Canards Solutions of Difference Equations with Small Step Size

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Canards Solutions of Difference Equations with SmallStep SizeAbir El-rabih aa UFR de Mathématique et d'Informatique , Université Louis Pasteur , 7 rue René Descartes,Strasbourg cedex, 67084, FrancePublished online: 17 Sep 2010.

To cite this article: Abir El-rabih (2003) Canards Solutions of Difference Equations with Small Step Size, Journal of DifferenceEquations and Applications, 9:10, 911-931, DOI: 10.1080/1023619031000080862

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Canards Solutions of Difference Equationswith Small Step Size

ABIR EL-RABIH*,†

UFR de Mathematique et d’Informatique, Universite Louis Pasteur, 7 rue Rene Descartes,67084 Strasbourg cedex, France

(Received 1 June 2002; Revised 28 November 2002; In final form 11 December 2002)

We consider difference equations of the form

yð1; x þ 1Þ ¼ xyð1; xÞ þ 1f ð1; x; að1Þ; yð1; xÞÞ;

where 1 . 0 and small, a(1) is a scalar function of 1 to be determined and f is an analytic function of 1, x, a and y, in aneighborhood of ð0; 1; a0; 0Þ [

4: Under some transversality condition with respect to a, we show the existence of

values of a(1) for which the difference equation admits canard solutions y(1, x), i.e. solutions on certain domainscontaining x ¼ 1 that are bounded uniformly with respect to 1. We study the asymptotic behavior of the canards as1! 0:

We also give an example of linear difference equations of the above form where f is replaced by að1Þ þ gðxÞ; gsome piecewise continuous function on some interval containing 1. Here we study solutions on discrete sets andcompare their properties with the analytic case.

The main tool of the proof is the construction, on suitable spaces of analytic functions, of a right inverse of anoperator L1 of the form

L1ða; yÞð1; xÞ ¼1

1½ yð1; x þ 1Þ2 ðx þ 1AðxÞÞyð1; xÞ�2 BðxÞað1Þ;

where Bð1Þ – 0:

Keywords: Difference equation; Difference operator; Inverse operator; Canard solution; Sum of a function

AMS Classification: 39A; 47B39

INTRODUCTION

We consider a (family of) difference equations of the form

yð1; x þ 1Þ ¼ xyð1; xÞ þ 1f ð1; x; að1Þ; yð1; xÞÞ; ð1Þ

where

. 1 [�0; 10], 10 being some fixed small positive number,

. a is some scalar function of 1 to be determined,

ISSN 1023-6198 print/ISSN 1563-5120 online q 2003 Taylor & Francis Ltd

DOI: 10.1080/1023619031000080862

*Corresponding author. Tel.: þ33-03-90-24-02-68. Fax: þ 33-03-90-24-03-28.E-mail: [email protected]

†Scholarship holder of the National Council of Scientific Research in Lebanon.

Journal of Difference Equations and Applications,

Vol. 9, No. 10, October 2003, pp. 911–931

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. y is some complex function of 1 and the complex variable x and

. f is an analytic function of 1 in a neighborhood of [0, 10], and of the variables a, x, y in a

neighborhood of ða0; 1; 0Þ [ 3:

Equation (1) is closely related to the static bifurcation problem [7]

ynþ1 ¼ xyn þ 1f ð1; x; að1Þ; ynÞ; ð2Þ

where y0 is arbitrary, x is a real or complex parameter as well as the variable y and f is some

C 1 or analytic function. For x – 1 and sufficiently small 1, Eq. (2) admits, by the implicit

function theorem, a unique fixed point y* close to 0. This fixed point is stable and attractive if

0 , x , 1; but unstable and repulsive if x . 1 (cf. Ref. [4], p.22). If x ¼ 1; a fixed point

might exist for small 1 and suitably chosen a(1), but its stability properties depend upon

properties of f. In any case, x ¼ 1 is a parameter value associated to a stability loss of the

fixed point and thus bifurcation might occur; this is the so-called static bifurcation because

x remains fixed.

The study of Eq. (1) is equivalent to a certain dynamic bifurcation problem [11], if

considered on the discrete set of all x ¼ x0 þ n1; where x0 is arbitrarily fixed. Putting

xn ¼ x0 þ n1; yn ¼ yðxnÞ; Eq. (1) can be written as:

xnþ1 ¼ xn þ 1; ynþ1 ¼ xnyn þ 1f ð1; xn; að1Þ; ynÞ: ð3Þ

Here, the parameter value x changes slowly with n; the question is how closely this

dynamic bifurcation problem is related to the static one. We refer to Ref. [11] for a more

detailed discussion of static and dynamic bifurcation.

In the present article, we are interested in finding “canard” [2,3,5,7,9] values and

solutions of Eq. (1), i.e. values a(1) for which Eq. (1) admits well-bahaved (see below)

solutions that are defined on sets of x-values containing points x . 1 and x , 1: These

canards can be studied for discrete subsets x0; . . .; xN of the real axis, x0 , 1 , xN or for

complex domains containing x ¼ 1: The existence of a canard value and solution for a

domain containing x ¼ 1 is much more restrictive than the existence for just discrete

subsets; it requires that the canard value can be chosen the same for all discrete subsets of

the domain.

More precisely, our aim is to show the existence of functions a(1) for which Eq. (1) admits

analytic solutions y bounded for small 1 . 0 and on certain bounded x-domains containing

the “critical” point x ¼ 1; see theorem 7, section 4.

In the discrete (real) case, only a linear form of Eq. (1) is considered. We prove

the existence of canards in this case and show that the interval of possible canard values

has exponentially small length for a given discrete set x0; . . .; xN : In an example, we

show that for non analytic f, the canard values cannot be chosen independent of the

discrete set. We also study the asymptotic behavior as 1! 0 of the family y1 of

solutions of Eq. (1).

Observe that we chose to discuss the specific family of Eqs. (1) to simplify matters but our

results could be easily generalized to families of difference equations of the form

yð1; x þ 1Þ ¼ gðxÞyð1; xÞ þ 1f ð1; x; að1Þ; yð1; xÞÞ; ð4Þ

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having the same properties as Eq. (1) and where g(x) is some analytic function

such that gðxÞ2 1 has a simple zero. The domain of existence of canard depends now

on g.

Some recent work on solutions for difference equations and invariant curves can be found

in Refs. [8,10,11]. However, it seems that no work has been yet done on “ephemeral canard”

solutions that are discussed in this article. By “ephemeral canard” solutions, we mean

“canard” solutions that are short-lived in the sense that the a-interval of existence has only

exponentially small length.

The principal result of this work is the existence of analytic solutions a(1), y(1, x) bounded

on certain x-domains to be discussed later. The proof uses some right inverse of a certain

difference operator and the fixed point theorem.

