Canada a drabek p fonda a (eds ) handbook of differential equations ordinary differential equations vol 1

The paper by Agarwal and O’Regan deals with singular initial andboundary value problems the nonlinear term may be singular in its dependent variableand is allowed to change sign.. Thepap

H ANDBOOK

Department of Mathematical Analysis, Faculty of Sciences,

University of Granada, Granada, Spain

P DRÁBEK

Department of Mathematics, Faculty of Applied Sciences,

University of West Bohemia, Pilsen, Czech Republic

A FONDA

Department of Mathematical Sciences, Faculty of Sciences,

University of Trieste, Trieste, Italy

2004

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Handbook of differential equations

Ordinary differential equations: Vol 1

Ordinary differential equations is a wide mathematical discipline which is closely related toboth pure mathematical research and real world applications Most mathematical formula-tions of physical laws are described in terms of ordinary and partial differential equations,and this has been a great motivation for their study in the past In the 20th century theextremely fast development of Science led to applications in the fields of chemistry, bi-ology, medicine, population dynamics, genetic engineering, economy, social sciences andothers, as well All these disciplines promoted to higher level and new discoveries weremade with the help of this kind of mathematical modeling At the same time, real worldproblems have been and continue to be a great inspiration for pure mathematics, particu-larly concerning ordinary differential equations: they led to new mathematical models andchallenged mathematicians to look for new methods to solve them

It should also be mentioned that an extremely fast development of computer sciencestook place in the last three decades: mathematicians have been provided with a tool whichhad not been available before This fact encouraged scientists to formulate more complexmathematical models which, in the past, could hardly be resolved or even understood Even

if computers rarely permit a rigorous treatment of a problem, they are a very useful tool

to get concrete numerical results or to make interesting numerical experiments In the field

of ordinary differential equations this phenomenon led more and more mathematicians

to the study of nonlinear differential equations This fact is reflected pretty well by thecontributions to this volume

The aim of the editors was to collect survey papers in the theory of ordinary differentialequations showing the “state of the art”, presenting some of the main results and methods

to solve various types of problems The contributors, besides being widely acknowledgedexperts in the subject, are known for their ability of clearly divulging their subject We areconvinced that papers like the ones in this volume are very useful, both for the experts andparticularly for younger research fellows or beginners in the subject The editors wouldlike to express their deepest gratitude to all contributors to this volume for the effort made

in this direction

The contributions to this volume are presented in alphabetical order according to thename of the first author The paper by Agarwal and O’Regan deals with singular initial andboundary value problems (the nonlinear term may be singular in its dependent variableand is allowed to change sign) Some old and new existence results are established andthe proofs are based on fixed point theorems, in particular, Schauder’s fixed point theo-rem and a Leray–Schauder alternative The paper by De Coster and Habets is dedicated tothe method of upper and lower solutions for boundary value problems The second orderequations with various kinds of boundary conditions are considered The emphasis is put

v

on well ordered and non-well ordered pairs of upper and lower solutions, connection tothe topological degree and multiplicity of the solutions The contribution of Došlý dealswith half-linear equations of the second order The principal part of these equations is rep-

resented by the one-dimensional p-Laplacian and the author concentrates mainly on the

oscillatory theory The paper by Jacobsen and Schmitt is devoted to the study of radial

solutions for quasilinear elliptic differential equations The p-Laplacian serves again as a

prototype of the main part in the equation and the domains as a ball, an annual region,the exterior of a ball, or the entire space are under investigation The paper by Llibre isdedicated to differential systems or vector fields defined on the real or complex plane Theauthor presents a deep and complete study of the existence of first integrals for planar poly-nomial vector fields through the Darbouxian theory of integrability The paper by Mawhintakes the simple forced pendulum equation as a model for describing a variety of nonlinearphenomena: multiplicity of periodic solutions, subharmonics, almost periodic solutions,stability, boundedness, Mather sets, KAM theory and chaotic dynamics It is a review pa-per taking into account more than a hundred research articles appeared on this subject Thepaper by Srzednicki is a review of the main results obtained by the Wa˙zewski method inthe theory of ordinary differential equations and inclusions, and retarded functional dif-ferential equations, with some applications to boundary value problems and detection ofchaotic dynamics It is concluded by an introduction of the Conley index with examples ofpossible applications

