FORMAL AND ANALYTIC SOLUTIONS FOR A QUADRIC ITERATIVE FUNCTIONAL EQUATION

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1 Electronic Journal of Differential Equations, Vol. 202 (202), No. 46, pp. 9. ISSN: URL: or ftp ejde.math.txstate.edu FORMAL AND ANALYTIC SOLUTIONS FOR A QUADRIC ITERATIVE FUNCTIONAL EQUATION PINGPING ZHANG Abstract. In this article, we study a quadric iterative functional equation. We prove the existence of formal solutions, and that every formal solution yields a local analytic solution when the eigenvalue of the linearization for the auxiliary function lying inside the unit circle, lying on the unit circle with a Brjuno number, or a root of.. Introduction Solving iterative functional equations is difficult since the unknown arises in the iteration [5, 2]. Using Schauder fixed point theorem, Zhang [22] proved the existence and uniqueness of solutions for a general iterative functional equation, the so-called polynomial-like iterative functional equation, λ x(t) + λ 2 x 2 (t) + + λ n x n (t) = F (t), t R. Later various properties of solutions of iterative functional equations, such as continuity, differentiability, monotonicity, convexity, analyticity, stability, have received much more attention; see e.g. [8] [5], [7] [20], [23]. Among these studies, the existence of analytic solutions caused more concerns since it is closely related to small divisors problem. In [3], analytic invariant curves for a planar map were obtained by solving the iterative functional equation x(z + x(z)) = x(z) + G(z) + H(z + x(z)), z C. We notice that [3] and [] are all based on eigenvalue of the linearization θ is inside the unit circle or a Diophantine number by using Schröder conversion and majorant series. On the other hand, Reich and his co-authors [8]-[0] have studied the formal solutions of a quadric iterative functional equation, called the generalized Dhombres functional equation, f(zf(z)) = ϕ(f(z)), z C, in the ring of formal power series C[[z]]. They described the structure of the set of all formal solutions when the eigenvalue θ of linearization is not a root of, and also showed every formal solutions yield a local analytic solutions when θ is not on 2000 Mathematics Subject Classification. 39B22, 34A25, 34K05. Key words and phrases. Iterative functional equation; analytic solution; small divisor; Brjuno condition. c 202 Texas State University - San Marcos. Submitted December 2, 20. Published March 23, 202.

2 2 P. ZHANG EJDE-202/46 the unit circle or a Diophantine number, as well as represent analytic solutions by infinite products for θ ling in the unit circle. In 2008, Xu and Zhang [8] studied the analytic solutions of a q-difference equation k j=0 t= C t,j (z)(x(q j z)) t = G(z), z C, (.) they obtained local analytic solutions under Brjuno condition, and proved noexistence of analytic solutions when the eigenvalue θ of linearization satisfies Cremer condition. Following that, Si and Li [5] discussed analytic solutions of the (.) with a singularity at the origin. In this article, we study the quadric iterative functional equation x(az + bzx(z)) = H(z) (.2) in the complex field, where x(z) is unknown function, H(z) is a given holomorphic function, a and b are nonzero complex parameters. It is a more complicated equation than the involutory function x 2 (t) = t, which is the Babbage equation with n = 2. We discuss the existence of formal solutions for (.2) when a is arbitrary nonzero complex number. Moreover, every formal solution yields a local analytic solution when a is lying inside the unit circle, lying on the unit circle with a Brjuno number or a root of. Our idea comes from [5]. Let y(z) = az + bzx(z). Then Therefore, x(z) = x(y(z)) = y(z) az. bz y(y(z)) ay(z), by(z) Then (.2) is equivalent to the functional equation y(y(z)) ay(z) = by(z)h(z). (.3) Using the conversion y(z) = g(θ(g (z))), Equation (.3) transforms into the equation without functional iteration Suppose g(θ 2 z) ag(θz) = bg(θz)h(g(z)). (.4) g(z) = a n z n, H(z) = h n z n. (.5) n= n= Substituting (.5) into (.4), we obtain (θ 2 aθ)a z + (θ 2(n+) aθ n+ )a n+ z n+ = b n= n= j= i +i 2 + +im=j; m=,2,...,j Comparing coefficients, we obtain θ n+ j a n+ j h m a i a i2... a im z n+. (.6) (θ 2 aθ)a = 0, (.7)