In order to transform Eq. (1) into a fixed point equation, we first need to

construct, on suitable spaces of analytic functions, a right inverse of operators L1 of the

form

L1ða; yÞð1; xÞ ¼1

1½ yð1; x þ 1Þ2 ðx þ 1AðxÞÞyð1; xÞ�2 BðxÞað1Þ; ð5Þ

where A(x) and B(x) are holomorphic functions. Here we need the assumption that

Bð1Þ – 0: The construction of this inverse operator is done at the end of the third

section after several preparatory steps.

First, we construct a right inverse T1 of operators L1 given by

~L1yð1; xÞ ¼1

1½ yð1; x þ 1Þ2 ðx þ 1AðxÞÞyð1; xÞ�: ð6Þ

Next, we discuss x-domains for which T1 defines a bounded operator. We introduce

inverse operators T^1 of L1 that are bounded on certain x-domains, on the left,

respectively the right, of x ¼ 1; to be described later. Then, we show that, given g(x, 1),

the functions Tþ1 h and T2

1 h are the same for hðx; 1Þ ¼ BðxÞað1Þ þ gð1; xÞ provided a(1)

has a certain value; here we need that Bð1Þ – 0: This leads to an explicit formula for a

right inverse of L1.

In the fourth section, we use the newly constructed right inverse of L1 and the fixed point

principle to prove the existence of bounded analytic solutions of Eq. (1) in the general

(nonlinear) case.

This new result gives rise to open questions. We mention below some of them to be

pursued in the context of my PhD research.

. Prove local results on canards for difference equations using Gevrey

techniques analogous to those for differential equations in Ref. [3]. Then,

compare these results to the Gevrey results in the case of bifurcation delay found

in Ref. [1].

. Extend the discussions on the discrete solutions; for example, treat nonlinear

equations and prove the conjecture that all values of a(1,u), 0 # u # 1 lie in some

interval of length Oð1 kþ1Þ if f is C k.

. Show exponential closeness also in the complex case.

. Extend the result to infinite domains; here, as for example in Ref. [9], canard

solutions might be unique.

CANARDS SOLUTIONS OF DIFFERENCE EQUATIONS 913

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THE PROBLEM ON THE REAL LINE

The Real Linear Case

We consider the boundary value problem

yðx þ 1Þ ¼ xyðxÞ þ að1Þ þ 1f ð1; xÞ; yðaÞ ¼ yað1Þ; yðbÞ ¼ ybð1Þ; ð7Þ

where 0 , a , 1 , b and ya(1) and yb(1) are bounded. We also assume that f is bounded on

[a, b ]. We call a discrete solution (a(1), y(1, x)) of Eq. (7) on [a, b ] a value a(1) and a

function y(1, x) defined on the finite set {a;aþ 1; . . .;b2 1;b}: Observe that such a discrete

solution is only defined if b2 a is a multiple of 1. Moreover, we say that it is bounded if both

a(1) and y(1, x) are bounded on the set of all 1, x such that ðb2 aÞ=1 and ðx 2 aÞ=1 are

integers.

Theorem 1 There exists a discrete bounded solution (a(1), y(1, x)) of Eq. (7) on [a, b ].

Proof We set x0 ¼ a; 1 ¼ ðb2 aÞ=n1; n1 [ N and xn ¼ aþ n1; n ¼ 0; . . .; n1: Then

Eq. (7) admits a discrete solution given by

yð1; xnÞ ¼Xn21

k¼0

ðað1Þ þ 1f ð1; xkÞÞYn21

m¼kþ1

xm

" #þ yað1Þ

Yn21

m¼0

xm; n ¼ 1; . . .; n1; ð8Þ

where a(1) is determined by yðbÞ ¼ ybð1Þ: We denote

In UYn21

m¼1

xm; Jn UXn21

k¼0

Yn21

m¼kþ1

xm; I U In1; J U Jn1

; and ð9Þ

k fk U max1[�0;10�;x[½a;b�j f ð1; xÞj: ð10Þ

Then

að1Þ ¼1

Jybð1Þ2 x0yað1ÞI 2 1

Xn121

k¼0

f ð1; xkÞYn121

m¼kþ1

xm

!; ð11Þ

Obviously,

jað1Þj # 1k fk þ1

Jjybð1Þ2 x0·yað1Þ·Ij; and ð12Þ

jyð1; xnÞj # ðjað1Þj þ 1k fkÞJn þ x0·jyað1Þ·Inj: ð13Þ

In order to show that the corresponding solution (a(1), y(1, x)) is bounded, we put

gnðxkÞ ¼Yn21

m¼kþ1

xm; k ¼ 0; . . .; n 2 1:

Then gnðxk21Þ ¼ xkgnðxkÞ and gnðxn21Þ ¼ 1: It is natural to look for an analytic solution

of the homogeneous equation gðx 2 1Þ ¼ xgðxÞ: We introduce a new unknown z(1, x)

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given by

zð1; xÞ ¼ 1 log gðxÞ:

Then, z satisfies zð1; x 2 1Þ ¼ zð1; xÞ þ 1 log x: That is, z is a so-called sum of the complex

function 2 log x (a definition of a sum is given on page 10). It is known [6] that such a sum

satisfying additionally zð1; 1Þ ¼ Oð1Þ exists and is of the form

zð1; xÞ ¼ ~zðxÞ þ Oð1Þ; where ~zðxÞ ¼ x 2 x log x; ð14Þ

that is zð1; 0Þ ¼ Oð1Þ: Then gðxÞ ¼ expðð1=1Þzð1; xÞÞ; and gðxkÞ ¼ expðð1=1Þzð1; xkÞÞ:

Since gnðxn21Þ ¼ 1; we have gnðxkÞ ¼ expðð1=1Þðzð1; xkÞ2 zð1; xn21ÞÞÞ: Moreover,

I ¼ gn1ðx0Þ ¼ O exp

1

1ð~zðx0Þ2 ~zðxn121ÞÞ

� �� ¼ O exp

1

1ð~zðaÞ2 ~zðbÞÞ

� �� ;

and there exist positive constants L1, L2 such that

J ¼Xn121

k¼0

gn1ðxkÞ $ L1

Xn121

k¼0

exp1

1ð~zðxkÞ2 ~zðxn121ÞÞ

� �$ L2

Xn121

k¼0

exp1

1ð~zðxkÞ2 ~zðbÞÞ

� �:

As z is increasing on [a, 1] and decreasing on [1, b ], there exists L3 . 0 with

J $L3

1

ðba

exp1

1ð~zðtÞ2 ~zðbÞÞ

� �dt;

which, using Laplace’s method and ~zð1Þ ¼ 1; is shown to be greater than or equal to

L4121=2 expðð1=1Þð1 2 ~zðbÞÞÞ; with L4 . 0:

Since also ya and yb are bounded, we have

yb=J ¼ O 11=2 exp1

1ð~zðbÞ2 1Þ

� �� ; and ð15Þ

x0·ya·I=J ¼ O 11=2 exp1

1ð~zðaÞ2 1Þ

� �� : ð16Þ

With Eq. (12), this implies that jað1Þj # 1k fk þ gð1Þ; where g(1) is an exponentially small

function: gð1Þ ¼ Oð11=2 expð1=1ðmaxð~zðaÞ; ~zðbÞÞ2 1ÞÞÞ:

Then a is bounded since f is bounded on [a, b ]. Next we show that y is also bounded.

