A remark on k th order linear functional equations with constant coefficients

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EQUATIONS WITH CONSTANT COEFFICIENTS

JITKA LAITOCHOV ´A

Received 30 January 2006; Accepted 18 May 2006

Abel functional equations are associated to a linear homogeneous functional equation with constant coefficients. The work uses the spaceSof continuous strictly monotonic functions. Generalized terms are used, because of the spaceS, like composite function, it-erates of a function, Abel functional equation, and linear homogeneous functional equa-tion inSwith constant coefficients. The classical theory of linear homogeneous functional and difference equations is obtained as a special case of the theory in spaceS. Equivalence of points and orbits of a point are introduced to show the connection between the lin-ear functional and the linlin-ear difference equations inS. Asymptotic behavior at infinity is studied for a solution of the linear functional equation.

Copyright © 2006 Jitka Laitochov´a. 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

The linear functional equations are considered in the space of real-valued functions of a real variablex∈᏶,᏶=(−∞,∞). The setNdenotes the set of positive integers. The setZdenotes integers, and the setRdenotes real numbers. SymbolC0(᏶) is the set of

continuous functions on the interval᏶.

Definitions of the terms which we generalize in the spaceS can be found in [1–5]. In particular, we generalize the notions of iterates and linear homogeneous functional equations with constant coeﬃcients.

1.1. Definition of the spaceS. A function f ∈C0(᏶) belongs toSif and only if it maps

the interval᏶ one-to-one onto the interval (a,b), wherea∈R ora= −∞,b∈R or

b= ∞.

1.2. Multiplication inS. Let us choose inSan arbitrary functionX, a so-calledcanonical function, and letX∗be the inverse function toX. LetF,G∈S. The composite function

H=FX∗G(x) will be called aproductand denoted byH=F◦G.

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It is easy to show thatH∈S. The setSwith the operation of multiplication◦forms a noncommutative groupᏳ, where the canonical functionXis the neutral element. Fur-ther, to each elementF∈Sthere is an inverse elementF−1₌_XF∗_X_in_{Ᏻ, where}_F∗_∈_S_is

the inverse function toF∈S.

Boldface promotion of symbols will be used to denote elements ofᏳ. ThenF,G∈S

correspond toF,G∈ᏳandᏳ-multiplication is defined by composition:

F◦G(x)≡FX∗G(x). (1.1)

If f ∈C0(᏶),Φ∈S, then the product f ◦Φ(x) (not necessarily inᏳ) is defined to be

the composite function f(X∗(Φ(x)))∈C0(᏶). In general, f is not one-to-one; boldface

promotion of f is disallowed, because f ∈S.

1.3. Iteration inS. LetX∈Sbe the canonical function. LetΦ∈S. Theiteratesof the functionΦinSare given by group operations as follows:

Φ0₍_x₎₌_X₍_x_),

Φn+1₍_x₎₌_Φ_◦_Φn₍_x_), _x_∈_᏶_,_n₌_{0, 1, 2,}_{. . .}_, Φn−1₍_x₎₌_Φ−1_◦_Φn₍_x_), _x_∈_᏶,_n₌_0,₋_1,₋_2,_{. . .}_,

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whereΦ−1is the inverse element to the elementΦinSaccording to multiplication◦in groupᏳ.

2. Linear functional equations of thekth order with constant coeﬃcients

Letaj∈R,j=0, 1, 2,. . .,k. Then the equation

akf ◦Φk(x) +ak−1f◦Φk−1(x) +···+a1f◦Φ1(x) +a0f◦Φ0(x)=0 (2.1)

in Sis called a linear homogeneous functional equation of kth order with constant coef-ficients. Thecoeﬃcients are the constantsaj, j=0, 1, 2,. . .,k. It is assumed thatak=0. Asolutionof (2.1) is a function f ∈C0(᏶) that satisfies the equation for allx. It is not

assumed that f is one-to-one.

Letg(x)=X(x+ 1) and letΦ∈S. The generalized Abel functional equation

α◦Φ(x)=g◦α(x) (2.2)

is called theassociated functional equationto (2.1). Asolutionof (2.2) is a functionα∈S

such that (2.2) holds for allx. The algebraic equation

akλk+ak−1λk−1+···+a1λ+a0=0 (2.3)

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Theorem2.1. Let (2.1) be given. Letλ1,λ2,. . .,λkbe simple positive roots of the characteris-tic equation (2.3). Letαbe a continuous solution of the associated Abel functional equation (2.2). Then the functions

f1=λX

are linearly independent solutions of (2.1). Proof. Functions (2.4) have the form

f =λX∗α(x), (2.5)

Abel functional equation (2.2) implies

X∗α◦Φn₍_x₎₌_X∗_X_X∗_α₍_x_{) +}_n₌_X∗_α₍_x_{) +}_n_, _(2.7)

Thus the function (2.5) is a solution of (2.1) ifλis a root of the characteristic equation (2.3). Roots of the characteristic equation have to be positive for the functions (2.5) to be defined.