Last, but not least, we thank the Editors at Elsevier, who gave us the opportunity ofmaking available a collection of articles that we hope will be useful to mathematiciansand scientists interested in the recent results and methods in the theory and applications ofordinary differential equations

List of Contributors

Agarwal, R.P., Florida Institute of Technology, Melbourne, FL (Ch 1)

De Coster, C., Université du Littoral, Calais Cédex, France (Ch 2)

Došlý, O., Masaryk University, Brno, Czech Republic (Ch 3)

Habets, P., Université Catholique de Louvain, Louvain-la-Neuve, Belgium (Ch 2) Jacobsen, J., Harvey Mudd College, Claremont, CA (Ch 4)

Llibre, J., Universitat Autónoma de Barcelona, Bellaterra, Barcelona, Spain (Ch 5) Mawhin, J., Université Catholique de Louvain, Louvain-la-Neuve, Belgium (Ch 6) O’Regan, D., National University of Ireland, Galway, Ireland (Ch 1)

Schmitt, K., University of Utah, Salt Lake City, UT (Ch 4)

Srzednicki, R., Institute of Mathematics, Jagiellonian University, Kraków, Poland (Ch 7)

vii

1 A survey of recent results for initial and boundary value problems singular in the

R.P Agarwal and D O’Regan

2 The lower and upper solutions method for boundary value problems 69

C De Coster and P Habets

O Došlý

4 Radial solutions of quasilinear elliptic differential equations 359

J Jacobsen and K Schmitt

CHAPTER 1

A Survey of Recent Results for Initial and Boundary Value Problems Singular in the Dependent Variable

Ravi P Agarwal

Department of Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA

E-mail: agarwal@fit.edu

Donal O’Regan

Department of Mathematics, National University of Ireland, Galway, Ireland

E-mail: donal.oregan@nuigalway.ie

Contents

1 Introduction 3

2 Singular boundary value problems 8

2.1 Positone problems 8

2.2 Singular problems with sign changing nonlinearities 22

3 Singular initial value problems 51

References 67

Abstract

In this survey paper we present old and new existence results for singular initial and bound-ary value problems Our nonlinearity may be singular in its dependent variable and is allowed

to change sign

HANDBOOK OF DIFFERENTIAL EQUATIONS

Ordinary Differential Equations, volume 1

Edited by A Cañada, P Drábek and A Fonda

© 2004 Elsevier B.V All rights reserved

1

A survey of recent results for initial and boundary value problems 3

1 Introduction

The study of singular boundary value problems (singular in the dependent variable) isrelatively new Indeed it was only in the middle 1970s that researchers realized that largenumbers of applications [7,11,12] in the study of nonlinear phenomena gave rise to singularboundary value problems (singular in the dependent variable) However, in our opinion, itwas the 1979 paper of Taliaferro [20] that generated the interest of many researchers insingular problems in the 1980s and 1990s In [20] Taliaferro showed that the singularboundary value problem

0 t (1− t)q(t) dt < ∞ Problems of the form (1.1) arise frequently in the study of

nonlin-ear phenomena, for example in non-Newtonian fluid theory, such as the transport of coalslurries down conveyor belts [12], and boundary layer theory [11] It is worth remarking

here that we could consider Sturm–Liouville boundary data in (1.1); however since the

ar-guments are essentially the same (in fact easier) we will restrict our discussion to Dirichletboundary data

In the 1980s and 1990s many papers were devoted to singular boundary value problems

Almost all singular problems in the literature [8–10,14–18,21] up to 1994 discussed

posi-tone problems, i.e., problems where f : [0, 1]×(0, ∞) → (0, ∞) In Section 2.1 we present

the most general results available in the literature for the positone singular problem (1.2)

In 1999 the question of multiplicity for positone singular problems was discussed for thefirst time by Agarwal and O’Regan [2] The second half of Section 2.1 discusses multi-plicity In 1994 [16] the singular boundary value problem (1.2) was discussed when the

nonlinearity f could change sign Model examples are

f (t, y) = t−1e1

− (1 − t)−1 and f (t, y)=g(t)

y σ − h(t), σ > 0 which correspond to Emden–Fowler equations; here g(t) > 0 for t ∈ (0, 1) and h(t) may change sign Section 2.2 is devoted to (1.2) when the nonlinearity f may change sign The

results here are based on arguments and ideas of Agarwal, O’Regan et al [1–6], and Habets

and Zanolin [16] Section 3 presents existence results for the singular initial value problem

(1.3) where the nonlinearity f may change sign.

