Research Article On the Stability of a Functional Equation Associated with the Fibonacci Numbers

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1 Abstract and Applied Analysis, Article ID , 6 pages Research Article On the Stability of a Functional Equation Associated with the Fibonacci Numbers Cristinel Mortici,,2 Michael Th Rassias, 3 and Soon-Mo Jung 4 Valahia University of Târgovişte, Bulevardul Unirii 8, Târgovişte, Romania 2 Academy of Romanian Scientists, Splaiul Independenţei 54, Bucharest, Romania 3 Department of Mathematics, ETH Zürich, Raemistrasse 0, 8092 Zürich, Switzerland 4 Mathematics Section, College of Science and Technology, Hongik University, Sejong , Republic of Korea Correspondence should be addressed to Soon-Mo Jung; smjung@hongikackr Received 5 May 204; Accepted 8 July 204; Published 20 July 204 Academic Editor: Chengjian Zhang Copyright 204 Cristinel Mortici 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 We prove the Hyers-Ulam stability of the generalized Fibonacci functional equation F(x) g(x)f(h(x)) = 0, where g and h are given functions Introduction In 940, Ulam [] gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems Among them was the question concerning the stability of group homomorphisms Let G be a group and let G 2 be a metric group with the metric d(, ) Given > 0, does there exist a δ > 0 such that if a function h : G G 2 satisfies the inequality d(h(xy), h(x)h(y)) < δ, forallx, y G, then there exists a homomorphism H:G G 2 with d(h(x), H(x)) <, for all x G? The case of approximately additive functions was solved by Hyers [2] under the assumption that G and G 2 are Banach spacesindeed,heprovedthefollowingtheorem Theorem Let f:g G 2 be a function between Banach spaces such that f (x+y) f(x) f(y), () for some >0and for all x, y G Then,thelimit A (x) = lim n 2 n f(2 n x) (2) exists for each x G,andA:G G 2 is the unique additive function such that f (x) A(x), (3) for any x G Moreover,iff(tx) is continuous in t, foreach fixed x G, then the function A is linear Hyers proved that each solution of the inequality f(x + y) f(x) f(y) can be approximated by an exact solution; say an additive function In this case, the Cauchy additive functional equation, f(x+y)=f(x)+f(y),issaid to have the Hyers-Ulam stability Sincethen,thestabilityproblemsofalargevarietyof functional equations have been extensively investigated by several mathematicians (cf [3 4]) In this paper, we investigate the Hyers-Ulam stability of the functional equation F (x) g(x) F (h (x)) =0, (4) where g and h are given functions In Section 2, we prove that the functional equation (4) has a large class of nontrivial solutions Section 3 is devoted to the investigation of the Hyers-Ulam stability problems for (4) In the last section, we prove the Hyers-Ulam stability of (4) when g is a constant function, which is a generalization

