On
Boundary Value Problems
For Nonlinear Ordinary Differential Equations
I. Undamped Equations with Dirichlet Type Boundary Conditions
Hongyou
Wu
Submitted July 17, 2005 and, in revised form, September 12,
2005.
Abstract. This paper deals with Dirichlet type
boundary value problems for undamped nonlinear second-order ordinary
differential equations. Numerical experiments generate generic possibilities
for the number of solutions when the constant terms in boundary conditions
vary. In addition to the trivial situation, where such a problem never has a
solution for any values of the constants, there are three generic
possibilities: (i) the number of solutions is always positive and finite, (ii)
there are infinitely many solutions for all choices of the constants, and (iii)
there are no solutions for certain values of the constants and at least two
solutions otherwise. We prove that, subject to appropriate simple conditions,
each possibility is realized. Explicit examples show that the main conditions
in these results are sharp.
1991 Mathematics
Subject Classification. Primary 34B15.
Key words and
phrases. Nonlinear boundary
value problems, number of solutions, solutions
region, symmetries.
1.1. Problem. In this series of papers, we study boundary value
problems (BVP’s) of the form
for nonlinear
ordinary differential equations (ODE’s), where the functions ,
,
,
the points and the numbers ,
are given such that and do not have any constant term.
Such problems have a
wide range of applications. For example, the ODE’s involved describe physical
situations such as the motion of a mass attached to a nonlinear spring and a
nonlinear damper or the motion of a pendulum. The boundary conditions (BC’s) in
these cases consist of two requirements on the status of the motion at two or
more specific moments. More applications of such BVP’scan be found in [4] and Section 1.2 of [1].
Furthermore, special solutions of certain BVP’s for partial differential
equations can be obtained from this type of problems, see, for example, [3].
1.2. Questions. One of the main questions of theoretic
investigation is: for a fixed pair ,
how many solutions does such a BVP have?
We look at the
problem from a different point of view. Given an , consider the region in the
plane consisting of the points such
that the BVP has at least solutions, then: what does look
like? How does change
when, for example, varies?
These are natural
and fundamental questions. Even though in many situations, one is mainly
interested in the number of solutions of the BVP for a fixed pair , however, it is generally helpful to know the
global information about the number of solutions. Moreover, this information,
at least part of it, is necessary when the stability of the number of solutions
under perturbations of the constants and is also
concerned.
1.3. Current state of knowledge. The study of the ’s can be regarded as a beginning part of the
research on the dependence of the solution set of a nonlinear BVP on the
functions ,
,
and parameters ,
,
,
,
defining the problem. We believe that as the
understanding of nonlinear BVP’s deepens, such research will attract more and
more attention.
To the best of our
knowledge, the ’s
are not explicitly known for any nonlinear BVP. Only partial information for
certain problems is available in the literature. See, for example, Chapter 1
Section 4 in [1]. Some results about the and
of a
class of BVP’s were recently obtained by L. Kong and Q. Kong in [8], [9], [10], [11] and [7].
1.4. Overview. In this first paper of the series, we study the ’s of the problems consisting of undamped equations
and Dirichlet type BC’s, i.e., BVP’s of the form
We start with
numerical experiments dealing with the ’s
of such a problem. Experiments suggest that for the ’s,
there are three generic possibilities and a couple of non-generic possibilities,
in addition to the trivial possibility that for
each . The three generic possibilities are: either
the BVP has solutions for all points , and possibly multiple, but finitely many,
solutions for certain pairs , i.e., and for all
sufficiently large ; or the BVP has infinitely many solutions for
all pairs , i.e., for
every ; or there is a simple curve going to infinity
such that the BVP has no solution for all points on one
side of the
curve and two solutions for all pairs on the
other side , i.e., (the closure
of )
and .
Symmetries of the ’s
are used to demonstrate the accuracy of the numerical experiments. To further
verify the accuracy, three exact examples are then worked out. These examples
also motivate the theoretic results to be presented.
