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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 945010, 7 pages doi:10.1155/2008/945010 Research Article A Fixed Point Approach to the Stability of a Functional Equation of the Spiral of Theodorus Soon-Mo Jung1 and John Michael Rassias2 1 Mathematics Section, College of Science and Technology, Hong-Ik University, 339-701 Chochiwon, South Korea 2 Mathematics Section, Pedagogical Department, National and Capodistrian University of Athens, 4 Agamemnonos Street, Aghia Paraskevi, Attikis, 15342 Athens, Greece Correspondence should be addressed to John Michael Rassias, jrassias@primedu.uoa.gr Received 2 April 2008; Accepted 26 June 2008 Recommended by Fabio Zanolin Cădariu and Radu applied the fixed point method to the investigation of Cauchy and Jensen functional equations. In this paper, we adopt the idea of Cădariu and √ Radu to prove the stability of a functional equation of the spiral of Theodorus, fx  1  1  i/ x  1fx. Copyright q 2008 S.-M. Jung and J. M. Rassias. 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 In 1940, Ulam 1 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 those was the question concerning the stability of group homomorphisms: let G1 be a group and let G2 be a metric group with the metric d·, ·. Given ε > 0, does there exist a δ > 0 such that if a function h : G1 → G2 satisfies the inequality dhxy, hxhy < δ for all x, y ∈ G1 , then there exists a homomorphism H : G1 → G2 with dhx, Hx < ε for all x ∈ G1 ? The case of approximately additive functions was solved by Hyers 2 under the assumption that G1 and G2 are Banach spaces. Indeed, he proved that each solution of the inequality fx  y − fx − fy ≤ ε, for all x and y, can be approximated by an exact solution, say an additive function. Later, the result of Hyers was significantly generalized for additive mappings by Aoki 3 see also 4 and for linear mappings by Rassias 5. It should be remarked that we can find in the books 6–8 a lot of references concerning the stability of functional equations see also 9–11. Recently, Jung √ and Sahoo 12 proved the generalized Hyers-Ulam stability of the functional equation f r 2  1  frarctan1/r which is closely related to the square root spiral, for the case that f1  0 and fr is monotone increasing for r > 0 see also 13, 14. 2 Fixed Point Theory and Applications In 2003, Cădariu and Radu 15 applied the fixed point method to the investigation of Jensen’s functional equation see 16–19. Using such a clever idea, they could present a short and simple proof for the stability of the Cauchy functional equation. In 20, Gronau investigated the solutions of the Theodorus functional equation   i fx  1  1  √ fx, x1 where i  √ 1.1 −1. The function T : −1, ∞ → C defined by √ ∞  1  i/ k T x  √ k1 1  i/ x  k 1.2 is called the Theodorus function. Theorem 1.1. The unique solution f : −1, ∞ → C of 1.1 satisfying the additional condition that lim n→∞ fx  n 1 fn 1.3 for all x ∈ 0, 1 is the Theodorus function. Theorem 1.2. If f : −1, ∞ → C is a solution of 1.1 such that f0  1, |fx| is monotonic and argfx is monotonic and continuous, then f is the Theodorus function. Theorem 1.3. If f : −1, ∞ → C is a solution of 1.1 such that f0 √  1, |fx| and argfx are monotonic and such that argfn  1  argfn  arg 1  i/ n  1 for any n ∈ {0, 1, 2, . . .}, then f is the Theodorus function. In this paper, we will adopt the idea of Cădariu and Radu and apply a fixed point method for proving the Hyers-Ulam-Rassias stability of the Theodorus functional equation 1.1. 2. Preliminaries Let X be a set. A function d : X × X → 0, ∞ is called a generalized metric on X if and only if d satisfies M1  dx, y  0 if and only if x  y; M2  dx, y  dy, x for all x, y ∈ X; M3  dx, z ≤ dx, y  dy, z for all x, y, z ∈ X. Note that the only substantial difference of the generalized metric from the metric is that the range of generalized metric includes the infinity. We now introduce one of fundamental results of fixed point theory. For the proof, refer to 21. Theorem 2.1. Let X, d be a generalized complete metric space. Assume that Λ : X → X is a strictly contractive operator with the Lipschitz constant L < 1. If there exists a nonnegative integer k such that dΛk1 f, Λk f < ∞ for some f ∈ X, then the following are true. S.