Handbook of mathematics for engineers and scienteists part 135 pps

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 β1 = 0... Equations reducib

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) = 

|βk|≤σ

hk 1

s=0

C ks x s e β k x ≡ 

|βk|≤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

β1 = 0 In this case, the coefficients of equation (17.2.4.1) should satisfy the condition

a m + a m–1+· · · + a1 + a0=0

Corollary 2 There is only one solution y(x) ≡ 0 in the class [1, σ] only if σ <

min{|β1|,|β2|, }

17.2.4-2 Linear nonhomogeneous difference equations

1◦ A linear 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) +· · · + a1 y (x + h1) + a0y (x + h0) = f (x), (17.2.4.6)

where a0a m≠ 0, m≥ 1, and h 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 solution2y(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 2y(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

2y(x) = x μn

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 nin (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 pxn

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

2y(x) = e px x μn

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 nin (17.2.4.10) can be found by the method of indefinite coefficients

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) +· · ·

+ a1f (x + h1)y(x + h1) + a0f (x + h0)y(x + h0) = 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) +· · · + a1 u (x + h1) + a0u (x + h0) = 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 + h1, x + h0

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

A set I for which (17.3.1.1) holds is called a submodulus set for the function f (x) If

we have f (I) = I, then I 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)), xI, 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)), xI, n=0, –1, –2, , (17.3.1.3)

where f–1denotes the function inverse to f 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)

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) if f (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 xU , then ξ 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, let I 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 ξ m [I] (briefly S ξ m ) 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 xI , x≠ξ; (17.3.1.5)

(f (x) – ξ)(ξ – x) <0 for xI , x≠ξ, (17.3.1.6)

where ξI

Denote by R m ξ [I] (briefly R m ξ ) the class of functions f (x) belonging to S ξ mand strictly

increasing on I Fig 17.3 represents a function in S ξ m and Fig 17.4 represents a function

in R m ξ

O

x ξ

ξ

y

y=x

y=f x( )

Figure 17.3 A function belonging to S ξ m.

O

x ξ

ξ

y=f x( )

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 xI Analogously, if ξ = – ∞, then (17.3.1.5) is replaced by f(x) < x

for xI.

The following statements hold:

(i) If f (x)S m

ξ , then ξ is a fixed point of f (x).

(ii) If f (x)S m

ξ , then for any x0 I the sequence f[n] (x0) is monotonic (and strictly

monotonic whenever x0 ≠ξ) and

lim

n→∞ f

[n] (x0) = ξ.

17.3.1-3 Asymptotic properties of iterations in a neighborhood of a fixed point.

1◦ Let f (0) =0,0< f (x) < x for0< x < x0, 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), where1< k < m and a, b >0 Then the following limit relation holds:

lim

n→∞ n

1

k–1 f[n] (x) = [a(k –1)] k–11 , 0< x < x0 (17.3.1.7)

Example Consider the function f (x) = sin x We have 0< sin x < x for 0 < x < ∞ and sin x =

x– 16x3+ 1201 x5+ o(x5), which corresponds to the values a = 16 and k =3 Substituting these values into (17.3.1.7), we obtain

lim

n→∞

√

nsin[n] x=√

3 , 0< x < ∞.

2◦ Let f (0) =0,0< f (x) < x for0< x < x0, 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), where0< λ <1and1< k < m Then the following limit relation holds:

lim

n→∞

f[n] (x) – λ n x

ax k

λ – λ k, 0< x < x0

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 integer0 ≤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 jwith the help of

formal power series and the relations f (f[n] (x)) = f[n] (f (x)).

For a1≠ 0and|a1| ≠ 1, the first three coefficients of the series (17.3.1.9) have the form

A n1 = a n1, A n2= a

2n

1 – a n1

a2

1– a1 a2,

A n3 = a

3n

1 – a n1

a3

1– a1 a3+2(a2n1 – a n1)(a n1 – a1)

(a31– a1)(a21– a1) a2, n=2, 3,

For a1=1, we have

A n1=1, A n2 = na2, A n3 = na3+ n(n –1)a22, n=2, 3,

The series (17.3.1.8) has a nonzero radius of convergence if and only if there exist

constants A >0and B >0such 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

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)

where f (x) and g(x)≠ 0are known functions and the function y(x) is to be found.

Let f (x)R0

ξ , where x I and ξ I , and let g(x) be continuous on I, g(x)≠ 0for

xI , 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 I0⊂ I such that

lim

n→∞ G n (x) =0

uniformly on I0

(iii) Neither of the cases (i) or (ii) occurs

THEOREM In case (i), the homogeneous functional equation (17.3.2.1) has a

one-parameter family of continuous solutions on I For any 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

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, and M 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◦ Let s >0, s≠ 1and 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

 ,

whereΘ(z) is an arbitrary1-periodic function, is also a solution of equation (17.3.2.5)

2◦ Let f (x) be defined on a submodulus set I and let h(x) be a one-to-one mapping of I

onto a set I1 Let

p (x) = h(f (h–1(x))), where h– 1is the inverse function of h If ψ(x) is a solution of the equation

ψ (p(x)) = sψ(x)

on I1, 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 If ξ = ∞, we take h(x) =1/x Thus, we can assume that0

is a fixed point of f (x).

3◦ Suppose that f (x)

R2

0[I],0I, 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

This solution is given by the formula

y (x) = σ lim

n→∞ s

–n f[n] (x), (17.3.2.7)

and for σ≠ 0it 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–1y (z).

5◦ For s >0, s≠ 0, the replacement y(x) = s u(x)/creduces 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:

Solutions of this equation are called automorphic functions If f (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 xI and ξI , f (x)R0

ξ Suppose that g(x) and F (x) are continuous functions on

I and g(x)≠ 0for x I , x≠ξ In accordance with the investigation of the corresponding