báo cáo hóa học:" Research Article Limit Properties of Solutions of Singular Second-Order Differential Equations" doc

Volume 2009, Article ID 905769, 28 pagesdoi:10.1155/2009/905769 Research Article Limit Properties of Solutions of Singular Second-Order Differential Equations Irena Rach ˚unkov ´a,1 Svat

Volume 2009, Article ID 905769, 28 pages

doi:10.1155/2009/905769

Research Article

Limit Properties of Solutions of Singular

Second-Order Differential Equations

Irena Rach ˚unkov ´a,1 Svatoslav Stan ˇek,1 Ewa Weinm ¨uller,2

1 Department of Mathematical Analysis, Faculty of Science, Palack´y University,

Tomkova 40, 779 00 Olomouc, Czech Republic

2 Institute for Analysis and Scientific Computing, Vienna University of Technology,

Wiedner Hauptstrasse 8-10, 1040 Wien, Austria

Correspondence should be addressed to Irena Rach ˚unkov´a,rachunko@inf.upol.cz

Received 23 April 2009; Accepted 28 May 2009

Recommended by Donal O’Regan

We discuss the properties of the differential equation ut  a/tut  ft, ut, ut, a.e on

0, T, where a ∈ R\{0}, and f satisfies the L p-Carath´eodory conditions on0, T ×R2for some

p > 1 A full description of the asymptotic behavior for t → 0 of functions u satisfying the

equation a.e on0, T is given We also describe the structure of boundary conditions which

are necessary and sufficient for u to be at least in C10, T As an application of the theory, new

existence and/or uniqueness results for solutions of periodic boundary value problems are shown.Copyrightq 2009 Irena Rach ˚unkov´a et al This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited

where a ∈ R \ {0}, u : 0, T → R, and the function f is defined for a.e t ∈ 0, T and for

allx, y ∈ D ⊂ R × R The above equation is singular at t  0 because of the first term in the right-hand side, which is in general unbounded for t → 0 In this paper, we will also

alow the function f to be unbounded or bounded but discontinuous for certain values of the time variable t ∈ 0, T This form of f is motivated by a variety of initial and boundary

value problems known from applications and having nonlinear, discontinuous forcing terms,such as electronic devices which are often driven by square waves or more complicated

discontinuous inputs Typically, such problems are modelled by differential equations where

f has jump discontinuities at a discrete set of points in 0, T, compare 1

This study serves as a first step toward analysis of more involved nonlinearities, where

typically, f has singular points also in u and u Many applications, compare2 12, showingthese structural difficulties are our main motivation to develop a framework on existenceand uniqueness of solutions, their smoothness properties, and the structure of boundary

conditions necessary for u to have at least continuous first derivative on 0, T Moreover,

using new techniques presented in this paper, we would like to extend results from13,14

based on ideas presented in 15 where problems of the above form but with appropriately

smooth data functionf have been discussed.

Here, we aim at the generalization of the existence and uniqueness assertions derived

in those papers for the case of smooth f We are especially interested in studying the limit properties of u for t → 0 and the structure of boundary conditions which are necessary andsufficient for u to be at least in C10, T.

To clarify the aims of this paper and to show that it is necessary to develop anew technique to treat the nonstandard equation given above, let us consider a modelproblem which we designed using the structure of the boundary value problem describing amembrane arising in the theory of shallow membrane caps and studied in10; see also 6,9,



t3ut t3

1

T  1, a  −3, ft, u, u

 −

1

8u2 −a0

u  b0t 2γ−4



Function f is not defined for u  0 and for t  0 if γ ∈ 1, 2 We now briefly discuss a

simplified linear model of1.4,

The question which we now pose is the role of the boundary conditions1.3, more

precisely, are these boundary conditions necessary and su fficient for the solution u of 1.6 to beunique and at least continuously differentiable, u ∈ C10, 1? To answer this question, we can

use techniques developed in the classical framework dealing with boundary value problems,exhibiting a singularity of the first and second kind; see15,16, respectively However, in

these papers, the analytical properties of the solution u are derived for nonhomogeneous

terms being at least continuous Clearly, we need to rewrite problem1.6 first and obtain itsnew form stated as,