-M. Jung and J. M. Rassias 3 a The sequence {Λn f} converges to a fixed point F of Λ; b F is the unique fixed point of Λ in   X ∗  g ∈ X | dΛk f, g < ∞ ; 2.1 c If h ∈ X ∗ , then dh, F ≤ 1 dΛh, h. 1−L 2.2 3. Main results In the following theorem, by using the idea of Cădariu and Radu see 15, 16, we will prove the Hyers-Ulam-Rassias stability of the functional equation 1.1 for the spiral of Theodorus. Theorem 3.1. Given a constant a > 0, suppose ϕ : a, ∞ → 0, ∞ is a function and there exists a constant L, 0 < L < 1, such that ϕx  1  √ 1 x1 ϕx ≤ Lϕx for all x ≥ a. If a function f : a, ∞ → C satisfies the inequality       fx  1 − 1  √ i fx ≤ ϕx  x1 3.1 3.2 for all x ≥ a, then there exists a unique solution F : a, ∞ → C of 1.1, which satisfies   Fx − fx ≤ 1 ϕx 1−L 3.3 1 x  jm 3.4 for all x ≥ a. More precisely, F is defined by  Fx  lim n→∞ n k1 k  −ik 1≤j1 ≤···≤jk ≤n1−k m1 fx  n − k  fx  n for all x ≥ a. Proof. We set X  {h | h : a, ∞ → C is a function} and introduce a generalized metric on X as follows:     dg, h  inf C ∈ 0, ∞ | gx − hx ≤ Cϕx, ∀x ≥ a . 3.5 First, we will verify that X, d is a complete space. Let {gn } be a Cauchy sequence in X, d. According to the definition of Cauchy sequences, there exists, for any given ε > 0, a positive integer Nε such that dgm , gn  ≤ ε for all m, n ≥ Nε . From the definition of the generalized metric d, it follows that ∀ ε > 0 ∃ Nε ∈ N ∀ m, n ≥ Nε ∀ x ≥ a : |gm x − gn x| ≤ εϕx. 3.6 4 Fixed Point Theory and Applications If x ≥ a is fixed, 3.6 implies that {gn x} is a Cauchy sequence in C, |·|. Since C, |·| is complete, {gn x} converges in C, |·| for each x ≥ a. Hence we can define a function g : a, ∞ → C by gx  lim gn x. 3.7 n→∞ If we let m increase to infinity, it follows from 3.6 that for any ε > 0, there exists a positive integer Nε with |gn x − gx| ≤ εϕx for all n ≥ Nε and all x ≥ a, that is, for any ε > 0, there exists a positive integer Nε such that dgn , g ≤ ε for any n ≥ Nε . This fact leads us to the conclusion that {gn } converges in X, d. Hence X, d is a complete space cf. the proof of 22, Theorem 3.1 or 16, Theorem 2.5. We now define an operator Λ : X → X by Λhx  hx  1 − √ i x1 hx x ≥ a 3.8 for any h ∈ X. We assert that Λ is strictly contractive on X. Given g, h ∈ X, let C ∈ 0, ∞ be an arbitrary constant with dg, h ≤ C, that is, |gx − hx| ≤ Cϕx 3.9 for all x ≥ a. It then follows from 3.1 and 3.8 that       Λgx − Λhx ≤ gx  1 − hx  1  √ 1 gx − hx x1 C ϕx ≤ Cϕx  1  √ x1 ≤ LCϕx 3.10 for every x ≥ a, that is, dΛg, Λh ≤ LC. Hence we conclude that dΛg, Λh ≤ Ldg, h, for any g, h ∈ X. Next, we assert that dΛf, f < ∞. In view of 3.2 and the definition of Λ, we get   Λfx − fx ≤ ϕx 3.11 dΛf, f ≤ 1. 3.12 for each x ≥ a, that is, By using mathematical induction, we now prove that Λ fx  n n k1 k  k −i 1≤j1 ≤···≤jk ≤n1−k m1 1 x  jm fx  n − k  fx  n 3.13 S.-M. Jung and J. M. Rassias 5 for all n ∈ N and all x ≥ a. Since f ∈ X, the definition 3.8 implies that 3.13 is true for n  1. Now, assume that 3.13 holds true for some n ≥ 1. It then follows from 3.8 and 3.13 that      i Λn1 f x  Λn f x  1 − √ Λn f x x1 n  n   Λ f x  1  −ik1 1j1 ≤···≤jk1 ≤n1−k k1 k1  × 1 x  jm m1 fx  n − k − √ i x1 fx  n n−1    Λn f x  1  −ik1 1j1 ≤···≤jk1 ≤n1−k k1 k1  × 1 x  jm m1  n fx  n − k  −i k  −ik 1≤j1 ≤···≤jk ≤n1−k k1  fx  1  n  n m1 − √  n 1j1 ≤···≤jk ≤n2−k  fx  n  −i k  −ik k1 2≤j1 ≤···≤jk ≤n2−k  fx  n  1  n 1≤j1 ≤···≤jn1 ≤1 k1 k  −ik 1≤j1 ≤···≤jk ≤n2−k m1 n1 m1 n1 x1 fx − √ i x1 fx  n 1 x  jm fx  n  1 − k fx x1 1 fx  n  1 − k x  jm 1j1 ≤···≤jk ≤n2−k m1 1 fx  1  n − k k  n1   −in1 n1 m1 √ 1 −ik k1  x  1  jm  k  n1 x1 √ 1 −ik k2 i  n1 1 x  jm 1 x  jm m1 1 x  jm fx  n  1 − k fx fxn1−k fxn 1, 3.14 which is the case when n is replaced by n  1 in 3.13. Considering 3.12, if we set k  0 in Theorem 2.1, then Theorem 2.1a implies that there exists a function F ∈ X, which is a fixed point of Λ, such that dΛn f, F → 0 as n → ∞. Hence, we can choose a sequence {Cn } of positive numbers with Cn → 0 as n → ∞ such that dΛn f, F ≤ Cn for each n ∈ N. In view of definition of d, we have  n    Λ f x − Fx ≤ Cn ϕx x ≥ a 3.15 for all n ∈ N. This implies the pointwise convergence of {Λn fx} to Fx for every fixed x ≥ a. Therefore, using 3.4, we can conclude