2 2 Abstract and Applied Analysis of the papers [4, 7, 4] More precisely, Jung [7] provedthe Hyers-Ulam stability of the generalized Fibonacci functional equation f (x) =pf(x ) qf(x 2) (5) in the class of functions f:r X,whereX is a real (or complex) Banach space Theorem 2 (see [7, Theorem 3]) Assume that the quadratic equation x 2 px+q=0has real solutions a and b with 0< b < < a Ifafunctionf:R Xsatisfies the inequality f (x) pf(x ) +qf(x 2) (6) for all x Rand for some >0, then there exists a unique solution F:R Xof (5) such that f (x) F(x) a b a b ( a )( b ) Asimilarcasefor0 < b < < a with b =/2was investigated by Brzdęk et al [4] andtrif[4] whoobtained the estimate f (x) F(x) 4 2 b (2 a ) (8) If either 0 < b < /2 and a > 3/2 b or /2 < b < 3/4 and a > (5 6 b )/(6 8 b ), then the inequality (7)issharper than that of (8) In Section 4 of this paper, we improve the results of papers [4, 7, 4] inthesensethatweestimate f(x) F(x) even when both a and b are larger or smaller than Moreover, we deal with a functional equation (4) that is regarded as a more generalized form of the Fibonacci functional equation (5) In this paper, R, Z, andn stand for the sets of real numbers, integers, and positive integers, respectively 2 Solutions of (4) Evidently, (4) admits the trivial solution F=0 In order to avoid the trivial case, we search in this section for a class of nontrivial solutions of (4) Let D be a subset of RAfunctionh:D Dis said to be of disjoint iterated images, shortly (DII)-function, if (i) there exists a partition (7) D= D n ; (9) n (ii) h maps bijectively D n onto D n+ for each integer n As an example for a (DII)-function, we introduce a function h : (0, ] (0, ] defined by h (x) = n+2 (nx + n+ ), x D n := ( n+, n ], (0) for all n N Foreveryn N, this function is linear on D n and it transforms each D n onto D n+ We are now in a position to prove that the set of all solutions of (4)isnotemptybutitisaninfiniteset Theorem 3 Let h:d Dbe a (DII)-function and g:d R \{0} There is a one-to-one correspondence between the set of all solutions F:D R of the functional equation (4) and the set of all functions φ:d R Proof Given a φ:d R, we define a function F on D as F (x) =φ(x), () for all x D Assume that F is defined on D n for some n 2Ifx D n,thenh (x) D n and we put F (x) = g(h (x)) F (h (x)), (2) for all x D n By this inductive procedure, F is completely defined We now show that F is a solution of (4) Let z be any point of D and let n 2be an integer such that h(z) D n Put x=h(z)in (2)toget F (h (z)) = F (z), (3) g (z) which is (4) Conversely, we associate to every solution F of (4) the function φ=f D We notice that a (DII)-function h is injective as we see the following: if x, y D n for some n Nwith x =ybut h(x) = h(y), thenh(x) = h(y) D n+ and, hence, x=ybecause h maps bijectively D n onto D n+, a contradiction If x D m and y D n for some m, n N with m =n,itisthenobvious that h(x) = h(y) because h(x) D m+, h(y) D n+,and D m+ D n+ =0Buth is not surjective, since Imh = D \ D We now study the set of solutions of (4) under the assumption that h:d Dis a bijection For any pair of points x, y D, weusethenotationx yif there exists a k Zwith y=h k (x)since isanequivalencerelationin D,let D= Δ i (4) i I be the corresponding partition in -equivalence classes Δ i = x i (i I);thatis, Δ i ={h k (x i ) k Z} (5) Theorem 4 Given a subset D of R, leth : D D be a bijective function and g:d R \{0} Assume that D= Δ i (6) i I

3 Abstract and Applied Analysis 3 is a partition of D corresponding to the equivalence relation with the property (5) Then, there exists a one-to-one correspondence between the set of all solutions F:D R of the functional equation (4) and the set of all real sequences {y i } i I Proof For any real sequence {y i } i I, we define F(x i ) = y i for all i I,whereI is the index set for the partition corresponding to the equivalence relation with the property (5) We further define the function F:D R by F(h(x i )) = g(x i ) F(x i)= g(x i ) y i, F(h (x i )) = g (h (x i )) F (x i )=g(h (x i )) y i (7) In general, if F is defined at h k (x i ) and h k (x i ),thenf is defined at h k+ (x i ) and h k (x i ) by F(h k+ (x i )) = g(h k (x i )) F(hk (x i )), F(h k (x i )) = g (h k (x i )) F (h k (x i )) (8) For each i I, we can use such an inductive procedure to define the function F on Δ i and we see that F Δ i is uniquely determined by the value of y i Conversely, every solution F : D R of (4) canbe associated to the real sequence {F(x i )} i I Corollary 5 Given a subset D of R, leth : D D be a bijective function and α R \{0} Assume that D= Δ i (9) i I is a partition of D corresponding to the equivalence relation with the property (5) Then there exists a one-to-one correspondence between the set of all solutions F:D R of the functional equation and the set of all real numbers {y i } i I 3 Hyers-Ulam Stability of (4) F (x) αf(h (x)) =0 (20) The above conditions imposed on the function h were necessary for showing that the functional equations (4) and (20) havelargeclassesofnontrivialsolutionsthestability results presented in the sequel are valid also under weaker conditions as we shall see in the following theorems Theorem 6 Given real numbers a and b with a<b,leth: (a, b) (a, b) and g : (a, b) I be given functions, where I (0, ) is an interval of length l Assume that a bounded functionf:(a,b) R satisfies the inequality f (x) g(x) f (h (x)) (2) for all x (a, b) and for some >0Then,foreveryα I, there exists a solution F:(a,b) R of (20) such that f (x) F(x) +l f α, (22) for any x (a, b),where f = sup x (a,b) f(x) Proof First, we prove that f (x) αf(h (x)) λ, (23) for all x (a, b),wherewesetλ=+l f Indeed, it follows from (2)that f (x) αf(h (x)) f (x) g(x) f (h (x)) + g (x) f (h (x)) αf(h (x)) + g (x) α f (h (x)) +l f =λ, (24) for every x (a, b) By replacing x with h k (x) and then multiplying with α k both sides of (23), we get αk f (h k (x)) α k+ f (h k+ (x)) λαk, (25) for all x (a, b) and k NSince we have f (x) αn f(h n (x)) n αk f(h k (x)) α k+ f(h k+ (x)) n λα k =λ αn α, (26) f (x) αn f(h n (x)) λ αn α, (27) for any x (a, b) and n N The inequality (25) shows that the sequence {α n f(h n (x))} n N is a Cauchy sequence for every x (a, b) Thus, we can define a function F:(a,b) R by F (x) = lim n αn f(h n (x)), (28) for all x (a, b) Hence, it follows from (23)that F (x) αf(h (x)) = lim n αn f(h n (x)) lim n αn+ f(h n+ (x)) = lim n αn (f (h n (x)) αf(h n+ (x))) (29) lim n αn λ=0, for each x (a, b), which implies that F is a solution of (20) Finally, inequality (22) is an immediate consequence of (27) if we take the limit as n

