906 DIFFERENCE EQUATIONS AND OTHER FUNCTIONAL EQUATIONS THEOREM. Any solution of equation (17.2.4.1) in the class of exponentially growing functions of a finite degree σ can be represented in the form y(x)= |β k |≤σ h k –1 s=0 C ks x s e β k x ≡ |β k |≤σ P k (x)e β k x ,( 17 .2.4.5) where the sum is over all zeroes of the characteristic function (17.2.4.2) in the circle |t| ≤ σ ; C ks are arbitrary constants, and P k (x) are arbitrary polynomials of degrees ≤ n k – 1 . Corollary 1. A necessary and sufficient condition for the existence of polynomial solutions of equation (17.2.4.1) is that the characteristic function (17.2.4.2) has zero root β 1 = 0. In this case, the coefficients of equation (17.2.4.1) should satisfy the condition a m + a m–1 + ···+ a 1 + a 0 = 0. Corollary 2. There is only one solution y(x) ≡ 0 in the class [1, σ] only if σ < min{|β 1 |, |β 2 |, }. 17.2.4-2. Linear nonhomogeneous difference equations. 1 ◦ .Alinear nonhomogeneous difference equation with constant coefficients, in the case of arbitrary differences, has the form a m y(x + h m )+a m–1 y(x + h m–1 )+···+ a 1 y(x + h 1 )+a 0 y(x + h 0 )=f(x), (17.2.4.6) where a 0 a m ≠ 0, m ≥ 1,andh i ≠ h j for i ≠ j. Let f(x) be a function of exponential growth of degree σ. Then equation (17.2.4.6) always has a solution y(x) in the class [1, σ]. The general solution of equation (17.2.4.6) in the class [1, σ] can be represented as the sum of the general solution (17.2.4.5) of the homogeneous equation (17.2.4.1) and a particular solution y(x) of the nonhomogeneous equation (17.2.4.6). 2 ◦ . Suppose that the right-hand side of the equation is the polynomial f(x)= n s=0 b s x s , b n ≠ 0, n ≥ 0.(17.2.4.7) Then equation (17.2.4.6) has a particular solution of the form y(x)=x μ n s=0 c s x s , c n ≠ 0,(17.2.4.8) where t = 0 is a root of the characteristic function (17.2.4.2), the multiplicity of this root being equal to μ. The coefficients c n in (17.2.4.8) can be found by the method of indefinite coefficients. 3 ◦ . Suppose that the right-hand side of the equation has the form f(x)=e px n s=0 b s x s , b n ≠ 0, n ≥ 0.(17.2.4.9) Then equation (17.2.4.6) has a particular solution of the form y(x)=e px x μ n s=0 c s x s , c n ≠ 0,(17.2.4.10) where t = p is a root of the characteristic function (17.2.4.2), its multiplicity being equal to μ. The coefficients c n in (17.2.4.10) can be found by the method of indefinite coefficients. 17.3. LINEAR FUNCTIONAL EQUATIONS 907 17.2.4-3. Equations reducible to equations with constant coefficients. 1 ◦ . The difference equation with variable coefficients a m f(x + h m )y(x + h m )+a m–1 f(x + h m–1 )y(x + h m–1 )+··· + a 1 f(x + h 1 )y(x + h 1 )+a 0 f(x + h 0 )y(x + h 0 )=g(x) can be reduced, with the help of the replacement y(x)=f (x)u(x), to a difference equation with constant coefficients a m u(x + h m )+a m–1 u(x + h m–1 )+···+ a 1 u(x + h 1 )+a 0 u(x + h 0 )=g(x). 2 ◦ . Two other difference equations with variable coefficients can be obtained from equa- tions considered in Paragraph 17.2.3-5 (Items 2 ◦ and 3 ◦ ), where the quantities x + m, x + m – 1, , x + 1, x should be replaced, respectively, by x + h m , x + h m–1 , , x + h 1 , x + h 0 . 17.3. Linear Functional Equations 17.3.1. Iterations of Functions and Their Properties 17.3.1-1. Definition of iterations. Consider a function f(x)defined on a set I and suppose that f(I) ⊂ I.