Under some technical assumptions, we prove
that the BVP always has a unique solution when is
increasing in . This is a special case of the first generic
possibility. If goes to
and sufficiently fast as approaches and , respectively, then the second generic possibility
shows up. When goes to
fast
enough as approaches and
stays bounded from below as goes to
, one has the third generic possibility. The
exact examples also imply that the main assumptions in these results and their
refinements are sharp.
1.5. Applications. As simple applications of some results of this
paper, we deduce information about the real eigenvalues of certain nonlinear
eigenvalue problems (EP’s). These problems are obtained from the classical
Sturm-Liouville problem via replacing its potential term by, for example,
nonlinear polynomials of the unknown function; see Example 6.8, Example 6.11, and Example 6.13.
1.6. Organization. The organization of this paper is as follows. In Section 2, we
give the precise assumptions on the BVP’s to be studied and state the main
results of this paper. Section 3 is devoted to two symmetry
principles; Section
4 discusses computer codes and numerical experiments; while Section 5 deals with some
exact examples. In Section
6, we give a few lemmas and show the main results together with their refinements.
2. Notation and main results.
2.1. Specifics. Even though most of the ideas of this paper apply
to much more general BVP’s, for simplicity, in this paper we only consider
BVP’s of the form
where
(1) are fixed such that ,
while is given and satisfies the Carathéodory
conditions, i.e.,
(a)
for each , is measurable on ,
(b)
for a.e., ,
is continuous on ,
and
(c)
for every ,
there is a non-negative function such that
and the local
Lipschitz condition, i.e.,
(d)
for each ,
there is a non-negative function such that
(2) the constants are arbitrary.
By a solution
of the BVP (2.1), we mean a
differentiable function such
that is
absolutely continuous on , satisfies the differential equation (DE) in (2.1) a.e., and the BC
in (2.1) is fulfilled by .
The Carathéodory
conditions and the local Lipschitz condition are standard assumptions
guaranteeing that for each and
every pair , the initial-value problem (IVP)
has a unique solution. See, for example, [5]. For some of the results of this
paper, the uniqueness of solutions is not needed. However, for the simplicity
of statements, we make the requirements (a)—(d) on throughout the whole paper.
Notation: for each and
every pair , denote by the
unique solution of the IVP (2.4), by the maximum
domain of , and by the
right-end point of . When is
given and does not change, we abbreviate , and as , , , respectively.
2.2. Main results. The following are the main results of this paper.
Theorem 2.1. If there are functions such that
then for every pair ,
the boundary value problem (2.1)
has a solution.
Theorem 2.2. If for almost every ,
is increasing on ,
then the boundary value problem (2.1) always has a unique solution.
Theorem 2.3. Assume that for each sufficiently
negative ,
is increasing on ,
and for every sufficiently large ,
is decreasing on .
If for each ,
there is a constant and a zero measure set such that
then for every pair ,
the boundary value problem (2.1)
has infinitely many solutions.
Theorem 2.4. Assume that
(i)
the differential equation has a solution defined on the whole ;
(ii)
for each ,
there are a constant and a zero measure set such that (2.7) holds;
(iii)
there is a function satisfying (2.6).
Then, there are a constant and a strictly decreasing continuous function
such that ,
and the boundary value problem (2.1)
has
(i)
no solution if either ,
or and ,
(ii) a solution if
and ,
(iii) at least two solutions if and .
It seems to us that Theorem 2.3 is the first result claiming infinite many
solutions. Theorem 2.4 partially improves and generalizes
the corresponding results in [8],
[9], [10], [11], and [7] for (2.1). For refinements of Theorem 2.3 and Theorem 2.4,
see Theorem 6.14, Theorem
6.18 and Theorem 6.22.
2.3. Relations. To end this section, we mention some relations
among the form of the DE in (2.1) and forms used by
other authors. When a DE of the form
is put into the form in (2.1),
one gets .