4 4 Abstract and Applied Analysis 4 When g Is Constant In the case of g(x) = s {, +} for all x R,weinvestigate the Hyers-Ulam stability of the functional equation F (x) sf(h (x)) =0, (30) where h:r R and F:R Xare functions and X is a real Banach space Theorem 7 Let X be a real Banach space and let s be a real number with s < If a function f:r X satisfies the inequality f (x) sf(h (x)), (3) for all x Rand for some >0, then there exists a solution F:R Xof (30) such that f (x) F(x) s, (32) Proof By replacing x with h k (x) and multiplying with s k both sides of (3), we get sk f(h k (x)) s k+ f(h k+ (x)) s k, (33) for all x R and k {0,,2,}By(33), we have f (x) sn f(h n (x)) n sk f(h k (x)) s k+ f(h k+ (x)) n s k = s n s, for all x R and n NHence,weget (34) f (x) sn f (h n (x)) s n s, (35) for any x Rand n N The inequality (33) shows that the sequence {s n f(h n (x))} n N is a Cauchy sequence for any fixed x R Thus, since X is a complete space, we can define a function F:R Xby F (x) = lim n sn f(h n (x)), (36) It follows from (3)that F (x) sf(h (x)) = lim n sn f(h n (x)) lim n sn+ f(h n+ (x)) = lim n sn f(hn (x)) sf(h n+ (x)) lim n sn =0, (37) which implies that F is a solution of (30) Finally, the inequality (32)immediatelyfollowsfrom(35) provided that we take the limit as n Assume now that h : R R is bijective A similar theorem can be proved when s > Theorem 8 Let X bearealbanachspace,leth:r R be a bijective function, and let s R be given with s > Ifa function f:r Xsatisfies the inequality f (x) sf(h (x)), (38) for all x Rand for some >0, then there exists a solution F:R Xof (30) such that f (x) F(x) s, (39) Proof By replacing x with h (x) and dividing by s both sides of (38), we get f (x) s f(h (x)) s, (40) for any x R Sincetheconstant/ s is less than, our assertion follows fromtheorem 7Inparticular,wehave F (x) = lim n s n f(h n (x)), (4) for each x R Corollary 9 Let X be a real Banach space, let h:r R be a bijective function, and let s Rbe given with s =Ifa function f:r Xsatisfies the inequality f (x) sf(h (x)), (42) for any x Rand for some >0, then there exists a solution F:R Xof (30) such that f (x) F(x) s, (43) By combining the results of Theorems 7 and 8, wecan present a stability result of the following functional equation F (x) pf(h (x)) +qf(h (h (x))) =0, (44) where h : R R is bijective and the range space of the function F:R Xis a real Banach space Theorem 0 Let p and q be given real numbers such that the quadratic equation x 2 px+q = 0has distinct real solutions a and b with a =and b = Assume that a bijective function h:r R is given and X is a real Banach space If a function f:r Xsatisfies f (x) pf(h (x)) +qf(h (h (x))), (45) for all x Rand for some >0, then there exists a solution F:R Xof (44) such that f (x) F(x) a b ( + ), a b (46)