(17.3.1.1) AsetI for which (17.3.1.1) holds is called a submodulus set for the function f(x). If we have f (I)=I,thenI is called a modulus set for the function f (x). For a function f (x)defined on a set I and satisfying the condition (17.3.1.1), by f [n] (x) we denote the nth iteration defined by the relations f [0] (x)=x, f [n+1] (x)=f(f [n] (x)), x I, n = 0, 1, 2, (17.3.1.2) For an invertible function f (x), one can define its iterations also for negative values of the iteration index n: f [n–1] (x)=f –1 (f [n] (x)), x I, n = 0,–1,–2, ,(17.3.1.3) where f –1 denotes thefunction inverse tof. In view of the relations(17.3.1.2) and(17.3.1.3), we also have f [1] (x)=f(x), f [n] (f [m] (x)) = f [n+m] (x). (17.3.1.4) 908 DIFFERENCE EQUATIONS AND OTHER FUNCTIONAL EQUATIONS 17.3.1-2. Fixed points of a function. Some classes of functions. A point ξ is called a fixed point of the function f (x)iff(ξ)=ξ. A point ξ is called an attractive fixed point of the function f(x) if there exists a neighborhood U of ξ such that lim n→∞ f [n] (x)=ξ for any x U. If, in addition, we have |f(x)–ξ| ≤ ε|x – ξ|, 0 < ε < 1, for x U,thenξ is called a strongly attractive fixed point. If f(x) is differentiable at a fixed point x = ξ and |f (ξ)| < 1, then ξ is a strongly attractive fixed point. For a < b,letI be any of the sets a < x < b, a ≤ x < b, a < x ≤ b, a ≤ x ≤ b. One or both endpoints a and b may be infinite. The closure of I is denoted by I. Denote by S m ξ [I](briefly S m ξ ) the class of functions f(x) satisfying the following conditions: 1 ◦ . f(x) has continuous derivatives up to the order m in I. 2 ◦ . f(x) satisfies the inequalities (f(x)–x)(ξ – x)>0 for x I, x ≠ ξ;(17.3.1.5) (f(x)–ξ)(ξ – x)<0 for x I, x ≠ ξ,(17.3.1.6) where ξ I. Denote by R m ξ [I](briefly R m ξ ) the class of functions f(x) belonging to S m ξ and strictly increasing on I. Fig. 17.3 represents a function in S m ξ and Fig. 17.4 represents a function in R m ξ . O x ξ ξ y yx= yfx= () Figure 17.3. A function belonging to S m ξ . O x ξ ξ y yx= yfx= () Figure 17.4. A function belonging to R m ξ . Remark. If ξ =+∞, then in the definition of the classes S m ∞ and R m ∞ , the condition (17.3.1.6) is superfluous and (17.3.1.5) is replaced by f(x)>x for x I. Analogously, if ξ =–∞, then (17.3.1.5) is replaced by f (x)<x for x I. The following statements hold: (i) If f (x) S m ξ ,thenξ is a fixed point of f(x). (ii) If f(x) S m ξ , then for any x 0 I the sequence f [n] (x 0 ) is monotonic (and strictly monotonic whenever x 0 ≠ ξ)and lim n→∞ f [n] (x 0 )=ξ. 17.3. LINEAR FUNCTIONAL EQUATIONS 909 17.3.1-3. Asymptotic properties of iterations in a neighborhood of a fixed point. 1 ◦ .Letf(0)=0, 0 < f (x)<x for 0 < x < x 0 , and suppose that in a neighborhood of the fixed point the function f(x) can be represented in the form f(x)=x – ax k + bx m + o(x m ), where 1 < k < m and a, b > 0. Then the following limit relation holds: lim n→∞ n 1 k–1 f [n] (x)=[a(k – 1)] 1 k–1 , 0 < x < x 0 .(17.3.1.7) Example. Consider the function f(x)=sinx.Wehave0 <sinx < x for 0 < x < ∞ and sin x = x – 1 6 x 3 + 1 120 x 5 + o(x 5 ), which corresponds to the values a = 1 6 and k = 3. Substituting these values into (17.3.1.7), we obtain lim n→∞ √ n sin [n] x = √ 3, 0 < x < ∞. 