For xn # 1; we have

jyð1; xnÞj # ð21kfk þ gÞJn þ ajyað1ÞjI: ð17Þ

We have as before

Jn ¼ O1

1

ðxn

a

exp1

1ð~zðtÞ2 ~zðxnÞÞ

� �dt

� �¼ Oð121=2Þ; and

In ¼ O exp1

1ð~zðaÞ2 ~zðxnÞÞ

� �� :

Since z is increasing on [a, 1], In ¼ Oð1Þ: This shows that y is bounded on [a, 1].


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The proof that y is bounded on [1, b ] is symmetric to the above case. We rewrite Eq. (7) as

yðxÞ ¼1

xyðx þ 1Þ2

1

xðað1Þ þ 1f ð1; xÞÞ: ð18Þ

We keep 1 and n1 as in the above case for xn # 1: However, we set x0 ¼ b and xn ¼ b2 n1;

n ¼ 0; . . .; n1: Then Eq. (7) admits a discrete solution given by

yð1; xnÞ ¼Xn

k¼1

21

xk

ðað1Þ þ 1f ð1; xkÞÞYn

m¼kþ1

1

xm

" #þ ybð1Þ

Yn

m¼1

1

xm

; n ¼ 1; . . .; n1: ð19Þ

Then

jyð1; xnÞj # C1

Xn21

k¼0

Yn21

m¼kþ1

1

xm

þ C2

Yn21

m¼1

1

xm

;

where C1 and C2 are constants.

By a similar argument to the case xn [ ½a; 1�; we have

Yn21

m¼1

1

xm

¼ O exp1

1ð~zðbÞ2 ~zðxnÞÞ

� �� ; and

Xn21

k¼0

Yn21

m¼kþ1

1

xm

¼ O1

1

ðxn

b

exp1

1ð~zðtÞ2 ~zðxnÞÞ

� �dt

� �;

where the function z is decreasing on [1, b ]. This shows that y is bounded on [1, b ]. A

Theorem 2 If (a1(1), y1(1, x)) and (a2(1), y2(1, x)) are two bounded discrete solutions of

Eq. (7), then a1 2 a2 is exponentially small.

Proof Let d ¼ y1 2 y2: Then d satisfies the boundary value problem

dðx þ 1Þ ¼ xdðxÞ þ a1ð1Þ2 a2ð1Þ; dðaÞ ¼ dað1Þ ¼ y1ð1;aÞ2 y2ð1;aÞ; dðbÞ ¼ dbð1Þ

¼ y1ð1;bÞ2 y2ð1;bÞ

Since da(1) and db(1) are bounded, we have by Eqs. (8) and (11)

dð1; xnÞ ¼ ða1ð1Þ2 a2ð1ÞÞJn þ adað1ÞIn; n ¼ 1; . . .; n1;

a1ð1Þ2 a2ð1Þ ¼dbð1Þ

J2 adað1Þ

I

J;

respectively.

By Eqs. (15) and (16), the quotients 1/J and I/J are exponentially small. This finishes the

proof. A

Next, we would like to see what happens if we change the discrete set of definition of our

solution by shifting it to the right, say to {aþ u1; aþ ðuþ 1Þ1; . . .;bþ ðu2 1Þ1; bþ u1},

where u [ ½0; k�; k, a fixed integer such that aþ u1 , 1: We have done some numerical

experiments that illustrate the behavior of a. We develop this in the example below.

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Example

We consider the boundary value problem

yðx þ 1Þ ¼ xyðxÞ þ að1Þ þ 1 signðx 2 1Þ; yðaÞ ¼ yðbÞ ¼ 0; ð20Þ

where signðx 2 1Þ ¼ 1 if x $ 1;21 if x , 1 and 0 , a , 1 , b: It is clear that there exits a

unique value of a(1) for which this boundary value problem admits a family of discrete

solutions y1ðxÞ1[�0;10� bounded on [a, b ] and that tends to the slow curve y0 ¼ 0 as 1! 0:

However, we are interested in the behavior of a as we move a little our boundary conditions.

In other words, what will become the new value of a if we consider the new boundary

conditions yðaþ u1Þ ¼ yðbþ u1Þ ¼ 0; where u [ ½0; k�; k a fixed integer such that

aþ u1 , 1? We denote such value by að1; uÞ ¼ auð1Þ ¼ au:

Numerical experiments show that a ¼ að1; uÞ depends on u in a “quasi-periodic” way, i.e.

að1; uþ 1Þ2 að1; uÞ is exponentially small (See Fig. 1 on page 7 and Fig. 2 on page 8).

Indeed, this “quasi-periodicity” is easily inferred from the above section. To see this, we

denote (au(1), yu(1, x)) the discrete solution of the boundary value problem

yðx þ 1Þ ¼ xyðxÞ þ að1Þ þ 1 signðx 2 1Þ; yðaþ u1Þ ¼ yðbþ u1Þ ¼ 0; ð21Þ

and ðauþ1ð1Þ; yuþ1ð1; xÞÞ the discrete solution of the boundary value problem

yðx þ 1Þ ¼ xyðxÞ þ að1Þ þ 1 signðx 2 1Þ; yðaþ ðuþ 1Þ1Þ ¼ yðbþ ðuþ 1Þ1Þ ¼ 0: ð22Þ

FIGURE 1 The function ai=10 2 a0 for 1 ¼ 0:017; a ¼ 0:3; b ¼ 2:


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Then both solutions can be extended to discrete solutions of the boundary value problem

on the interval ½aþ u1;bþ ðuþ 1Þ1�:

yðx þ 1Þ ¼ xyðxÞ þ að1Þ þ 1 signðx 2 1Þ; yðaþ u1Þ

¼ ya;u , 1; yðbþ ðuþ 1Þ1Þ ¼ yb;uþ1 , 1: ð23Þ

Then our claim is shown by Theorem 2.

In the following sections, we will construct analytic solutions of Eq. (1) defined on certain

bounded x-domains containing x ¼ 1; these contain, of course, real intervals [a, b ] of the

form studied above. As a consequence, the function a(1, u) is not only quasi-periodic in this

case, but—as a consequence of Theorem 2–exponentially close to some constant (i.e.

independent of u). This was one of the motivations to study the existence of such solutions

defined on domains.