5 Abstract and Applied Analysis 5 As we mentioned in the Introduction, our result extends Jung s result in [7], since for any x R,whereλ is a real number Then, it follows from (50)and(54)that a b ( + a = a b a b ) b ( a )( b ) (47) F λ (x) pf λ (h (x)) +qf λ (h (h (x))) = F λ (x) bf λ (h (x)) a(f λ (h (x)) bf λ (h (h (x)))) = ( λ)(u(x) bu(h (x)) a (U (h (x)) bu(h (h (x))))) when 0 < b < < a Moreover, Jung s result is a particular case of Theorem 0 when we set h(x) = x in (45) Proof of Theorem 0 If we set then the inequality (45)yields u (x) =f(x) af(h (x)), (48) u (x) bu(h (x)), (49) According to Corollary 9, there exists a solution U:R Xof with for any x R If we set U (x) bu(h (x)) =0 (50) u (x) U(x) then the inequality (45)yields b, (5) V (x) =f(x) bf(h (x)), (52) V (x) av (h (x)), (53) for any x R InviewofCorollary 9 again, there exists a solution V: R Xof V (x) av(h (x)) =0 (54) +λ(v (x) bv(h (x)) a(v (h (x)) bv(h (h (x))))) = λ(v (x) bv(h (x)) a(v (h (x)) bv(h (h (x))))) = λ (( b a )V(x) a( b )V(h (x))) a = λ( b a ) (V (x) av(h (x))) = 0, (57) for all x R, which implies that F λ is a solution of (44) for every fixed real number λ We now set λ= a (58) b a and assert that the function F (x) = b U (x) a V (x) (59) b a b a satisfies the requirements of this theorem Indeed, it follows from (5)and(55)that f (x) ( b U (x) a b a b a V (x)) = b a (b a) f (x) (bu (x) av(x)) b b a U (x) (f(x) af(h (x))) + a b a V (x) (f(x) bf(h (x))) (60) = b U (x) u(x) + a V (x) V (x) b a b a b b a b + a b a a with V (x) V(x) a, (55) = for any x R a b ( + a ), b We now define a function F:R Xby F λ (x) = ( λ) U (x) +λv(x), (56) Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper

6 6 Abstract and Applied Analysis Acknowledgments Soon-Mo Jung was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no 203RAA ) The work of Cristinel Mortici was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI Project no PN- II-ID-PCE Cristinel Mortici s final remarks for improving this paper were made during his visit at CERN laboratories in Geneva, Switzerland References [] S M Ulam, ACollectionofMathematicalProblems,Interscience, New York, NY, USA, 960 [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,vol27,pp ,94 [3] JBrzdękandS-MJung, Anoteonstabilityofalinearfunctional equation of second order connected with the Fibonacci numbers and Lucas sequences, Inequalities and Applications, vol 200, Article ID , 0 pages, 200 [4] J Brzdęk, D Popa, and B Xu, Note on nonstability of the linear recurrence, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg,vol76,pp83 89,2006 [5] JBrzdęk,DPopa,andBXu, Hyers-Ulamstabilityforlinear equations of higher orders, Acta Mathematica Hungarica, vol 20,no-2,pp 8,2008 [6] S Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, London, UK, 2002 [7] S-M Jung, Functional equation f(x) = pf(x ) qf(x 2) and its Hyers-Ulam stability, Inequalities and Applications, vol 2009, Article ID 8678, 0 pages, 2009 [8] S-M Jung, Hyers-Ulam stability of Fibonacci functional equation, Bulletin: Iranian Mathematical Society,vol35,no2, pp , 2009 [9]S-MJung,DPopa,andMTRassias, Onthestabilityof the linear functional equation in a single variable on complete metric groups, Global Optimization, vol 59, no, pp 65 7, 204 [0] S-M Jung and M T Rassias, A linear functional equation of third order associated with the Fibonacci numbers, Abstract and Applied Analysis,vol204,ArticleID37468,7pages,204 [] M Kuczma, Functional Equations in a Single Variable, Polish Scientific, Warszawa, Poland, 968 [2] Y-H Lee, S-M Jung, and M T Rassias, On an n-dimensional mixed type additive and quadratic functional equation, Applied Mathematics and Computation,vol228,pp3 6,204 [3] D Popa, Hyers-Ulam stability of the linear recurrence with constant coefficients, Advances in Difference Equations, no 2, pp0 07,2005 [4] T Trif, Hyers-Ulam-Rassias stability of a linear functional equation with constant coefficients, Nonlinear Functional Analysis and Applications,vol,no5,pp88 889,2006

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