2 ◦ .Letf(0)=0, 0 < f (x)<x for 0 < x < x 0 , and suppose that in a neighborhood of the fixed point the function f(x) can be represented in the form f(x)=λx + ax k + bx m + o(x m ), where 0 < λ < 1 and 1 < k < m. Then the following limit relation holds: lim n→∞ f [n] (x)–λ n x λ n = ax k λ – λ k , 0 < x < x 0 . 17.3.1-4. Representation of iterations by power series. Let f(x) be a function with a fixed point ξ = f(ξ) and suppose that in a neighborhood of that point f(x) can be represented by the series f(x)=ξ + ∞ j=1 a j (x – ξ) j (17.3.1.8) with a nonzero radius of convergence. For any integer N > 0, there is a neighborhood U of the point ξ in which all iterations f [n] (x) for integer 0 ≤ n ≤ N are defined and also admit the representation f [n] (x)=ξ + ∞ j=1 A nj (x – ξ) j .(17.3.1.9) The coefficients A nj can be uniquely expressed through the coefficients a j with the help of formal power series and the relations f (f [n] (x)) = f [n] (f(x)). For a 1 ≠ 0 and |a 1 | ≠ 1,thefirst three coefficients of the series (17.3.1.9) have the form A n1 = a n 1 , A n2 = a 2n 1 – a n 1 a 2 1 – a 1 a 2 , A n3 = a 3n 1 – a n 1 a 3 1 – a 1 a 3 + 2 (a 2n 1 – a n 1 )(a n 1 – a 1 ) (a 3 1 – a 1 )(a 2 1 – a 1 ) a 2 , n = 2, 3, 910 DIFFERENCE EQUATIONS AND OTHER FUNCTIONAL EQUATIONS For a 1 = 1,wehave A n1 = 1, A n2 = na 2 , A n3 = na 3 + n(n – 1)a 2 2 , n = 2, 3, The series (17.3.1.8) has a nonzero radius of convergence if and only if there exist constants A > 0 and B > 0 such that |a j | ≤ AB j–1 , j = 1, 2, Under these conditions, the series (17.3.1.8) is convergent for |x – ξ| < 1/B, and the series (17.3.1.9) is convergent for |x – ξ| < R n ,where R n = ⎧ ⎪ ⎨ ⎪ ⎩ A – 1 B(A n – 1) if A ≠ 1, 1 nB if A = 1. 17.3.2. Linear Homogeneous Functional Equations 17.3.2-1. Equations of general form. Possible cases. A linear homogeneous functional equation has the form y f(x) = g(x)y(x), (17.3.2.1) where f (x)andg(x) ≠ 0 are known functions and the function y(x) is to be found. Let f(x) R 0 ξ ,wherex I and ξ I,andletg(x) be continuous on I, g(x) ≠ 0 for x I, x ≠ ξ. Consider the sequence of functions G n (x)= n–1 k=0 g f [k] (x) , n = 1, 2, (17.3.2.2) Three cases may occur. (i) The limit G(x) = lim n→∞ G n (x)(17.3.2.3) exists on I. Moreover, G(x) is continuous and G(x) ≠ 0. (ii) There exists an interval I 0 ⊂ I such that lim n→∞ G n (x)=0 uniformly on I 0 . (iii) Neither of the cases (i) or (ii) occurs. T HEOREM. In case (i), the homogeneous functional equation (17.3.2.1) has a one- parameter family of continuous solutions on I .Forany a , there exists exactly one continuous function y(x) satisfying equation (17.3.2.1) and the condition y(ξ)=a. This solution has the form y(x)= a G(x) .( 17 .3.2.4) 17.3. LINEAR FUNCTIONAL EQUATIONS 911 In case (ii), equation (17.3.2.1) has a continuous solution depending on an arbitrary function, and every continuous solution y(x) on I satisfies the condition lim x→ξ y(x)=0 . In case (iii), equation (17.3.2.1) has a single continuous solution, y(x) ≡ 0 . In order to decide which of the cases (i), (ii), or (iii) takes place, the following simple criteria can be used. (i) This case takes place if ξ is a strongly attractive fixed point of f(x) and there exist positive constants δ, μ,andM such that |g(x)–1| ≤ M|x – ξ| μ for x I ∩ (ξ – δ, ξ + δ). (ii) This case takes place if |g(ξ)| < 1. (iii) This case takes place if |g(ξ)| > 1. Remark. For g(ξ)=1, any of the three cases (i), (ii), or (iii) may take place. 