Handbook of mathematics for engineers and scienteists part 135 pps

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