3 EJDE-202/46 FORMAL AND ANALYTIC SOLUTIONS 3 and (θ 2(n+) aθ n+ )a n+ = b j= i +i 2 + +im=j; m=,2,...,j θ n+ j a n+ j h m a i a i2... a im. (.8) Under a 0, the equality (.7) implies that θ = a, then (.8) turns into (θ n )θ n+2 a n+ = b θ n+ j a n+ j h m a i a i2... a im. (.9) j= i +i 2 + +im=j; m=,2,...,j This means the sequence {a n } n=2 can be determined successively from (.9) in a unique manner for any a 0; that is, (.4) has formal solution for arbitrary nonzero complex number a. Noticing that the function H(z) is holomorphic in a neighborhood of the origin, we assume h n. The reason for this, is that (.4) and hypothetic conditions g(0) = 0, g (0) = a still hold under the transformations H(z) = ρ F (ρ z), g(z) = ρ G(ρ z) for h n ρ n. We prove analyticity of solutions to (.4) under varius hypotheses: (A) (elliptic case) θ = e 2πiα, α R\Q is a Brjuno number; i.e., B(α) = k=0 <, where { p k q k } denotes the sequence of partial fraction log q k+ q k of the continued fraction expansion of α; (A2) (parabolic case) θ = e 2πi q p for some integer p N with p 2, q Z\{0}, and θ e 2πi l k for all k p, l Z\{0}. (A3) (hyperbolic case) 0 < θ <. 2. Existence of analytic solutions for (.4) When (A) is satisfied, that is, θ = e 2πiα with α irrational, small divisors arises inevitably. Since (θ n ) appears in the denominator and the powers of θ form a dense subset, there will be n such that θ n is arbitrarily large, see [6]. In 942, Siegel [6] showed a Diophantine condition that α satisfies α p q > γ q δ for some positive γ and δ. In 965, Brjuno [2] put forward Brjuno number which satisfies B(α) = log q n+ < q n n and improved Diophantine condition, he showed that as long as α is a Brjuno number, small divisors is still dealt with tactfully. In the sequel we discuss the analytic solution of (.4) with Brjuno number α. For this purpose, the Davie s Lemma is necessary. Lemma 2. (Davie s Lemma [3]). Assume K(n) = n log 2+ k(n) k=0 g k(n) log(2q k+ ), then the function K(n) satisfies

4 4 P. ZHANG EJDE-202/46 (a) There is a universal constant τ > 0 (independent of n and of α), such that ( k(n) K(n) n k=0 log q k+ q k ) + τ ; (b) for all n and n 2, we have K(n ) + K(n 2 ) K(n + n 2 ); (c) log θ n K(n) K(n ). Theorem 2.2. Under assumption (A), (.4) has an analytic solution of the form g(z) = a n z n, a 0. (2.) n= Proof. We prove the formal solution (2.) is convergent in a neighborhood of the origin. From (.9), we have θ n+ j a n+ b (θ n )θ n+2 a n+ j h m a i a i2... a im = b b j= j= j= i +i 2 + +im=j; m=,2,...,j i +i 2 + +im=j; m=,2,...,j i +i 2 + +im=j; m=,2,...,j θ n a n+ j h m a i a i2... a im θ n a n+ j a i a i2... a im. To construct a majorant series, we define {B n } n= by B = a and B n+ = b B n+ j B i B i2... B im, n =, 2,.... We denote Then Let j= G(z) = a z + i +i 2 + +im=j; m=,2,...,j G(z) = B n+ z n+ n= = a z + b = a z + b (2.2) B n z n. (2.3) n= n= j= i +i 2 + +im=j; m=,2,...,j n= j= G(z) G j+ (z) G(z) = a z + b G2 (z) ( z)g 3 (z) G 4 (z) ( z)( G(z))( G 2 (z)). B n+ j B i B i2... B im z n+ B n+ j z n+ j ζ 2 ( z)ζ 3 ζ 4 R(z, ζ) = ζ a z b ( z)( ζ)( ζ 2 = 0. (2.4) )

5 EJDE-202/46 FORMAL AND ANALYTIC SOLUTIONS 5 We regard (2.4) as an implicit functional equation, since R(0, 0) = 0, R ζ (0, 0) = 0. We know that (2.4) has a unique analytic solution ζ(z) in a neighborhood of the origin such that ζ(0) = 0, ζ (0) = a and R(z, ζ(z)) = 0, so we have G(z) = ζ(z). Naturally, there exists constant T > 0 such that B n T n, n =, 2,.... We now deduce by induction on n that a n+ B n+ e k(n), n 0. (2.5) In fact, a = B, since k(0) = 0. We assume that a i+ B i+, i < n, n =, 2,.... Then a n+ b b = b j= j= j= i +i 2 + +im=j; m=,2,...,j i +i 2 + +im=j; m=,2,...,j i +i 2 + +im=j; m=,2,...,j θ n a n+ j a i a i2... a im θ n B n+ je k(n j) B i e k(i ) B i2 e k(i2 )... B im e k(im ) θ n B n+ jb i B i2... B im e k(n m) = θ n B n+e k(n m) θ n B n+e k(n ) θ n B log θ n+e n +k(n) = B n+ e k(n), by means of Davie s Lemma, thus (2.5) is proved. Note that for some universal constant τ > 0. Then that is, K(n) n(b(α) + τ) a n+ T n+ e n(b(α)+τ) ; lim sup( a n+ ) /(n+) lim sup(t n+ e n(b(α)+τ) ) /(n+) = T e B(α)+τ. n n This implies that th radius of convergence for (2.) is at least (T e B(α)+τ ), the proof is complete. In what follows, we consider the case that the constant θ is not only on the unit circle, but also a root of unity. Denote the right side of (.9) as Λ(n, θ) = b θ n+ j a n+ j h m a i a i2... a im. j= i +i 2 + +im=j; m=,2,...,j Theorem 2.3. Assume (A2) holds and Λ(vp, θ) 0, v =, 2,.... (2.6)