17.3.2-2. Schr ¨ oder–Koenigs functional equation. Consider the Schr ¨ oder–Koenigs equation y(f(x)) = sy(x), s ≠ 0,(17.3.2.5) which is a special case of the linear homogeneous functional equation (17.3.2.1) for g(x)=s = const. 1 ◦ .Lets > 0, s ≠ 1 and let ¯y(x) be a solution of equation (17.3.2.5) satisfying the condition ¯y(x) ≠ 0. Then the function y(x)=¯y(x)Θ ln |¯y(x)| ln s , where Θ(z) is an arbitrary 1-periodic function, is also a solution of equation (17.3.2.5). 2 ◦ .Letf(x)bedefined on a submodulus set I and let h(x) be a one-to-one mapping of I onto a set I 1 .Let p(x)=h(f(h –1 (x))), where h –1 is the inverse function of h.Ifψ(x) is a solution of the equation ψ(p(x)) = sψ(x) on I 1 , then the function y(x)=ψ(h(x)) satisfies equation (17.3.2.5) on I. On the basis of this statement, a fixed point ξ can be moved to the origin. Indeed, if ξ is finite, we can take h(x)=x – ξ.Ifξ = ∞,wetakeh(x)=1/x. Thus, we can assume that 0 is a fixed point of f (x). 912 DIFFERENCE EQUATIONS AND OTHER FUNCTIONAL EQUATIONS 3 ◦ . Suppose that f (x) R 2 0 [I], 0 I,and f (0)=s, 0 < s < 1. Then for each σ (–∞, ∞), there exists a unique continuously differentiable solution of equation (17.3.2.5) on I satisfying the condition y (0)=σ.(17.3.2.6) This solution is given by the formula y(x)=σ lim n→∞ s –n f [n] (x), (17.3.2.7) and for σ ≠ 0 it is strictly monotone on I (it is an increasing function for σ > 0, and it is a decreasing function for σ < 0). 4 ◦ . For an invertible function f , (17.3.2.5) can be reduced to a similar equation with the help of the transformation z = f (x): y(f –1 (z)) = s –1 y(z). 5 ◦ .Fors > 0, s ≠ 0, the replacement y(x)=s u(x)/c reduces the Schr ¨ oder–Koenigs equation (17.3.2.5) to the Abel equation for the function u(x) (see equation (17.3.3.5), in which y should be replaced by u). In Subsection 17.3.3, Item 4 ◦ , there is a description of the method for constructing continuous monotone solutions of the Abel equation to within certain arbitrary functions. 17.3.2-3. Automorphic functions. Consider the special case of equation (17.3.2.5) for s = 1: y(f(x)) = y(x). (17.3.2.8) Solutions of this equation are called automorphic functions.Iff(x) is invertible on a modulus set I, then the general solution of equation (17.3.2.8) may be written in the form y(x)= ∞ n=–∞ ϕ(f [n] (x)), (17.3.2.9) where ϕ(x) is an arbitrary function on I such that the series (17.3.2.9) is convergent. 17.3.3. Linear Nonhomogeneous Functional Equations 17.3.3-1. Equations of general form. Possible cases. 1 ◦ . Consider a linear nonhomogeneous functional equation of the general form y(f(x)) = g(x)y(x)+F (x). (17.3.3.1) Let x I and ξ I, f(x) R 0 ξ . Suppose that g(x)andF (x) are continuous functions on I and g(x) ≠ 0 for x I, x ≠ ξ. In accordance with the investigation of the corresponding . sum of the general solution (17.2.4.5) of the homogeneous equation (17.2.4.1) and a particular solution y(x) of the nonhomogeneous equation (17.2.4.6). 2 ◦ . Suppose that the right-hand side of. a j with the help of formal power series and the relations f (f [n] (x)) = f [n] (f(x)). For a 1 ≠ 0 and |a 1 | ≠ 1,thefirst three coefficients of the series (17.3.1.9) have the form A n1 = a n 1 ,. function for σ > 0, and it is a decreasing function for σ < 0). 4 ◦ . For an invertible function f , (17.3.2.5) can be reduced to a similar equation with the help of the transformation