t3ut t3

b0t β

which suggest to introduce a new variable, vt : t3ut In a general situation, especially for

the nonlinear case, it is not straightforward to provide such a transformation, however We

now introduce zt : ut, vt T and immediately obtain the following system of ordinarydifferential equations:

b0t β3

where g ∈ C0, 1 According to 16, the latter system of equations has a continuous solution

if and only if the regularity condition Mz0  0 holds This results in

v0  0 ⇐⇒ lim

compare conditions1.3 Note that the Euler transformation, ζt : ut, tut T which isusually used to transform1.6 to the first-order form would have resulted in the followingsystem:

b0t β1

Here, w may become unbounded for t → 0, the condition Nζ0  0, or equivalently

limt→ 0tut  0 is not the correct condition for the solution u to be continuous on 0, 1.

From the above remarks, we draw the conclusion that a new approach is necessary tostudy the analytical properties of1.1

2 Introduction

The following notation will be used throughout the paper Let J ⊂ R be an interval

Then, we denote by L1J the set of functions which are Lebesgue integrable on J The

corresponding norm is u 1 : J |ut|dt Let p > 1 By L p J, we denote the set of functions whose pth powers of modulus are integrable on J with the corresponding norm given by

Analogously, AClocJ and AC1

locJ are the sets of functions being absolutely continuous on each compact subinterval I ⊂ J and having absolutely continuous first derivatives on each compact subinterval I ⊂ J, respectively.

As already said in the previous section, we investigate differential equations of theform

ut  a

t u

t  ft, u t, ut, a.e on 0, T, 2.1

where a∈ R \ {0} For the subsequent analysis we assume that

f satisfies the L p-Carath´eodory conditions on0, T × R × R, for some p > 1 2.2specified in the following definition

Definition 2.1 Let p > 1 A function f satisfies the L p -Carath´eodory conditions on the set 0, T ×

R × R if

i f·, x, y : 0, T → R is measurable for all x, y ∈ R × R,

ii ft, ·, · : R × R → R is continuous for a.e t ∈ 0, T,

iii for each compact set K ⊂ R × R there exists a function mKt ∈ L p 0, T such that

|ft, x, y| ≤ mKt for a.e t ∈ 0, T and all x, y ∈ K.

We will provide a full description of the asymptotical behavior for t → 0 of functions

u satisfying 2.1 a.e on 0, T Such functions u will be called solutions of 2.1 if they

additionally satisfy the smoothness requirement u∈ AC10, T; see next definition.

Definition 2.2 A function u : 0, T → R is called a solution of 2.1 if u ∈ AC10, T and

satisfies

ut  a

t u

t  ft, u t, ut a.e on 0, T. 2.3

conditions necessary for the solution to be at least continuous on 0, 1 These results are

modified for nonlinear problems inSection 4 InSection 5, by applying the theory developed

inSection 4, we provide new existence and/or uniqueness results for solutions of singularboundary value problems2.1 with periodic boundary conditions.

3 Linear Singular Equation

First, we consider the linear equation, a∈ R \ {0},

ut  a

t u

where h ∈ L p 0, T and p > 1.

As a first step in the analysis of3.1, we derive the necessary auxiliary estimates used

in the discussion of the solution behavior For c ∈ 0, T, let us denote by

Now, let a > 0, c > 0 Without loss of generality, we may assume that 1/p /  1 − a For 1/p 

1− a, we choose p∗∈ 1, p, and we have h ∈ L p∗0, T and 1/p∗> 1 − a.

First, let a ∈ 0, 1 − 1/p Then 1/q  1 − 1/p > a, 1 − aq > 0, and

c1−aq− t1−aq

1− aq

t

ds

s aq

1/q < 1− aq −1/q

c 1/q−a  t 1/q−a

Consequently,3.3, 3.6, and the H¨older inequality yield, t ∈ 0, T,

loc0, T satisfying 3.1 a.e on 0, T Remember that such function u does not need to

be a solution of3.1 in the sense ofDefinition 2.2

Lemma 3.1 Let a ∈ R \ {0}, c ∈ 0, T, and let ϕ a c, t be given by 3.2.

is the set of all functions u∈ AC1

loc0, T satisfying 3.1 a.e on 0, T.

is the set of all functions u∈ AC1

loc0, T satisfying 3.1 a.e on 0, T.