The existence results in this paper are based on fixed point theorems In particular weuse frequently Schauder’s fixed point theorem and a Leray–Schauder alternative We begin

of course with the Schauder theorem

THEOREM 1.1 Let C be a convex subset of a Banach space and F : C → C a compact,

continuous map Then F has a fixed point in C.

In applications to construct a set C so that F takes C back into C is very difficult and sometimes impossible As a result it makes sense to discuss maps F that map a subset of C into C One result in this direction is the so-called nonlinear alternative of Leray–Schauder.

THEOREM 1.2 Let E be a Banach space, C a convex subset of E, U an open subset of

C and 0 ∈ U Suppose F : U → C (here U denotes the closure of U in C) is a continuous,

compact map Then either

(A1) F has a fixed point in U ; or

(A2) there exists u ∈ ∂U (the boundary of U in C) and λ ∈ (0, 1) with u = λF (u).

PROOF Suppose (A2) does not occur and F has no fixed points in ∂U (otherwise we are

Clearly N : C → C is a continuous, compact map Theorem 1.1 guarantees the existence

of an x ∈ C with x = Nx Notice x ∈ U since 0 ∈ U As a result x = μ(x)F (x), so x ∈ A.

To conclude the introduction we present existence principles for nonsingular initial andboundary value problems which will be needed in Sections 2 and 3 First we use Schauder’sfixed point theorem and a nonlinear alternative of Leray–Schauder type to obtain a generalexistence principle for the Dirichlet boundary value problem

subinterval of (0, 1).

A survey of recent results for initial and boundary value problems 5

THEOREM1.3 Suppose the following two conditions are satisfied:

such that f (t, y) h(t) for a.e t ∈ (0, 1) and y ∈ R (1.9)

holds Then (1.4) has a solution.

PROOF (I) We begin by showing that solving (1.8) λ is equivalent to finding a solution

so y(0) = a Similarly integrate y(t) from x (x ∈ (0, 1)) to 1 and interchange the order

of integration to get y(1) = b Thus if y ∈ C[0, 1] satisfies (1.10) λ then y is a solution of

Then (1.10) λis equivalent to the fixed point problem

y = (1 − λ)p + λNy, where p = a(1 − t) + b ( 1.12) λ

It is easy to see that N : C[0, 1] → C[0, 1] is continuous and completely continuous Set

A survey of recent results for initial and boundary value problems 7

and

(I) Assume



for each r > 0 there exists h r ∈ L1[0, T ] such that

|y|  r implies f (t, y) h r (t) for a.e t ∈ (0, T ) (1.16)

holds In addition suppose there is a constant M > |a|, independent of λ, with |y|0=supt ∈[0,T ] |y(t)| = M for any solution y ∈ AC[0, T ] to

there exists h ∈ L1[0, T ] such that f (t, y) h(t)

for a.e t ∈ (0, T ) and y ∈ R (1.18)

holds Then (1.13) has a solution.

PROOF (I) Solving (1.17) λ is equivalent to finding a solution y ∈ C[0, T ] to

(II) Solving (1.13) is equivalent to the fixed point problem y = Ny where N is as in

( 1.20) It is easy to see that N : C[0, T ] → C[0, T ] is continuous and compact (since (1.18)

holds) The result follows from Schauder’s fixed point theorem 

2 Singular boundary value problems

In Section 2.1 we discuss positone boundary value problems Almost all singular papers inthe 1980s and 1990s were devoted to such problems In Theorem 2.1 we present probablythe most general existence result available in the literature for positone problems In thelate 1990s the question of multiplicity for singular positone problems was raised, and wediscuss this question in the second half of Section 2.1 Section 2.2 is devoted to singularproblems where the nonlinearity may change sign

Here the nonlinearity f may be singular at y = 0 and q may be singular at t = 0 and/or

t = 1 We begin by showing that (2.1) has a C[0, 1] ∩ C2( 0, 1) solution To do so we first establish, via Theorem 1.3, the existence of a C[0, 1] ∩ C2( 0, 1) solution, for each

m = 1, 2, , to the “modified” problem

0 f (t, y)  g(y) + h(y) on [0, 1] × (0, ∞) with

g > 0 continuous and nonincreasing on (0, ∞),

h  0 continuous on [0, ∞), and h

g

nondecreasing on (0, ∞),

(2.5)for each constant H > 0 there exists a function ψ

r

0

du

g(u) > b0 (2.7)

A survey of recent results for initial and boundary value problems 9

hold; here

b0= max

2 1

Then (2.1) has a solution y ∈ C[0, 1] ∩ C2( 0, 1) with y > 0 on (0, 1) and |y|0< r.

