(y) (n k) ε 8 3 2k 9 ( + 3 2 + 3 4 + + 3 ) 2(n k ) =(3 2k 3 2n )ε 3 2k ε (3) for all x R in which y=x/3 k.theaboverelationshows that the mentioned sequence is a Cauchy sequence and thus limit (5) exists Takingthatn tends to infinity in (2), we can see that inequality (6) holdsforallx R. Replacing x, y by x/3 n, y/3 n,respectively,in(4) and dividing both sides by 3 2n,wededucethat 2x + y 2x y 3 2n f( 3 n )+f( 3 n ) 2f (x/3n )f(y/3 n )[4f(y/3 n )+f(x/3 n )] (4f (y/3 n ) f(x/3 n )) 2 (4) 8ε 3 2(n+) holds for all x, y R.Allowingn in (4), we see that Q satisfies (2) forallx, y R.ToprovethatQ is a unique quadratic reciprocal function satisfying (2)subjectto(6), let us consider a Q : R R to be another quadratic reciprocal function which satisfies (2) and inequality (6). Clearly Q and Q satisfy (7)andusing(6), we get Q (x) Q (x) = lim n lim n 3 2n Q(x 3 n ) Q( x 3 n ) 3 2n [ Q(x 3 n ) f(x 3 n ) + f(x 3 n ) Q ( x 3 n ) ] 2ε lim =0, n 32n This shows the uniqueness of Q. (5)

Function Spaces 3 3. Gsvruua Stability of (2) Theorem 3. Let l {, }be fixed. Suppose that F:R R R is a function such that x F( 32l(n l) 3, x )< ln (6) 3ln n= Assume in addition that f:r R is a function which satisfies the functional inequality F(x,y) (7) holds for all x, y R. Then there exists a unique quadratic reciprocal function Q:R R which satisfies the Rassias equation (2) and the inequality f (x) Q(x) n= l+ /2 x F( 32l(n l) 3, x ln 3 ) ln (8) Proof. We prove the result only in the case that l=.another case is similar. Putting y=xin (7), we have f (3x) 3 2 f (x) F(x, x) (9) Replacingxby x/3 in the above inequality, we get f (x) 3 2 f (x 3 ) F(x 3, x 3 ) (20) Replacingx by x/3 n in (20) and then dividing both sides by 3 2n,wehave 3 2n f(x 3 n ) x f( 32(n+) 3 n+ ) 3 2n F(x 3 n, x ) (2) 3n for all x R and all nonnegative integers n.thusthesequence {(/3 2n )f(x/3 n )} is Cauchy by (6) and so this sequence is convergent. Indeed, lim n 3 2n f(x )=Q(x), (22) 3n On the other hand, by using (20) and applying mathematical induction to a positive integer n,weobtain f (x) 3 2n f ( x n 3 n ) 3 F ( x 2(k ) 3, x ) k 3k (23) k= for all x R and all n N.Lettingn in (23)andusing (22), one sees that inequality (8) holdsforallx R. Replacing x, y by x/3 n, y/3 n in (7) and dividing both sides by 3 2n,wededucethat 2x + y 2x y 3 2n f( 3 n )+f( 3 n ) 2f (x/3n )f(y/3 n )[4f(y/3 n )+f(x/3 n )] (4f (y/3 n ) f(x/3 n )) 2 3 2n F(x 3 n, x 3 n ) (24) holds for all x, y R.Takingn in (24), we see that Q satisfies (2) forallx, y R.Now,letQ : R R be another quadratic reciprocal function which satisfies (2)and inequality (8). Obviously Q and Q satisfy (7). Using (8), we get Q (x) Q (x) = 3 2n Q(x 3 n ) Q ( x 3 n ) 3 2n [ Q(x 3 n ) f(x 3 n ) 2 3 2n k= + f(x 3 n ) Q ( x 3 n ) ] x F( 32(k ) 3, x k+n 3 ) k+n 2 x = F( 32(k+n ) 3, x k+n 3 ) k+n k= 2 = 3 F(x 2(k ) 3, x k 3 ) k k=n (25) Takingn in the preceding inequality, we immediately find the uniqueness of Q.For l=,we obtain f (x) 32n f(3 n n x) 3 2(k+) F(3 k x, 3 k x) (26) k=0 fromwhichonecanprovetheresultbyasimilartechnique. This completes the proof. Corollary 4. Let α, r be nonnegative real numbers with r = 2.Supposethatf:R R is a function which satisfies the functional inequality f(2x+y)+f(2x y) α( x r + y r ) (27) for all x, y R. Then there exists a unique quadratic reciprocal function Q:R R that satisfies the Rassias equation (2) and the inequality f (x) Q(x) 8α x r (28) 3r+2