CONSTRUCTION OF AN INVERSE OF LL1

We first need to construct a right inverse of L1 given by Eq. (5). In other words, given 1 . 0

and some domain D , ; that we describe later, we want to construct a bounded linear

operator g1, on the banach space �HðDÞ; of the holomorphic bounded functions on D, with the

following property: given h [ �HðDÞ; then g1h satisfies L1g1h ¼ h:

We also consider the equation

~L1yð1; xÞ ¼ hð1; xÞ; ð24Þ

where L1 is defined by Eq. (6). In Refs. [6,10], equations of the form

D1yð1; xÞ ¼ hðxÞ; ð25Þ

FIGURE 2 The function ai=10 2 aði=10þ1Þ for 1 ¼ 0:017; a ¼ 0:3; b ¼ 2:

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where D1 is given by

D1y ¼yðx þ 1Þ2 yðxÞ

1; ð26Þ

were treated. For the convenience of the reader, we recall this discussion and some

definitions.

Definition 3 [10] (a) A complex domain D is called horizontally convex if for all x, x 0 in D

with same imaginary part the line[x, x 0] is a subset of D.

(b) A path g : I , R! is called c-ascending, c . 0 if

;t1; t2 [ I ðt1 , t2 ) Imðg ðt2Þ2 g ðt1ÞÞ $ cjg ðt2Þ2 g ðt1ÞjÞ:

(c) A simply connected domain D is called c-ascending if there are unique x þ and x 2 in

C1(D) (maybe infinite) that are of respective maximal and minimal imaginary parts in D and

if the boundary of D consists of two c-ascending paths from x 2 to x þ. The points x þ and x 2

are called the extreme points of D.

Here C1(D) denotes the closure of D.

Theorem 4 [10] Suppose that D , is a bounded simply connected domain that is

c-ascending with c # 1=2: We call its extreme points x^. We fix some 1 [�0; 1] and put

D1 ¼ {x þ 1sjx [ V; s [ ½21=2; 1=2�} ¼ D þ ½21=2; 1=2�:

Let �HðD1Þ be the space of the holomorphic and bounded functions h : D1 ! : Then there

exists a bounded linear operator V1, on �HðD1Þ, that is a right inverse of D1. This operator

can be chosen as follows:

. t is some number in [21/8, 1/8],

. g2x;t is an ascending path joining x2 þ 1t and x 2 1=2 (avoiding x 2 1 and x) and such

that Im j is increasing as j varies on it,

. gþx;t is an ascending path joining x 2 1=2 and xþ þ 1t (avoiding x 2 1 and x) and such

that Im j is increasing as j varies on it.

. With exð1; jÞ ¼ expðð2ip=1Þðj2 xÞÞ;

. ~UthðxÞ ¼ ð1=1ÞÐg2x;tðhðjÞ=1 2 exð1; jÞÞ dj2 ð1=1Þ

Ðgþx;tðhðjÞ=1 2 ð1=exð1; jÞÞÞ dj;

. ~SthðxÞ ¼Ðg2x;t

hðjÞ dj;

. ~VthðxÞ ¼ ~St hðxÞ2 1 ~UthðxÞ;

. and finally V1hðxÞ ¼ 4Ð 1=8

21=8~VthðxÞ dt:

Consider now the operator L1 defined by Eq. (6) and the equation

~L1yð1; xÞ ¼ hð1; xÞ: ð27Þ

We reduce this equation to the previous one Eq. (25) by “variation of constants”. First, we

find a solution Z(1, x) of:

Zð1; x þ 1Þ ¼Zð1; xÞ

x þ 1AðxÞ: ð28Þ

So, we let

uð1; xÞ ¼ 1 log Zð1; xÞ; ð29Þ


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then u satisfies

D1uð1; xÞ ¼ 2logðx þ 1AðxÞÞ: ð30Þ

Whenever such a function u exists and is uniformly bounded with respect to 1 on some

compact x-domain, we say that it is a sum [6] of the complex analytic function

2 log(x þ 1A(x)) on this x-domain. It is known that such a sum satisfying additionally

uð1; 1Þ ¼ Oð1Þ exists and satisfies

uð1; xÞ ¼ x 2 x logðxÞ2 1 þ Oð1Þ; ð31Þ

Using Eq. (29), Z(1, x) satisfies

Zð1; xÞ ¼ exp1

1ðx 2 x log x 2 1Þ þ Oð1Þ

� �: ð32Þ

Back to Eq. (27). It is equivalent to D1 ~y ¼ ~h; with

. ~yð1; xÞ ¼ Zð1; xÞyð1; xÞ; and

. ~hð1; xÞ ¼ ðZð1; xÞhð1; xÞ=ðx þ 1AðxÞÞÞ:

Thus we obtain the following result:

Given the same hypothesis as Theorem 4, but where D ] 1 is in the right half-plane

Re x . 0; there exists a linear operator T1, on �HðD1Þ; that defines a right inverse of ~L1 and it

is given by

T1hð1; xÞ ¼1

Zð1; xÞV1

Zð1; xÞhð1; xÞ

x þ 1AðxÞ

� �:

In other words,

T1 U I1 þ I2 þ I3; ð33Þ

with

I1hð1; xÞ ¼4

Zð1; xÞ

ð1=8

21=8

ðg2x ; t

Zð1; jÞ hð1; j Þ

jþ 1AðjÞdj dt ð34Þ

I2hð1; xÞ ¼24

Zð1; xÞ

ð1=8

21=8

ðg2x ; t

Zð1; j Þhð1; j Þ

ðjþ 1AðjÞÞð1 2 exð1; jÞÞdj dt ð35Þ

I3hð1; xÞ ¼4

Zð1; xÞ

ð1=8

21=8

ðgþ

x ; t


ðjþ 1AðjÞÞð1 2 e21x ð1; jÞÞ

dj dt ð36Þ

Discussion of Boundedness of T1

Contrary to the case of V1, T1 will not be uniformly bounded with respect to 1 on �HðD1Þ:

As explained below, this is due to the presence of the new function Z(1, x) that enters into

the integrals defining T1. Indeed I1 is not anymore bounded on �HðD1Þ: We have by Eq. (32)

Zð1; jÞ

Zð1; xÞ¼ exp

1

1ðR0ðxÞ2 R0ðj ÞÞ þ Oð1Þ

� �; ð37Þ

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where R0 is the corresponding relief function, i.e. the surface described by

R2 ! R; ða; bÞ 7! R0ða þ ibÞ;

where

R0ðxÞ ¼ Reðx log x 2 x þ 1Þ: ð38Þ

Considering the relief of R0(x) in the right half-plane (see Fig. 3 on page 11), we see that

the level curves passing through the point x ¼ 1 divide the right half-plane into four regions,

two mountains and two valleys. In Figs. 3–8, the mountains are in light and the valleys in

dark gray. The mountains are to the left and the right of x ¼ 1: We denote the mountain

situated to the left of x ¼ 1 by MW and the one to its right by ME. We also call VN the valley

situated to the north of x ¼ 1 and VS the one to its south. As already mentioned, D is some

bounded complex domain, to be constructed in the right half plane Re x . 0 and containing