PROOF Choose ε > 0, ε < r, with

Let n0∈ {1, 2, } be chosen so that 1

n0 < ε and let N0= {n0, n0+ 1, } To show (2.2) m,

m ∈ N0, has a solution we examine

where 0 < λ < 1 Let y be a solution of (2.11) m λ Then y 0 on (0, 1) and y  1

mon[0, 1] Also there exists t m ∈ (0, 1) with y 0 on (0, t m ) and y 0 on (t m , 1) For x ∈ (0, 1) we

 t m

0

xq(x) dx.

1



.

This together with (2.9) implies |y|0= r Then Theorem 1.3 implies that (2.10) m has a

solution y mwith|y m|0 r In fact (as above),

1

m  y m (t) < r for t ∈ [0, 1].

Next we obtain a sharper lower bound on y m, namely we will show that there exists a

constant k > 0, independent of m, with

To see this notice (2.6) guarantees the existence of a function ψ r (t)continuous on[0, 1] and positive on (0, 1) with f (t, u)  ψ r (t) for (t, u) ∈ (0, 1) × (0, r] Now, using the Green’s function representation for the solution of (2.10) m, we have

A survey of recent results for initial and boundary value problems 11

Now it is easy to check (as in Theorem 1.3) that

2k1t (1− t) for t ∈ [0, ε] Similarly there is

a constant k2, independent of m, with −Φ

r ( 1)  k2 Thus there is a δ > 0 with Φ r (t)

1

2k2(1− t) 1

2k2t (1− t) for t ∈ [1 − δ, 1] Finally since Φ r (t )

t (1−t) is bounded on[ε, 1 − δ] there is a constant k, independent of m, with Φ r (t)  kt(1 − t) on [0, 1], i.e., (2.15) is true.

Next we will show

{y m}m ∈N0 is a bounded, equicontinuous family on[0, 1]. (2.17)

Returning to (2.12) (with y replaced by y m) we have

We now show inf{tm : m ∈ N0} > 0 If this is not true then there is a subsequence S of

N0with t m → 0 as m → ∞ in S Now integrate (2.19) from 0 to t mto obtain

for m ∈ S Since t m → 0 as m → ∞ in S, we have from (2.22) that y m (t m ) → 0 as m → ∞

in S However since the maximum of y mon[0, 1] occurs at t m we have y m → 0 in C[0, 1]

as m → ∞ in S This contradicts (2.15) Consequently inf{t m : m ∈ N0} > 0 A similar

argument shows sup{tm : m ∈ N0} < 1 Let a0and a1be chosen as in (2.21) Now (2.19),

It is easy to see that v ∈ L1[0, 1] Let I : [0, ∞) → [0, ∞) be defined by

m ∈N0 is a bounded, equicontinuous family on[0, 1]. (2.24)

The equicontinuity follows from (here t, s ∈ [0, 1])

The Arzela–Ascoli theorem guarantees the existence of a subsequence N of N0and a

function y ∈ C[0, 1] with y mconverging uniformly on[0, 1] to y as m → ∞ through N Also y(0) = y(1) = 0, |y|0 r and y(t)  kt(1 − t) for t ∈ [0, 1] In particular y > 0 on

( 0, 1) Fix t ∈ (0, 1) (without loss of generality assume t =1

2) Now y m , m ∈ N, satisfies

the integral equation

y m (x) = y m

12



+ y m

12



x−12

+

x

1 (s − x)q(s)f s, y m (s)

ds

for x ∈ (0, 1) Notice (take x = 2

3) that {y m (12) }, m ∈ N, is a bounded sequence since

ks(1− s)  y m (s)  r for s ∈ [0, 1] Thus {y

m (12)}m ∈Nhas a convergent subsequence; for

A survey of recent results for initial and boundary value problems 13

convenience let{y

m (12)}m ∈N denote this subsequence also and let r0∈ R be its limit Now

for the above fixed t ,

y m (t) = y m

12



+ y m

12



t−12

+



+ r0



t−12

+

t

1 (s − t)q(s)f s, y(s)

ds.