Proof Let a / − 1 Note that 3.1 is linear and regular on 0, T Since the functions u1

h t  1 and u2

h t  t a1 are linearly independent solutions of the homogeneous equation ut −

a/tut  0 on 0, T, the general solution of the homogeneous problem is

u h t  c1 c2t a1, c1, c2∈ R. 3.12

Moreover, the function u p t  c

t ϕ a c, sds is a particular solution of 3.1 on 0, T Therefore,

the first statement follows Analogous argument yields the second assertion

We stress that by 3.8, the particular solution u p  c

i If a / − 1, then



c1 c2t a1−

t0

ϕ a 0, sds, c1, c2∈ R, t ∈ 0, T



3.13

is the set of all functions u∈ AC1

loc0, T satisfying 3.1 a.e on 0, T.

ii If a  −1, then



c1 c2ln t−

t0

ϕ−10, sds, c1, c2∈ R, t ∈ 0, T



3.14

is the set of all functions u∈ AC1

loc0, T satisfying 3.1 a.e on 0, T.

Proof Let us fix c ∈ 0, T and define

ϕ a 0, sds,

pt  −ϕ a c, t  ϕ a 0, t

 −t a

c0

h s

s a ds, d1: pc − d2c a1. 3.17

For a −1 we have

d2: −c0

sh sds, d1:

c0

ϕ−10, sds − d2ln c, 3.18which completes the proof

Again, by3.9, the particular solution,

u p t  −

t0

of3.1 for a < 0 satisfies u p ∈ C10, 1 Main results for the linear singular equation 3.1 arenow formulated in the following theorems.

Theorem 3.3 Let a > 0 and let u ∈ AC1

loc0, T satisfy equation 3.1 a.e on 0, T Then

lim

t→ 0u t ∈ R, lim

Moreover, u can be extended to the whole interval 0, T in such a way that u ∈ AC10, T.

Proof Let a function u be given Then, by3.10, there exist two constants c1, c2∈ R such that

Therefore u∈ L10, T, and consequently u ∈ AC10, T.

It is clear from the above theorem, that u ∈ AC10, T given by 3.21 is a solution of

3.1 for a > 0 Let us now consider the associated boundary value problem,

where B0, B1 ∈ R2×2are real matrices, and β ∈ R2 is an arbitrary vector Then the followingresult follows immediately fromTheorem 3.3.

Theorem 3.4 Let a > 0, p > 1 Then for any ht ∈ L p 0, T and any β ∈ R2there exists a unique solution u∈ AC10, 1 of the boundary value problem 3.26a and 3.26b if and only if the following

Proof Let u be a solution of3.1 Then u satisfies 3.21, and the result follows immediately

by substituting the values,

u 0  c1

c0

into the boundary conditions3.26b

Theorem 3.5 Let a < 0 and let a function u ∈ AC1

loc0, T satisfy equation 3.1 a.e on 0, T For

a ∈ −1, 0, only one of the following properties holds:

i limt→ 0u t ∈ R, lim t→ 0ut  0,

ii limt→ 0u t ∈ R, lim t→ 0ut  ±∞.

For a ∈ −∞, −1, u satisfies only one of the following properties:

i limt→ 0u t ∈ R, lim t→ 0ut  0,

ii limt→ 0u t  ∓∞, lim t→ 0ut  ±∞.

In particular, u can be extended to the whole interval 0, T with u ∈ AC10, T if and only if

ϕ a 0, sds for t ∈ 0, T. 3.29

Hence

ut  c2a  1t a − ϕ a 0, t for t ∈ 0, T. 3.30

Let c2  0, then it follows from 3.9 limt→ 0ut  0 Also, by 3.29, limt→ 0u t  c1 ∈ R.