FIGURE 4 Real part of x log x 2 x þ 1 þ 2ipx:

FIGURE 3 Real part of x log x 2 x þ 1:


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x ¼ 1; and that is horizontally convex and c-ascending with extreme points x 2 and x þ of

minimal and maximal imaginary parts, respectively. By Fig. 3 and since x 2 is the extreme

point of D with minimal imaginary part, the integration path of I1 Eq. (34), g2x;t; joining

x2 þ 1t and x 2 1=2 (in D1), cannot be generally chosen such that R0(j) decreases as j varies

on it. That is why I1 and subsequently T1 are not anymore bounded on �HðD1Þ: To obtain

boundedness of T1, further conditions are to be imposed on D and new related inverse

operators are to be introduced on subdomains of D. We give below a heuristic discussion

about this. A detailed description of the final domains and a proof of boundedness can be

found in section “Bounded Right Inverses of ~L1”.

Adding to T1 a constant times the homogeneous solution of ~L1yð1; xÞ ¼ 0; which is

1=Zð1; xÞ; still yields a right inverse of ~L1 on �HðD1Þ: Therefore, we can modify the

integration path g2x;t of I1 into a new one so as to keep T1 a right inverse of ~L1 on �HðD1Þ:

This new path will be chosen such that R0(x) decreases on it. Then we also want the

maximum of R0(x) on D to be attained on the initial point of this new path, say at some point

x0 to be determined later (This requirement is crucial of the choice of D). However, since

1 [ D is a saddle point (we have two mountains) (see Fig. 3), this point x0 may be chosen

either to the left of 1 in ME (that we call x20 in this case) or to its right in MW (that we call xþ0 in

this case). Indeed, these two possibilities of the choice of x0 (i.e. x20 and xþ0 ) are crucial.

FIGURE 5 Real part of x log x 2 x þ 1 2 2ipx:

FIGURE 6 The domain D.

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Then D must contain the two points x20 and xþ0 : Moreover, we will consider two integration

paths of I1 : g2x20

corresponding to x20 and gþxþ

0

corresponding to xþ0 : This will give us two

new operators I21 and Iþ1 instead of I1 and subsequently two new right inverse operators of~L1 on �HðD1Þ; say T2

1 and Tþ1 : T2

1 h (respectively, Tþ1 h) will be bounded for h holomorphic

and bounded on a subdomain D2 , MW < VN < VS (respectively, Dþ , ME < VN < VS)

of D to the left (respectively to the right) of the point 1 and to be well described later. We set

x20 [ R to be close to zero and less than one and xþ0 [ R to be greater than 1 but finite.

We define:

T21 U I21 þ I2 þ I3; and ð39Þ

Tþ1 U Iþ1 þ I2 þ I3; ð40Þ

where I2 is given by Eq. (35), I3 is given by Eq. (36), I21 is given by

I21 hð1; x Þ ¼

ðg2

x20


Zð1; xÞðjþ 1Aðj ÞÞdj; with ð41Þ

g2x20

being some integration path joining x20 and x 2 1=2 and along which the function R0

decreases (As Fig. 3 shows and as discussed earlier, such a path always exists in D21 if

x2 [ VS and xþ [ VN ; see Fig. 6 in the sequel), and where Iþ1 is given by

Iþ1 hð1; xÞ ¼

ðgþ

xþ0

Zð1; jÞhð1; jÞ

Zð1; xÞðjþ 1AðjÞÞdj; with ð42Þ

FIGURE 7 The domain D 2.

FIGURE 8 The domain D þ.


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gþxþ

0

being some integration path joining xþ0 and x 2 1=2 and along which the function R0

decreases (As Fig. 3 shows and as discussed earlier, such a path always exists in Dþ1 if

x2 [ VS and xþ [ VN ; see Fig. 6 in the sequel).

We have the following results:

. T21 is a right inverse operator of ~L1 on �HðD1Þ:

. Tþ1 is a right inverse operator of ~L1 on �HðD1Þ:

. I21 is bounded on �HðD21 Þ:

. Iþ1 is bounded on �HðDþ1 Þ:

These results will be proved later in theorem 5 of section “Bounded Right Inverses of ~L1”

when the domains D, D 2 and D þ are well-defined, but the main key of the proof is the

following:

. Since T21 (respectively Tþ

1 ) differs from T1 only by a constant times the homogeneous

solution of ~L1yð1; xÞ ¼ 0 and which is 1/Z(1, x), T21 (respectively Tþ

1 ) is also a right

inverse operator of ~L1:

. the quotient ðZð1; jÞ=Zð1; xÞÞ is bounded. (This will be guaranteed by the

choice of the domains D, D 2, D þ and the choice of the paths of integration of I21 and Iþ1 )

So far, the requirements on D are not enough to ensure the boundedness of I2 (35) and I3

(36) on �HðD1Þ: We consider the cases where x is sufficiently far away from x 2 and x þ. The

cases where x is near x 2 or x þ is treated later in section “Bounded Right Inverses of ~L1”.

For Im x . Im x2 þ 1c=8; since ð1=1 2 exð1; jÞÞ ¼ Oðexpðð2p=1ÞImðj2 xÞÞÞ and by

Eqs. (37) and (38), we obtain

Zð1; jÞ

Zð1; xÞð1 2 exð1; jÞÞ¼ O exp

1

1ðR1ðxÞ2 R1ðjÞÞ

� �� ;

where

R1ðxÞ ¼ Reðx log x 2 x þ 1 þ 2ipxÞ ¼ R0ðxÞ2 2p Im x: ð43Þ

As the relief R1(x) shows (see Fig. 4 on page 11), I2 can be bounded if x2 is chosen to lie

on or above (vertically) level curve of R1, that passes through xþ0 :

Similarly, for Im x , Im xþ 2 1c=8; since ð1=1 2 e21x ð1; jÞÞ ¼ Oðexpðð2p=1ÞImðx 2 jÞÞÞ

and by Eqs. (37) and (38), we obtain

Zð1; jÞ

Zð1; xÞð1 2 e21x ð1; jÞÞ

¼ O exp1


� �� ;

where

R2ðxÞ ¼ Reðx log x 2 x þ 1 2 2ipxÞ ¼ R0ðxÞ þ 2p Im x: ð44Þ

Consequently, ðxþþ1t

x21=2

Zð1; jÞhð1; jÞ

Zð1; xÞðjþ 1AðjÞÞð1 2 e21x ð1; jÞÞ

dj;

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can be bounded if x þ is chosen to lie on or below (vertically) the level curve of R2

(see Fig. 5 on page 12) that passes through xþ0 : Recall that x 2 is of minimal imaginary part

and x þ of maximal imaginary part.