that 3.4 is true. 6 1.1. Fixed Point Theory and Applications Moreover, because F is a fixed point of Λ, definition 3.8 implies that F is a solution to Since k  0 see 3.12 and f ∈ X ∗  {g ∈ X | df, g < ∞} in Theorem 2.1, by Theorem 2.1c and 3.12, we obtain df, F ≤ 1 1 dΛf, f ≤ , 1−L 1−L 3.16 that is, the inequality 3.3 is true for all x ≥ a. Assume that inequality 3.3 is also satisfied with another function G : a, ∞ → C which √ is a solution of 1.1. As G is a solution of 1.1, G satisfies that Gx  Gx  1 − i/ x  1Gx  ΛGx for all x ≥ a. In other words, G is a fixed point of Λ. In view of 3.3 with G and the definition of d, we know that df, G ≤ 1 < ∞, 1−L 3.17 that is, G ∈ X ∗  {g ∈ X | df, g < ∞}. Thus, Theorem 2.1b implies that F  G. This proves the uniqueness of F. Indeed, Cădariu and Radu proved a general theorem concerning the Hyers-UlamRassias stability of a generalized equation for the square root spiral   f p−1 px  k  fx  hx 3.18 see 23, Theorem 3.1. References 1 S. M. Ulam, A Collection of Mathematical Problems, vol. 8 of Interscience Tracts in Pure and Applied Mathematics, Interscience, New York, NY, USA, 1960. 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, 1941. 3 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. 4 D. G. Bourgin, “Classes of transformations and bordering transformations,” Bulletin of the American Mathematical Society, vol. 57, pp. 223–237, 1951. 5 Th. 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, 1978. 6 S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002. 7 D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, vol. 34 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, 1998. 8 S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, Fla, USA, 2001. 9 G. L. Forti, “Hyers-Ulam stability of functional equations in several variables,” Aequationes Mathematicae, vol. 50, no. 1-2, pp. 143–190, 1995. 10 D. H. Hyers and Th. M. Rassias, “Approximate homomorphisms,” Aequationes Mathematicae, vol. 44, no. 2-3, pp. 125–153, 1992. 11 Th. M. Rassias, “On the stability of functional equations and a problem of Ulam,” Acta Applicandae Mathematicae, vol. 62, no. 1, pp. 23–130, 2000. 12 S.-M. Jung and P. K. Sahoo, “Stability of a functional equation for square root spirals,” Applied Mathematics Letters, vol. 15, no. 4, pp. 435–438, 2002. S.-M. Jung and J. M. Rassias 7 13 S.-M. Jung, “A fixed point approach to the stability of an equation of the square spiral,” Banach Journal of Mathematical Analysis, vol. 1, no. 2, pp. 148–153, 2007. 14 S.-M. Jung and J. M. Rassias, “Stability of general Newton functional equations for logarithmic spirals,” Advances in Difference Equations, vol. 2008, Article ID 143053, 5 pages, 2008. 15 L. Cădariu and V. Radu, “Fixed points and the stability of Jensen’s functional equation,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 4, 7 pages, 2003. 16 L. Cădariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,” in Iteration Theory (ECIT ’02), vol. 346 of Grazer Mathematische Berichte, pp. 43–52, Karl-Franzens-Univ., Graz, 2004. 17 V. Radu, “The fixed point alternative and the stability of functional equations,” Fixed Point Theory, vol. 4, no. 1, pp. 91–96, 2003. 18 J. M. Rassias, “Alternative contraction principle and Ulam stability problem,” Mathematical Sciences Research Journal, vol. 9, no. 7, pp. 190–199, 2005. 19 J. M. Rassias, “Alternative contraction principle and alternative Jensen and Jensen type mappings,” International Journal of Applied Mathematics & Statistics, vol. 4, no. M06, pp. 1–10, 2006. 20 D. Gronau, “The spiral of Theodorus,” The American Mathematical Monthly, vol. 111, no. 3, pp. 230–237, 2004. 21 J. B. Diaz and B. Margolis, “A fixed point theorem of the alternative, for contractions on a generalized complete metric space,” Bulletin of the American Mathematical Society, vol. 74, pp. 305–309, 1968. 22 S.-M. Jung and T.-S. Kim, “A fixed point approach to the stability of the cubic functional equation,” Boletı́n de la Sociedad Matemática Mexicanae. Tercera Serie, vol. 12, no. 1, pp. 51–57, 2006. 23 L. Cădariu and V. Radu, “Fixed point methods for the generalized stability of functional equations in a single variable,” Fixed Point Theory and Applications, vol. 2008, Article ID 749392, 15 pages, 2008.