We can do this argument for each t ∈ (0, 1) and so y(t) +q(t)f (t, y(t)) = 0 for 0 < t < 1.

Finally it is easy to see that|y|0< r(note if|y|0= r then following essentially the

Next we establish the existence of two nonnegative solutions to the singular second orderDirichlet problem

here our nonlinear term g + h may be singular at y = 0 Next we state the fixed point result

we will use to establish multiplicity (see [13] for a proof)

THEOREM2.2 Let E

r, R are constants with 0 < r < R Suppose A : Ω R ∩ K → K (here Ω R

R }) is a continuous, compact map and assume the following conditions hold:

x = λA(x) for λ ∈ [0, 1) and x ∈ ∂ E Ω r ∩ K (2.26)

and



there exists a v ∈ K\{0} with x = A(x) + δv

for any δ > 0 and x ∈ ∂ E Ω R ∩ K. (2.27)

Then A has a fixed point in K

REMARK2.2 In Theorem 2.2 if (2.26) and (2.27) are replaced by

x = λA(x) for λ ∈ [0, 1) and x ∈ ∂ E Ω R ∩ K ( 2.26) 

and



there exists a v ∈ K\{0} with x = A(x) + δv

then A has also a fixed point in K

THEOREM 2.3 Let E

increasing with respect to K Also r, R are constants with 0 < r < R Suppose A : Ω R∩

K → K (here Ω R

following conditions hold:

x = λA(x) for λ ∈ [0, 1) and x ∈ ∂ E Ω r ∩ K (2.28)

and

Then A has a fixed point in K

PROOF Notice (2.29) guarantees that (2.27) is true This is a standard argument and for completeness we supply it here Suppose there exists v ∈ K\{0} with x = A(x) + δv for some δ > 0 and x ∈ ∂ E Ω R

since δv ∈ K,

a contradiction The result now follows from Theorem 2.2 

REMARK2.3 In Theorem 2.3 if (2.28) and (2.29) are replaced by

x = λA(x) for λ ∈ [0, 1) and x ∈ ∂ E Ω R ∩ K ( 2.28) 

and

then A has a fixed point in K

Now E = (C[0, 1], | · |0)(here|u|0= supt ∈[0,1] |u(t)|, u ∈ C[0, 1]) will be our Banach

A survey of recent results for initial and boundary value problems 15

THEOREM2.4 Let y ∈ K (as in (2.30)) Then there exists t0∈ [0, 1] with y(t0) = |y|0and

y(t)  θ(t, t0) |y|0 t(1 − t)|y|0 for t ∈ [0, 1].

PROOF The existence of t0is immediate Now if 0 t  t0then since y(t) is concave on [0, 1] we have

y(t0) = θ(t, t0) |y|0 t(1 − t)|y|0.

A similar argument establishes the result if t0 t  1. From Theorem 2.1 we have immediately the following existence result for (2.25)

THEOREM2.5 Suppose the following conditions are satisfied:

q ∈ C(0, 1), q > 0 on (0, 1) and

1 0

Then (2.25) has a solution y ∈ C[0, 1] ∩ C2( 0, 1) with y > 0 on (0, 1) and |y|0< r.

PROOF The result follows from Theorem 2.1 with f (t, u) = g(u) + h(u) Notice (2.6) is

THEOREM2.6 Assume (2.31)–(2.34) hold Choose a ∈ (0,1

2) and fix it and suppose there exists R > r with

here 0  σ  1 is such that

Then (2.25) has a solution y ∈ C[0, 1] ∩ C2( 0, 1) with y > 0 on (0, 1) and r < |y|0 R.

PROOF To show the existence of the solution described in the statement of Theorem 2.6

we will apply Theorem 2.3 First however choose ε > 0 and ε < r with

has a solution y m for each m ∈ N0 with y m > m1 on (0, 1) and r  |y m|0 R To show

( 2.39) m has such a solution for each m ∈ N0, we will look at

REMARK2.4 Notice g  (u)  g(u) for u > 0.