4 Function Spaces Proof. Letting F(x, y) = α( x r + y r ) in Theorem 3, wecan get the result. Corollary 5. Let α, r, s be nonnegative real numbers such that ρ=r+s= 2.Supposethatf:R R is a function which satisfies the functional inequality α x r y s (29) for all x, y R. Then there exists a unique quadratic reciprocal function Q:R R that satisfies the Rassias equation (2) and the inequality f (x) Q(x) 9α x ρ (30) 3ρ+2 Proof. Defining F(x, y) = α x r y s and applying Theorem 3, one can obtain the desired result. The proof of the following corollary is similar to the above results, so it is omitted. Corollary 6. Let α, r be nonnegative real numbers with r =.Supposethatf:R R is a function which satisfies the functional inequality α( x r y r + x 2r + y 2r ) (3) for all x, y R. Then there exists a unique quadratic reciprocal function Q:R R that satisfies the Rassias equation (2) and the inequality f (x) Q(x) 27α x 2r (32) 32r+2 Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgment The authors sincerely thank the anonymous reviewers for their careful reading, constructive comments, and fruitful suggestions to improve the quality of the first draft. References [] S. M. Ulam, Problems in Modern Mathematics, chapter6,john Wiley & Sons, New York, NY, USA, 940. [2] 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.222 224,94. [3] D. G. Bourgin, Classes of transformations and bordering transformations, Bulletin of the American Mathematical Society, vol. 57, pp. 223 237, 95. [4] P.M.Gruber, Stabilityofisometries, Transactions of the American Mathematical Society,vol.245,pp.263 277,978. [5] T. Aoki, On the stability of the linear transformation in Banach spaces, the Mathematical Society of Japan,vol.2,pp. 64 66, 950. [6] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297 300, 978. [7] P. Găvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, Mathematical Analysis and Applications,vol.84,no.3,pp.43 436,994. [8] F. Skof, Local properties and approximation of operators, Rendiconti del Seminario Matematico e Fisico di Milano,vol.53, pp. 3 29, 983. [9]P.W.Cholewa, Remarksonthestabilityoffunctionalequations, Aequationes Mathematicae, vol.27,no.-2,pp.76 86, 984. [0] A. Bodaghi and I. A. Alias, Approximate ternary quadratic derivations on ternary Banach algebras and C -ternary rings, Advances in Difference Equations,vol.202,article,202. [] S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg,vol.62,pp.59 64,992. [2] M.EshaghiGordjiandA.Bodaghi, OntheHyers-Ulam-Rassias stability problem for quadratic functional equations, East Journal on Approximations,vol.6,no.2,pp.23 30,200. [3]J.R.Lee,J.S.An,andC.Park, Onthestabilityofquadratic functional equations, Abstract and Applied Analysis,vol.2008, ArticleID62878,8pages,2008. [4] T. M. Rassias, On the stability of the quadratic functional equation and its applications, Studia Universitatis Babes-Bolyai,vol. 43, no. 3, pp. 89 24, 998. [5] K. Ravi and B. V. Senthil Kumar, Ulam-Gavruta-Rassias stability of Rassias Reciprocal functional equation, Global Applied Mathematics and Mathematical Sciences,vol.3,no.-2, pp. 57 79, 200. [6] K. Ravi, J. M. Rassias, and B. V. Senthil Kumar, Generalized Hyers-Ulam stability and examples of non-stability of reciprocal type functional equation in several variables, Indian Science,vol.2,no.3,pp.3 22,203. [7] K. Ravi, J. M. Rassias, and B. V. Senthil Kumar, Stability of reciprocal difference and adjoint functional equations in paranormed spaces: direct and fixed point methods, Functional Analysis, Approximation and Computation,vol.5,no.,pp.57 72, 203. [8] K. Ravi, J. M. Rassias, and B. V. Senthil Kumar, Example for the stability of reciprocal type functional equation controlled by product of different powers in singular case, Advances in Pure Mathematics,vol.6,pp.682 6825,203. [9] K.Ravi,J.M.Rassias,andB.V.SenthilKumar, Ulamstabilityof generalized reciprocal functional equation in several variables,

Function Spaces 5 International Applied Mathematics & Statistics,vol.9, no.d0,pp. 9,200. [20] K.Ravi,J.M.Rassias,andB.V.SenthilKumar, Afixedpoint approach to the generalized Hyers-Ulam stability of reciprocal difference and adjoint functional equations, Thai Mathematics,vol.8,no.3,pp.469 48,200. [2] K.Ravi,J.M.Rassias,andB.V.SenthilKumar, Ulamstabilityof reciprocal difference and adjoint functional equations, The AustralianJournalofMathematicalAnalysisandApplications, vol.8,no.,article3,pp. 8,20. [22] K. Ravi, E. Thandapani, and B. V. Senthil Kumar, Stability of reciprocal type functional equations, Panamerican Mathematical Journal,vol.2,no.,pp.59 70,20.

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