6 6 P. ZHANG EJDE-202/46 Then (.4) has an analytic solution of the form g(z) = a z + ζ vp z n + b n z n, a 0, N = {, 2,... } (2.7) n=vp, v N n vp, v N in a neighborhood of the origin for some ζ vp. solutions in any neighborhood of the origin. Otherwise, (.4) has no analytic Proof. In this parabolic case θ = e 2πi q p, the eigenvalue θ is a pth root of unity. If Λ(vp, θ) 0, for some natural number v, then (.9) does not hold since θ vp = 0, naturally, (.4) has no formal solutions. If Λ(vp, θ) 0, for all natural number v, (.4) has formal solution (2.). To prove (2.) yields a local analytic solution, we define the sequence {C n } n= satisfies C = a and C n+ = b Γ j= i +i 2 + +im=j; m=,2,...,j C n+ j C i C i2... C im, n =, 2,..., (2.8) where Γ = max{, θ i : i =, 2,..., p }. Clearly, the convergence of series n= C nz n can be proved similar as in Theorem 2.2. When (2.6) holds for all natural number v, the coefficients a vp have infinitely many choices in C, choose a vp = ζ vp arbitrarily such that Furthermore, we can prove a vp C vp, v =, 2,.... (2.9) a n C n, n vp. (2.0) In fact, a = C. If we suppose that a i+ C i+, i < n (n vp), then a n+ b b Γ j= j= = C n+. i +i 2 + +im=j; m=,2,...,j i +i 2 + +im=j; m=,2,...,j θ n+ j (θ n )θ n+2 a n+ j h m a i a i2... a im C n+ j C i C i2... C im From (2.9), (2.0) and the convergence of series n= C nz n, the formal solution (2.) yields a local analytic solution (2.7) in a neighborhood of the origin. This completes the proof. Theorem 2.4. Suppose (A3) holds, then (.4) has an analytic solution of the form g(z) = a n z n, a 0. n=

7 EJDE-202/46 FORMAL AND ANALYTIC SOLUTIONS 7 Proof. We prove the formal solution (2.) is convergent in a neighborhood of the origin. Since 0 < θ <, so lim n θ n =, From (.9), we have θ n+ j a n+ b (θ n )θ n+2 a n+ j h m a i a i2... a im b j= j= i +i 2 + +im=j; m=,2,...,j i +i 2 + +im=j; m=,2,...,j Let {D n } n= be defined by D = a and Denote Then Let D n+ = b j= F (z) = a z + i +i 2 + +im=j; m=,2,...,j D n+ z n+ n= = a z + b = a z + b n= j= n= j= θ +j a n+ j a i a i2... a im. θ +j D n+ jd i D i2... D im, n =, 2,.... F (z) = (2.) D n z n. (2.2) n= i +i 2 + +im=j; m=,2,...,j θf 2 (z) = a z + b (θ 2 )(θ F (z)). θ +j D n+ jd i D i2... D im z n+ θ +j F (z) F j+ (z) D n+ j z n+ j F (z) θξ 2 Q(z, ξ) = ξ a z b (θ 2 = 0. (2.3) )(θ ξ) Since Q(0, 0) = 0, Q ξ (0, 0) = 0, then (2.3) has a unique analytic solution ξ(z) in a neighborhood of the origin such that ξ(0) = 0, ξ (0) = a and Q(z, ξ(z)) = 0, so we have F (z) = ξ(z). Similar as in Theorem 2.3, we can prove a n D n, n =, 2,..., (2.4) by induction. Then the local analytic solution (2.) is existent in a neighborhood of the origin by means of the convergence of n= D n and inequality (2.4). This completes the proof. 3. Formal solutions and analytic solutions of (.2) In this section we prove the existence of formal solutions and analytic solutions of (.2).