Fix m ∈ N0 Let E = (C[0, 1], | · |0)and

K=u ∈ C[0, 1]: u(t)  0, t ∈ [0, 1] and u(t) concave on [0, 1]. (2.41)

Clearly K is a cone of E Let A : K → C[0, 1] be defined by

Ay(t)= 1

m+

1 0

G(t, s)q(s)

g  y(s)

+ h y(s)

A survey of recent results for initial and boundary value problems 17

A standard argument implies A : K → C[0, 1] is continuous and completely continuous Next we show A : K → K If u ∈ K then clearly Au(t)  0 for t ∈ [0, 1] Also notice that

Suppose this is false, i.e., suppose there exists y ∈ K ∩ ∂Ω1and λ ∈ [0, 1) with y = λAy.

We can assume λ = 0 Now since y = λAy we have

m on[0, 1] there exists t0∈ (0, 1) with y 0 on (0, t0),

y 0 on (t0, 1) and y(t0) = |y|0= r (note y ∈ K ∩ ∂Ω1) Also notice

To see this let y ∈ K ∩ ∂Ω2so|y|0= R Also since y(t) is concave on [0, 1] (since y ∈ K)

we have from Theorem 2.4 that y(t)  t(1 − t)|y|0 t(1 − t)R for t ∈ [0, 1] Also for

since y(s)  a(1 − a)R > 1

m0 for s ∈ [a, 1 − a] Note in particular that

y(s)∈a(1− a)R, R for s ∈ [a, 1 − a]. (2.50)

With σ as defined in (2.37) we have using (2.50) and (2.36),

A survey of recent results for initial and boundary value problems 19

Now Theorem 2.3 implies A has a fixed point y m ∈ K ∩ (Ω2\Ω1) , i.e., r  |y m|0 R.

In fact|y m|0> r (note if|y m|0= r then following essentially the same argument from

( 2.45)–(2.48) will yield a contradiction) Consequently (2.40) m (and also (2.39) m) has a

Next we will show

{y m}m ∈N0 is a bounded, equicontinuous family on[0, 1]. (2.53)

Returning to (2.45) (with y replaced by y m) we have

We now show inf{tm : m ∈ N0} > 0 If this is not true then there is a subsequence S of

N0with t m → 0 as m → ∞ in S Now integrate (2.55) from 0 to t mto obtain

for m ∈ S Since t m → 0 as m → ∞ in S, we have from (2.58) that y m (t m ) → 0 as m → ∞

in S However since the maximum of y mon[0, 1] occurs at t m we have y m → 0 in C[0, 1]

as m → ∞ in S This contradicts (2.52) Consequently inf{t m : m ∈ N0} > 0 A similar

argument shows sup{tm : m ∈ N0} < 1 Let a0and a1be chosen as in (2.57) Now (2.55),

m ∈N0 is a bounded, equicontinuous family on[0, 1]. (2.60)

The equicontinuity follows from (here t, s ∈ [0, 1])

I y m (t)

− I y m (s)

t s

y

m (x) g(y m (x)) dx



1+h(R)

g(R)

t s

The Arzela–Ascoli theorem guarantees the existence of a subsequence N of N0 and

a function y ∈ C[0, 1] with y mconverging uniformly on[0, 1] to y as m → ∞ through

N Also y(0) = y(1) = 0, r  |y|0 R and y(t)  t(1 − t)r for t ∈ [0, 1] In particular

y > 0 on (0, 1) Fix t ∈ (0, 1) (without loss of generality assume t =1

2) Now y m , m ∈ N,

satisfies the integral equation

y m (x) = y m

12



+ y

m

12



x−12

for x ∈ (0, 1) Notice (take x = 2

3) that {y m (12) }, m ∈ N, is a bounded sequence since

rs(1− s)  y m (s)  R for s ∈ [0, 1] Thus {y

m (12)}m ∈Nhas a convergent subsequence; for

A survey of recent results for initial and boundary value problems 21

convenience let{y

m (12)}m ∈N denote this subsequence also and let r0∈ R be its limit Now

for the above fixed t ,

y m (t) = y m

12



+ y

m

12



t−12



+ r0



t−12

+

... < /p>

xq(x) dx. < /p>

1 < /p>

 < /p>

. < /p>

This together with (2. 9) implies... < /p>

12 < /p>

 < /p>

+ y m < /p>

12 < /p>

 < /p>

t−1< /sup>2 < /p>

+ < /p>

 < /p>

+ r0 < /p>

 < /p>

t−1< /sup>2... r < /p>

0 < /p>

du < /p>

g(u) > b0 (2. 7) < /p>

A