To show the solutions (2.4) are linearly independent, consider a linear combination

c1f1(x) +···+ckfk(x)=0 for allx. Choose samplesx=X∗(x0), x=X∗(Φ(x0)), x= Therefore,c=0and the functions are independent. The proof is complete.

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n≤x < n+ 1, which satisfiesp(x)=p(x+ 1) orp=p◦ΦforΦ(x)=x+ 1. Others can be constructed from this example.

Theorem2.2. Let (2.1) be given having solution f0. Let p be a continuous automorphic function,p=p◦Φ. Then f(x)=p(x)f0(x)is also a solution of (2.1).

A more general solution is of the form

c1f1+c2f2+···+ckfk, (2.10)

wherec1,c2,. . .,ckare automorphic functions overΦand f1,. . .,fkare solutions of (2.1). Proof. The details to show f0satisfies (2.1) are as follows:

k

n=0 an

p f0

◦Φn₍_x₎₌ k

n=0 an

p◦Φn₍_x₎_f

0◦Φn(x)

=

k

n=0 an

p(x)f0◦Φn(x)

=p(x) k

n=0 an

f0◦Φn(x)

=0.

(2.11)

The proof is complete.

Theorem2.3. Let (2.1) be given. Letλ0 be a positive real root of characteristic equation (2.3), of multiplicitys,1≤s≤k. Letα(x)be a continuous solution of Abel functional equa-tion (2.2) and letX∗be the inverse function to canonical functionX. Then the functions

fr=

X∗α(x)rλ0X∗α(x), 0≤r < s, (2.12)

are independent solutions of (2.1).

Proof. Let p(λ) denote the characteristic polynomial. Assume for the first part of the proof thatXis the identity map. Define f(x)=(α(x))r_λα(x)

0 . The following lemmas will

be applied to complete the proof.

Lemma2.4. IfL=λ(d/dλ), thenLq₍_p₍_λ₎₎₌k

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Lemma2.6. {(α(x))r_λα(x)

Thendet(A)=0and the powers ofLemma 2.6are independent.

The proofs of the lemmas: forLemma 2.4use induction onq. ForLemma 2.5the left side of the equation isLq₍_p₍_λ_{)) for}_q₌_r₋_m_≥_{0. Apply}_{Lemma 2.4}_{. Expand}_Lq₍_p₍_λ_{)) by} calculus to verify that being zero atλ=λ0is possible because the derivatives (d/dλ)np(λ)

are zero atλ=λ0 for 0≤r < s, due to multiplicity of the rootλ0. ForLemma 2.6write

down a linear combination equal to zero. Cancelλα0(x). ForLemma 2.7the determinant is

a Vandermonde determinant, known to be nonzero for distinct sample valuest1,. . .,ts. The connection to the Abel equation is made by choosing sample values x=x0, x=

Φ(x0), and so forth, in the linear combination c1+c2α(x) +···+cs(α(x))s−1=0, and then writing the systemAc=0for vectorcto showc=0.

The proof that functions (2.12) satisfy the equation proceeds by insertingf into (2.1). The binomial formula (a+b)r₌r

the last step is justified byLemma 2.5.

Independence follows directly from Lemmas2.6and2.7. The details of proof for gen-eralXparallel the above steps, essentially replacingαbyX∗α. The proof is complete.

Theorem2.8. If the characteristic equation (2.3) has conjugate complex rootsλ1=λ¯2= r(cosω+isinω), then the corresponding linear homogeneous functional equation possesses two solutions in the form

f1=rX

∗_α₍_x₎

cosωX∗α(x), f2=rX

∗_α₍_x₎

sinωX∗α(x). (2.15)

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3. An application

LetSbe the space of all continuous functions which map the interval (−∞,∞) one-to-one onto itself. Let the canonical function beX(x)=x(identity), thenX∗(x)=x. Let Φ(x)=x+ 1. The Abel functional equation is then

α(x+ 1)=α(x) + 1. (3.1)

A linear homogeneous functional equation with constant coeﬃcients is

akfΦk(x) +ak−1fΦk−1(x) +···+a1fΦ1(x) +a0,fΦ0(x)=0, (3.2)

where fΦj₍_x_{) is a composite function}_f₍_Φj₍_x_)),_j₌_{1, 2,}_{. . .}_,_k_{, and}_Φj_{is defined by} suc-cessive composition. Supposeλ >0 is a root of the characteristic equation. By the theo-rems above, it has a solution of the form

f =λα(x), (3.3)

whereαsatisfies Abel equation (3.1).