8 8 P. ZHANG EJDE-202/46 Theorem 3.. Equation (.3) has a formal solution y(z) = g(θ g (z)) in a neighborhood of the origin, where g(z) is formal solution of (.4). Under one of the conditions in Theorems , every formal solution yields an analytic solution of the form y(z) = g(θ g (z)), where g(z) is analytic solution of (.4). Proof. Since g (0) = a 0, the inverse g (z) exists in a neighborhood of g(0) = 0. If we define y(z) = g(θ g (z)), then y(y(z)) ay(z) = g(θ(g (gθ(g (z))))) ag(θ(g (z))) = g(θ 2 (g (z))) ag(θ(g (z))) = b(gθ(g (z))h(g (z)) = by(z)h(z). (3.) as required, so (.3) has a formal solution y(z) = g(θ g (z)) in a neighborhood of the origin. Under one of the conditions in Theorems , the inverse g (z) exists and is analytic in a neighborhood of g(0) = 0, we obtain analytic solutions of (.3) in a neighborhood of the origin. The proof is completed. Suppose that y(z) = θz + b 2 z 2 + b 3 z , since a = θ and x(z) = y(z) az bz, it follows that x(z) = b 2 b z + b 3 b z2 + b 4 b z (3.2) That is, (.2) has a unique formal solution with the form (3.2) in a neighborhood of the origin. The formal solution also is analytic solution when y(z) is analytic in a neighborhood of the origin. References [] T. Carletti, S. Marmi; Linearization of analytic and non-analytic germs of diffeomorphisms of (C, 0), Bull. Soc. Math. France. 28(2000), [2] A. D. Brjuno; On convergence of transforms of differential equations to the normal form, Dokl. Akad. Nauk SSSR. 65(965), [3] A. M. Davie; The criticalunction for the semistandard map, Nonlinearity. 7(994), [4] J. Dhombres; Some aspects of functional equation, Chulalongkorn University Press, Bangkok, 979. [5] M. Kuczma; Functional equations in a single variable, Polish Scientific Publ., Warsaw, 968. [6] R. E. Lee DeVille; Brjuno Numbers and Symbolic Dynamics of the Complex Exponential, Qualitative theory of dynamical systems. 5(2004), [7] S. Marmi; An introduction to samll divisors problems, UniversitÀ di pisa dipartmento di math. 27(2000). [8] L. Reich, J. Smítal, M. Štefánková; Local analytic solutions of the generalized Dhombres functional equation II, J. Math. Anal. Appl. 355(2009), [9] L. Reich, J. Smítal; On generalized Dhombres equations with nonconstant polynomial solutions in the complex plane, Aequat. Math. 80(200), [0] L. Reich, J. Tomaschek; Some remarks to the formal and local theory of the generalized Dhombres functional equation, Results. Math. DOI 0.007/S [] R. E. Rice, B. Schweizer, A. Sklar; When is f(f(z)) = az 2 + bz + c? Amer. Math. Monthly. 87(980), [2] J. Si, W. Zhang, Analytic solutions of a functional equations for invariant curves, J. Math. Anal. Appl. 259(200),

9 EJDE-202/46 FORMAL AND ANALYTIC SOLUTIONS 9 [3] J. Si, X. P. Wang, W. Zhang; Analytic invariant curves for a planar map, Appl. Math. Lett. 5(2002), [4] J. Si, X. Li, Small divisors problem in dynamical systems and analytic solutions of the Shabat equation, J. Math. Anal. Appl. 367(200), [5] J. Si, H. Zhao; Small divisors problem in dynamical systems and analytic solutions of a q-difference equation with a singularity at the origin, Results. Math. 58(200), [6] C. L. Siegel; Iteration of analytic functions, Ann. of Math. 43(942), [7] B. Xu, W. Zhang; Decreasing solutions and convex solutions of the polynomial-like iterative equation, J. Math. Anal. Appl. 329(2007), [8] B. Xu, W. Zhang; Small divisor problem for an analytic q-difference equation, J. Math. Anal. Appl. 342(2008), [9] X. Wang, J. Si; Continuous solutions of an iterative function, Acta Math. Sinica. 42(5)(999), (in Chinese). [20] P. Zhang, L. Mi; Analytic solutions of a second order iterative functional differential equation, Appl. Math. Comp. 20(2009), [2] J. Zhang, L. Yang; Discussion on iterated roots of piecewise monotone function, Acta Math. Sinica. 26(983), (in Chinese). [22] W. Zhang; Discussion on the iterated equation P n i= λ if i (x) = F (x), Chin. Sci. Bulletin. 2(987), [23] W. Zhang; Stability of the solution of the iterated equation P n i= λ if i (x) = F (x), Acta. Math. Sci. 8(988), Pingping Zhang Department of Mathematics and Information Science, Binzhou University, Shandong , China address: zhangpingpingmath@63.com