Description of D, D 2 and D 1

So far, D ] 1 is some bounded complex domain, in the right half plane Re x . 0;

that is horizontally convex and c-ascending. Moreover, D is chosen to contain four points x20 ;

xþ0 ; x 2, x þ with the following properties:

. x20 is some real point, lying in MW, at which R0(x) is maximal in D > MW :

. xþ0 is some real point, lying in ME, at which R0(x) is maximal in D > ME:

. x 2 and x þ are the extreme points of D of minimal and maximal imaginary parts,

respectively.

. x2 [ VS is chosen to lie on or above (vertically) the level curve of R1 that passes

through xþ0 :

. x þ [ VN is chosen to lie on or below (vertically) the level curve of R2 that passes

through xþ0 :

. As an example (cf. Fig. 6), we might use x20 ¼ 0:01; xþ0 ¼ 5; x2 ¼ 1=2 2 5=2i;

xþ ¼ 1=2 þ 5=2i:

. y2N [�xþ; xþ0 ½>VN such that the segment ½1 2 d; y2N � is c-ascending.

. y2S [�x2; xþ0 ½>VS such that the segment ½ y2S ; 1 2 d� is c-ascending.

. yþN [�x20 ; xþ½>VS such that the segment ½1 þ d; yþN � is c-ascending.

. yþS [�x20 ; x2½>VS such that the segment ½ yþS ; 1 þ d� is c-ascending.

Then we define

. D to be the domain enclosed within the four segments ½x20 ; xþ�; ½xþ; xþ0 �; ½xþ0 ; x2� and

½x2; x20 �: (See Fig. 6 on page 12)

. D 2 to be the domain enclosed within the six segments ½x20 ; xþ�; ½xþ; y2N �; ½y2N ; 1 2 d�;

½1 2 d; y2S �; ½ y2S ; x2� and ½x2; x20 �: (See Fig. 7 on page 13)

. D þ to be the domain enclosed within the six segments ½xþ0 ; xþ�; ½xþ; yþN �; ½yþN ; 1 þ d�;

½1 þ d; yþS �; ½ yþS ; x2� and ½x2; xþ0 �: (See Fig. 8 on page 13)

To clear any later ambiguity, the above segments are not considered as parts of the

domains!

Bounded Right Inverses of ~LL1

Keeping the same terminology as before, we put D1 ¼ D þ ½21=2; 1=2� and

D^1 ¼ D^ þ ½21=2; 1=2�; respectively. Let x [ D1; t [ ½21=8; 1=8�; and ðT2

1 ; Tþ1 Þ be

given by Eqs. (39) and (40), i.e.

T^1 hð1; xÞ ¼

ðg^

x^0


Zð1; xÞðjþ 1Aðj ÞÞdj2 4

ð1=8

21=8

ð~g2x;t


Zð1; xÞðjþ 1AðjÞÞð1 2 exð1; jÞÞdj dt

þ 4

ð1=8

21=8

ð~g þx;t


Zð1; xÞðjþ 1Aðj ÞÞð1 2 e21x ð1; j ÞÞ

dj dt respectively;


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and where

. g^x^

0

are some integration paths from x^0 to x 2 1=2; respectively.

. ~g2x;t is some c-ascending path joining x2 þ 1t to x 2 1=2 (avoiding x 2 k1; k [ Z) and

along which the function R1 is decreasing.

. ~gþx;t is some c-ascending path joining x 2 1=2 to xþ þ 1t (avoiding x 2 k1; k [ Z) and

along which the function R2 is increasing.

We show in the result below the existence of such paths and the boundedness of T^1 on

�HðD^1 Þ, respectively.

Theorem 5 The linear operators T^1 define right inverse bounded operators of ~L1 on

�HðD^1 Þ:

Proof First, the two operators T^1 are also right inverses of ~L1 on �HðD^

1 Þ (indeed on�HðD1Þ):

ðT1 2 T21 Þðhð1; xÞÞ ¼ ðI1 2 I21 Þðhð1; xÞÞ ¼ ðI1Þðhð1; xÞÞ2 4

ð1=8

21=8

ðI21 Þðhð1; xÞÞ dt

¼4

Zð1; xÞ

ð1=8

21=8

ðg2x;t

Zð1; jÞ hð1; j Þ

jþ 1AðjÞdj dt ¼ C·Zð1; xÞ;

where C is some constant. That is, T21 differs from T1 only by a constant times a

homogeneous solution of ~L1hð1; xÞ ¼ 0 and which is 1/Z(1, x). It follows that, T21 is also

a right of ~L1 on �HðD21 Þ: Similar argument shows that Tþ

1 is also a right of ~L1 on�HðDþ

1 Þ:

Next, T^1 hð1; xÞ is bounded on �HðD^

1 Þ; respectively:

We show that I21 ; Iþ1 ; I2 and I3 are bounded. For x [ D^1 ; we can, without loss of

generality, assume that x 2 1=2 [ D^1 : Otherwise, ðx þ 1Þ2 1=2 ¼ x þ 1=2 [ D^

1 and the

estimate for T^1 hð1; x þ 1Þ together with T^

1 hð1; xÞ ¼ ðT^1 hð1; x þ 1Þ2 1hð1; xÞÞ=ðx þ 1AðxÞÞ

yields the estimate for T^1 hð1; xÞ.

We write x 2 1=2 ¼ �x þ 1 �a where �x [ D^ and �a [ ½21=2; 1=2�:

I^1 is bounded on �HðD^1 Þ :

By Fig. 3 and since R0(x) attains its maximum in D ^ at the point x^0 ; there exists a path

�g^x^

0

; joining x^0 to �x and along which R0 decreases. Then, the path g^x^

0

is chosen in D^1 to

be close to �g^x^

0

; with a distance at most of the order of 1. By Eq. (37), I^1 becomes bounded on�HðD^

1 Þ:

I2 is bounded on �HðD1Þ :

We distinguish between two cases:

Case 1 Im x . Im x 2 þ 1c/8

Since D is a c-ascending domain, there exists a c-ascending path g2�x joining x 2 to �x: Then ~g2x;tcan be chosen close to g2�x with a distance of the order of 1. Eqs. (37) and (38) together with

ð1=1 2 exð1; j ÞÞ ¼ Oðexpðð2p=1Þ Imðj2 xÞÞÞ imply that:

Zð1; jÞ


1

1ðR1ðxÞ2 R1ðj ÞÞ

� �� ;

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As Fig. 4 (page 11) shows, the path g2�x may also be chosen such that R1(j) decreases as j

varies on it. Then I2(h) becomes bounded for bounded h.

Case 2 Im x [ ½Im x2; Im x2 þ 1c=8�

Here the above path ~g2x;t is no longer sufficient for the estimates. Some polygonal paths must

be instead considered. Because these paths can be chosen exactly as in (cf. Ref. [11], p. 24),

we will not treat this case and we refer the reader there for a detailed discussion.