Linear system (3.2) is a linear diﬀerence equation with constant coeﬃcients of the form

akf(x+k) +ak−1f(x+k−1) +···+a1f(x+ 1) +a0f(x)=0, (3.4)

becauseΦ(x)=x+ 1. Abel equation (3.1) has a solutionα(x)=x. Ifλis a positive root of the characteristic equation, then the function f =λx_{is a solution of (}_3.4_).

3.1. Equivalence of points in᏶, orbit of a point inS. LetΦ∈S. Two pointsx,y∈᏶are Φ-equivalentif and only if there are numbersμ,ν∈Zsuch that

Φμ₍_x₎₌_Φν₍_y₎_. _(3.5)

The equivalence is reflexive, symmetric, and transitive. It means that there is a decompo-sition of the set᏶into disjoint sets of equivalent points.

The setOΦ(x0) of all pointsΦ-equivalent to a pointx0∈Jis called theΦ-orbitof the

pointx0.

For the classical diﬀerence equation (3.4), in whichX(x)=X∗(x)=x and Φ(x)=

x+ 1, the orbit of 0 is the setZof integers and the orbit ofx0is the translate of this set by x0:OΦ(x0)= {x0+μ}∞μ=−∞.

Lemma3.1. Letx0∈᏶,μ,ν∈Z. Let

xμ=X∗Φμx0

. (3.6)

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Proof. BecauseΦ0=Xandxμ∈᏶, thenx0=X∗Φ0(x0). Letp,q∈Z. To show that each

two pointsxp,xq defined by (3.6) areΦ-equivalent in ᏶, we will show that there are numbersμ,ν∈Zsuch that

Φμ_x

p=Φνxq. (3.7)

Use formula (3.6) to show that (3.7) is equivalent to

Φμ_X∗_Φp_x 0

=Φν_X∗_Φq_x 0

. (3.8)

The definition of multiplication◦implies the above is equivalent to

Φμ+p_x 0

=Φν+q_x 0

. (3.9)

Chooseμ=q,ν=pto obtain (3.7).

3.2. Behavior of solutions at infinity. Consider a linear homogeneouskth-order func-tional equation with constant coeﬃcients (2.1). Let the characteristic equation (2.3) have roots satisfying

λ1> λ2>···> λk>0. (3.10)

Letα∈Sbe a solution of the Abel functional equation (2.2) and letX∗be the inverse function to canonical functionX. Assumec1,. . .,ckare continuousΦ-automorphic func-tions onJ, that is,cj=cj◦Φ. Then a solution of (2.1) is given by

f(x)=c1(x)λX

∗_α₍_x₎

1 +c2(x)λX

∗_α₍_x₎

2 +···+ck(x)λX

∗_α₍_x₎

k . (3.11)

The right side of (3.11) can be rearranged into the expression

f(x)=λX1∗α(x)

c1(x) +c2(x)

_λ

2 λ1

X∗_α₍_x₎

+···+ck(x) _λ

k

λ1

X∗_α₍_x₎

. (3.12)

The inequalities

0<λj

λ1 <1, j=2, 3,. . .,k, (3.13)

imply

lim x→∞cj(x)

_λ j

λ1

X∗α(x)

=0. (3.14)

Formally, at least,

lim

x→∞f(x)=xlim→∞c1(x)λ

X∗_α₍_x₎

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We will discuss the meaning of (3.15). The following situations can occur.

are the solutions evaluated along the Φ-orbit. Because α∈S satisfies Abel functional equation (2.2), then the sequences can be written in the form

Therefore, we have a solution formula

fx0+n

Using the ideas above, the limits at infinity are determined as follows. (1) Ifλ1>1, then{λα1(x0)+n}diverges to∞and limn→∞f(x0+n)= ∞.

Note 3.2. The limit forx→ −∞can be determined using the same techniques of analysis. Note 3.3. The results for behavior of solutions for diﬀerence equations can be obtained from the results for (2.1). The idea is to evaluate along the orbitOΦ(x0),x0∈᏶.

References

[1] S. N. Elaydi,An Introduction to Diﬀerence Equations, 2nd ed., Undergraduate Texts in Mathe-matics, Springer, New York, 1999.

[2] A. O. Gel’fond,Differenzenrechnung, Hochschulbücher für Mathematik, vol. 41, VEB Deutscher Verlag der Wissenschaften, Berlin, 1958.

[3] M. Kuczma, Functional Equations in a Single Variable, Monografie Matematyczne, vol. 46, Pa ´nstwowe Wydawnictwo Naukowe, Warsaw, 1968.

[4] M. Kuczma, B. Choczewski, and R. Ger,Iterative Functional Equations, Encyclopedia of Mathe-matics and Its Applications, vol. 32, Cambridge University Press, Cambridge, 1990.

[5] F. Neuman,Funkcion´aln´ı Rovnice, SNTL, Prague, 1986.

Jitka Laitochová: Mathematical Department, Faculty of Education, Palacký University, ˇZiˇzkovo námˇesti 5, Olomouc 77140, Czech Republic