Let c2/ 0 Then 3.9, 3.29, and 3.30 imply that

ϕ−10, sds for t ∈ 0, T, 3.32

ut  c21

If c2  0, then limt→ 0ut  0 by 3.9, and it follows from 3.32 that limt→ 0u t  c1 ∈ R

Let c2/ 0 Then we deduce from 3.9, 3.32, and 3.33 that

In particular, for a < 0, u can be extended to 0, T in such a way that u ∈ C10, T if and only

if c2  0 Then, the associated boundary conditions read u0  c1and u0  0 Finally, for

Again, it is clear that u given by3.29 for a ∈ −1, 0 and a < −1, and u given by 3.32

for a −1 is a solution of 3.1, and u ∈ AC10, 1 if and only if u0  0 Let us now considerthe boundary value problem

Theorem 3.6 Let a < 0, p > 1 Then for any ht ∈ L p 0, T and any b2, β ∈ R there exists a unique

solution u∈ AC10, 1 of the boundary value problem 3.38a and 3.38b if and only if b0 b1/  0.

Proof Let u be a solution of3.1 Then u satisfies 3.29 for a ∈ −1, 0 and a < −1, and 3.32

for a −1 We first note that, by 3.9, for all a < 0,

ϕ a 0, sds, uT  −ϕ a 0, T, 3.40

into the boundary conditions3.38b

To illustrate the solution behaviour, described by Theorems 3.3 and 3.5, we havecarried out a series of numerical calculations on a MATLAB software package bvpsuitedesigned to solve boundary value problems in ordinary differential equations The solver

is based on a collocation method with Gaussian collocation points A short description ofthe code can be found in17 This software has already been used for a variety of singularboundary value problems relevant for applications; see, for example,18

The equations being dealt with are of the form

Finally, we expect limt→ 0u t  ±∞, and therefore we solve 3.41 subject to the

terminal conditions u1  α, u1  β See Figures1,2, and3

−3

−2

−1 0 1 2 3 4

holds that u0  0 for each choice of α and β.

4 Limit Properties of Functions Satisfying Nonlinear

Singular Equations

In this section we assume that the function u ∈ AC1

loc0, T satisfying differential equation

2.1 a.e on 0, T is given The first derivative of such a function does not need to be continuous at t  0 and hence, due to the lack of smoothness, u does not need to be a solution

of 2.1 in the sense of Definition 2.2 In the following two theorems, we discuss the limit

properties of u for t → 0

Theorem 4.1 Let us assume that 2.2 holds Let a > 0 and let u ∈ AC1

loc0, T satisfy equation

2.1 a.e on 0, T Finally, let us assume that that

and u can be extended on 0, T in such a way that u ∈ AC10, T.

Proof Let h t : ft, ut, ut for a.e t ∈ 0, T By 2.2, there exists a function mK ∈

L p 0, T such that |ft, ut, ut| ≤ mKt for a.e t ∈ 0, T Therefore, h ∈ L p 0, T Since the equality ut  a/tut  ht holds a.e on 0, T, the result follows immediately due to

toTheorem 3.5a solution u satisfies u0  ∞ or u0  −∞ or u0  0 in dependence of values α and

β In order to precisely recover a solution satisfying u0  0, the respective simulation was carried out as

an initial value problem with u 0  0 and u0  0

−100

−50 0 50 100

Theorem 4.2 Let us assume that condition 2.2 holds Let a < 0 and let u ∈ AC1

and u can be extended on 0, T in such a way that u ∈ AC10, T.

Proof Let h ∈ L p 0, T be as in the proof ofTheorem 4.1 According toTheorem 3.5and4.1,

u satisfies4.3 both for a ∈ −1, 0 and a ∈ −∞, −1.

Conditions5.1b can be written in the form 3.26b with B0  I, B1  −I, and β  0.

Then, matrix3.27 has the form

a characterization of a class of nonlinear periodic problems5.1a and 5.1b which haveonly one solution We begin the investigation of problem5.1a and 5.1b with a uniquenessresult

Theorem 5.2 uniqueness Let a > 0 and let us assume that condition 2.2 holds Further, assume

that for each compact set K ⊂ R × R there exists a nonnegative function hK∈ L10, T such that

4 Limit Properties of Functions Satisfying Nonlinear

Singular Equations

In this... with Gaussian collocation points A short description ofthe code can be found in17 This software has already been used for a variety of singularboundary value problems relevant for applications;... first derivative of such a function does not need to be continuous at t  and hence, due to the lack of smoothness, u does not need to be a solution

of 2.1 in the sense of Definition