I3 is bounded on �HðD1Þ :

We also distinguish between two cases:

Case 1 Im x , Im xþ 2 1c=8

Since D is a c-ascending domain, there exists a c-ascending path gþ�x joining �x to x þ. Then ~gþx;tcan be chosen close to gþ�x with a distance of the order of 1. Eqs (37) and (38) together with

ð1=1 2 e21x ð1; jÞÞ ¼ Oðexpðð2p=1Þ Imðx 2 jÞÞÞ imply that:

Zð1; jÞ


1


� �� ;

As Fig. 5 (page 12) shows, the path gþ�x may also be chosen such that R2(j) increases as j

varies on it. Then I3(h) becomes bounded for bounded h.

Case 2 Im x [ ½Im xþ 2 1c=8; Im x þ]

Here also some polygonal paths must be considered exactly as in (cf. Ref. [11], p. 24). A

As a direct consequence, we get: given hð1; xÞ [ �HðD1Þ; Eq. (24) admits, on D, two analytic

solutions y^ð1; xÞ ¼ T^1 hð1; xÞ that are bounded on D ^ respectively.

A Right Inverse of LL1

Replacing h(1, x), in the linear “canard” Eq. (24), by BðxÞað1Þ þ gð1; xÞ; we obtain

yð1; x þ 1Þ ¼ ðx þ 1AðxÞÞyð1; xÞ þ 1BðxÞað1Þ þ 1gð1; xÞ; ð45Þ

where A;B : D ! are holomorphic and bounded with Bð1Þ – 0 and g :�0; 10� £ D! is

also holomorphic and bounded. Here D is as described in section “Description of D, D 2 and

D þ” (see Fig.6). This equation has, thanks to theorem 5, two analytic solutions y^ð1; xÞ ¼

T^1 ðBðxÞað1Þ þ gð1; xÞÞ that are bounded on D ^ respectively. We want to find a(1) for

which Eq. (45) has an analytic bounded solution on D. Eq. (45) and its converse y ð xÞ ¼

ð1=ðx þ 1AÞÞ ð y ð x þ 1Þ þ 1Ba 2 1gÞ ensure analytic continuation of y 2, respectively, y þ,

on D. However, they may be unbounded on D. Therefore, it is natural to choose a such that

yþ ¼ y2; i.e.

T21 ðBðxÞað1Þ þ gð1; xÞÞ ¼ Tþ

1 ðBðxÞað1Þ þ gð1; xÞÞ:

Since

ðT21 2 Tþ

1 Þhð1; xÞ ¼

ðxþ0

x20

Zð1; jÞhð1; jÞ dj

jþ 1AðjÞ;


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we obtain

að1Þ ¼ 2

ðxþ0

x20

Zð1; j Þgð1; j Þ dj

jþ 1Aðj Þ

�ðxþ0

x20

Zð1; jÞBðjÞ dj

jþ 1AðjÞ: ð46Þ

We define two new operators T1 and T2 on �HðD1Þ by T1ðgÞ U a and

T2ðgÞ U T21 ðBa þ gÞ ¼ Tþ

1 ðBa þ gÞ; where a is given by Eq. (46). Then we have the

following result:

Theorem 6 The operator g1, given by g1gð1; xÞ ¼ ðT1g; T2gÞ defines a linear bounded

right inverse operator of L1 on �HðD1Þ.

Proof Using Laplace integral and by Eq. (32), the numerator on the right hand side of

Eq. (46) is bounded above by ð2p1Þ1=2kgk expðOð1ÞÞ=min1[�0;10�;j[½x20;xþ

0�jjþ 1AðjÞj and its

denominator behaves as ð2p1Þ1=2Bð1Þ expðOð1ÞÞ as 1!0. Then there exists some constant k

such that jað1Þj # kkgk; ;1 [�0; 10�: So, T1 is bounded on �HðD1Þ: Recall that the two

operators T21 and Tþ

1 ; both defined on �HðD1Þ; are bounded on �HðD21 Þ and �HðDþ

1 Þ;

respectively. Moreover, for this value of a(1), as given by Eq. (46), these two operators T21

and Tþ1 act on BðxÞað1Þ þ gð1; xÞ in the same way. Then the common value T2ðgð1; xÞÞ U

T21 ðBðxÞað1Þ þ gð1; xÞÞ ¼ Tþ

1 ðBðxÞað1Þ þ gð1; xÞÞ is an analytic solution of Eq. (45), on the

whole domain D and it is bounded on D2 < Dþ: It remains to show that it is bounded on

D* U D \ ðD2 < DþÞ: We choose an appropriate subdomain ~D of D containing D* and

such that › ~D , D2 < Dþ: To see why this is possible, we set xN ¼ ½y2N ; 1 2 d�> ½yþN ; 1 þ d�

and xS ¼ ½ y2S ; 1 2 d�> ½ yþS ; 1 þ d�: Then D* consists of the four segments ½1 2 d; xN�;

½xN ; 1 þ d�; ½1 þ d; xS�; ½xS; 1 2 d� and the region enclosed within them; (recall that the

domains D, D 2 and D þ are open) the existence of ~D becomes now obvious.

Applying the maximum modulus principle on ~D; we obtain that T2g is an analytic bounded

solution of Eq. (45) on D. This completes the proof. A

Consequently, Eq. (45) has an analytic bounded solution (a(1), y(1, x)) given by a ¼ T1g;

y ¼ T2g:

ANALYTIC SOLUTIONS OF NON LINEAR EQUATIONS ON BOUNDED

DOMAINS

We first have to discuss approximations of solutions of Eq. (1) as we want our solution to

behave nicely as 1! 0 þ : Putting formally 1 ¼ 0; we find that necessarily y ¼ Oð1Þ:

Comparing the coefficients of 1, we find that y1ðxÞ ¼ lim 1! 0þð1=1Þ y ð1; xÞ and

a0 ¼ lim1!0það1Þ; if they exist, have to satisfy

y1ðxÞ ¼ xy1ðxÞ þ f ð0; x; a0; 0Þ: ð47Þ

Hence a necessary condition for the existence of a well behaved solution is

f ð0; 1; a0; 0Þ ¼ 0: We assume in the sequel that such a value of a0 exists and has been

chosen. Without loss of generality, we can assume that a0 ¼ 0: Then y1ðxÞ ¼

f ð0; x; 0; 0Þ=ð1 2 xÞ is analytic also at x ¼ 1: Thus we will prove.

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Theorem 7 Let 10 . 0 be given and D be some bounded domain satisfying the conditions

of section “Description of D, D 2 and D þ” such that f is defined and analytic on

a neighborhood of ½0; 10� £ clðDÞ £ {ð0; 0Þ}: We assume that f ð0; 1; 0; 0Þ ¼ 0 and

ð›f=›aÞð0; 1; 0; 0Þ – 0: Then, for 11 . 0 sufficiently small, Eq. (1) admits a family

{ða1; y1Þ}1[�0;11� of solutions such that y1 are analytic on D and satisfy y1ðxÞ ¼ Oð1Þ

uniformly on D.

Proof We can assume, without loss of generality, that f ð0; x; 0; 0Þ ¼ 0 for all x. This can

always be achieved by putting y ¼ 1y1ðxÞ þ ~y with y1(x) as defined above the theorem.

Indeed, the equation for ~y is now

~yð1; x þ 1Þ ¼ x~yð1; xÞ þ 1~f ð1; x; að1Þ; ~yð1; xÞÞ;

where ~f ð1; x; a; ~yÞ ¼ f ð1; x; a; 1y1ðxÞ þ ~yÞ þ xy1ðxÞ2 y1ðx þ 1Þ; and therefore, ~f ð0; x; 0; 0Þ ¼

0 for all x by Eq. (47).

As already mentioned, we want to solve Eq. (1) using the fixed point theorem. Therefore,

we rewrite it as

yð1; x þ 1Þ ¼ ðx þ 1AðxÞÞyð1; xÞ þ 1BðxÞað1Þ þ 1hð1; x; að1Þ; yð1; xÞÞ; ð48Þ

where

. AðxÞ ¼ ð›f=›yÞð0; x; 0; 0Þ;

. BðxÞ ¼ ð›f=›aÞð0; x; 0; 0Þ;

. hð1; x; að1Þ; yð1; xÞÞ ¼ f ð1; x; að1Þ; yð1; xÞÞ2 AðxÞyð1; xÞ2 BðxÞað1Þ:

With the notation c ða; yÞð1; xÞ ¼ hð1; x; að1Þ; yð1; xÞÞ; Eq. (1) reads

L1ða; yÞð1; xÞ ¼ c ða; yÞ ð1; xÞ; ð49Þ

where the left-hand side of the equation denotes the main linear part of Eq. (1) and the right-

hand side contains the remaining terms.

In the preceding section, we constructed a right inverse g1 of L1 on D under the condition

that Bð1Þ ¼ ð›f=›aÞð0; 1; 0; 0Þ – 0 which is satisfied here. Then it is sufficient to solve the

fixed point equation ða; yÞ ¼ g1cða; yÞ:

We construct below a closed bounded subset B of a certain Banach space E on which we

show that g1c is a contraction. First, we define

. E ¼ {a :�0; 11�! R; a is bounded},

. F ¼ {y :�0; 11� £ K! ; y is holomorphic and bounded}, both endowed with the usual

norm.

. E ¼ E £ F;

. B ¼ {ða; yÞ [ E; jaj # r; jyj # r};

where 11, r are to be determined below.

We have

c ða2; y2Þ2 c ða1; y1Þ ¼

ð1

0

›c

›yða1; y1 þ s ð y2 2 y1ÞÞ ð y2 2 y1Þ ds

þ

ð1

0

›c

›aða1 þ sða2 2 a1Þ; y2Þða2 2 a1Þ ds;


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with

›c

›yða; yÞh : ð1; xÞ 7!

›f

›yð1; x; að1Þ; yð1; xÞÞ2 AðxÞ

� �hð1; xÞ;

and

›c

›aða; yÞb : ð1; xÞ 7!

›f

›að1; x; að1Þ; yð1; xÞÞ2 BðxÞ

� �bð1Þ:

Thus, by the definition of A and B and by the compactness of clðDÞ; we obtain for

ða; yÞ [ B

›c

›yða; yÞ ¼ Oðmaxðj1j; jaj; jyjÞÞ # Cðj1j þ jaj þ jyjÞ # Cð11 þ 2rÞ;

and

›c

›aða; yÞ ¼ Oðmaxðj1j; jaj; jyjÞÞ # Cðj1j þ jaj þ jyjÞ # Cð11 þ 2rÞ;

where C is some constant (independent of 11, r).

Now denote the norm L ¼ jg1j: Choosing r and 11 such that 2Cð11 þ 2rÞ # ð1=2LÞ;

e.g. 11, r # 112LC

; we obtain

jc ða2; y2Þ2 cða1; y1Þj #1

2Lmaxðjy2 2 y1j; ja2 2 a1jÞ

for all ða1; y1Þ; ða2; y2Þ [ B: As f ð0; x; 0; 0Þ ¼ 0 for all x and D is bounded, we

can also choose 11 such that c ð0; 0Þ : ð1; xÞ 7! f ð1; x; 0; 0Þ has a F-norm smaller than

r/(2L).

This implies that g1c : B!B is a contraction with a contraction factor at most 1/2. Thus

the existence of a fixed point (a, y) of g1c and hence of a solution of Eq. (1) under the

conditions of the theorem is proved. A

Acknowledgements

The author would like to thank her advisor Reinhard Schafke for his continuous support and

Augustin Fruchard for his remarks. The author is especially indebted to the National Council

of Scientific Research in Lebanon for financially supporting her PhD work and this research.

References

[1] C. Baesens, Gevrey series and dynamic bifurcations for analytic slow–fast mappings, Nonlinearity, 8 (1995),179–201.

[2] E. Benoıt, J. L. Callot, F. Diener and M. Diener, Chasse au canard, Collect. Math., 31(1–3) (1981), 37–119.[3] M. Canalis-Durand, J.-P. Ramis, R. Schafke and Y. Sibuya, Gevrey solutions of singularly perturbed differential

equations, J. Reine Angew. Math., 518 (2000), 95–129.[4] S. N. Elaydi, An Introduction To Difference Equations, Springer, Berlin, 1991.[5] A. Fruchard, Canards et rateaux, Ann. Inst. Fourier, Grenoble, 42(1–2) (1992), 825–855.[6] A. Fruchard, The sum of a function, Analysis, 16 (1996), 65–88.[7] A. Fruchard, Sur l’equation aux differences affine du premier ordre unidimensionnelle, Ann. Inst. Fourier,

Grenoble, 46(1) (1996), 139–181.

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[8] A. Fruchard and R. Schafke, Exponentially small splitting of separatrices for difference equations with smallstep size, J. Dynam. Control Systems, 2(no. 2) (1996), 193–238.

[9] A. Fruchard and R. Schafke, Exceptional complex solutions of the forced van der Pol equation, FunkcialajEkvacioj, 42 (1999), 201–223.

[10] A. Fruchard and R. Schafke, Analytic solutions of difference equations with small step size, J. Diff. Eq. Appl.,7 (2001), 651–684.

[11] A. Fruchard and R. Schafke, Bifurcation delay and difference equations, preprint IRMA 7, rue Rene Descartes67084 Strasbourg Cedex, France, 2001.


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Canards Solutions of Difference Equations with Small Step Size