Some papers in the literature are devoted to the situation where .
This corresponds to ,
and after replacing in (2.1) by ,
we have a new satisfying that . A few authors prefer DE’s on certain
normalized intervals, for example, some consider DE’s of the form
The DE in (2.1) can always be brought
into this form with ,
where .
So, varying in (2.1) is
the same as changing the parameter in its corresponding BVP with a DE of the
form (2.10).
3. Symmetry Principles
3.1. Terminology. We call defined in the introduction the existence
region of (2.1), the multiple solutions region of (2.1), and in general, with the solutions region of (2.1). So, the existence
region, in the -plane, of (2.1)
consists of the points such that (2.1)
has a solution.
3.2. Symmetry principles. In this section, we give two principles on
symmetries of the regions .
Symmetries of these regions will be used in Section 4 to check the accuracy of numerical
approximations, and the ideas will be employed in Section 6 to prove the main results.
The following useful
lemma will be called the Reversion Principle.
Lemma 3.1. If is a solution of the boundary value problem (2.1), then is a solution of the boundary value problem
and vice versa.
The new BVP (3.1) will be called the
reverse problem of (2.1). As a consequence of the Reversion Principle (and a
translation of the independent variable), we have the following fact, which
will be named the Reflectional Symmetry Principle.
Lemma 3.2. If there
is a zero measure subset of such that
then each region of the boundary value problem (2.1) is symmetric with
respect to the line ,
i.e., invariant under the reflection .
Example
3.3. If is independent of ,
then the Reflectional Symmetry Principle applies to the BVP (2.1).
The following result
will be called the Negation Principle.
Lemma 3.4. If is a solution of the boundary value problem (2.1), then is a solution of the boundary value problem
and vice versa.
The new BVP (3.3) will be called the
negated problem of (2.1). As a consequence of the Negation Principle, we
have the following fact, which will be named the Rotational Symmetry Principle.
Lemma 3.5. If there
is a zero measure subset of such that
then each region of the boundary value problem (2.1) is symmetric with
respect to the origin, i.e., invariant under the rotation about the origin by .
Example
3.6. If is always odd in ,
then the Rotational Symmetry Principle applies to the BVP (2.1).
There are more
general versions of the above principles, and they will be given in future papers
of this series. So, here we omit the details.
4. Numerical Experiments
4.1. Overview. In this section, we discuss numerical experiments
dealing with BVP’s of the form (2.1). The main ideas of
this section apply to general nonlinear BVP’s and will be used in forthcoming papers
of this series.
The theoretic base
of our numerical approach is the uniqueness of solutions of the IVP’s (2.4). To explain the
main ideas in our codes for numerical experiments, first fix an . For each , there is a unique solution of the
DE in (2.1) satisfying and . Moreover, every solution of the DE fulfilling
the first equation in the BC in (2.1) is among these solutions.
In this way, we get a one-to-one parameterization of the solutions making the
first equation in the BC true, and the only thing left is to find out how many
of these solutions satisfy the second equation in the BC. To simplify the rest
of the explanation, we assume that for each , is
defined on the whole . Assume further that the monotonicity of changes
at most finitely many times. The key point of the codes is that for every value
of at
which the monotonicity of changes, is
a boundary point of some , see Figure 1 for an illustration. When has a
finite limit as approaches
or , is also
a boundary point. Moreover, these are all boundary points of the ’s
with as
their first coordinate at which the number of solutions of the BVP changes. (It
is possible to have boundary points at which the number of solutions does not
change when only varies,
see Example 4.1 below. However, these missing
boundary points are accumulation points of the other boundary points. So, the
boundary curves are almost not affected by not plotting them.)
Figure 1. Changes of monotonicity of and boundary points of ’s.
By varying afterwards, we get all the boundary points of the ’s (at which the number of solutions of the BVP
changes), and hence the boundaries of the ’s
(minus a few accumulation points). Using the monotonicity information of for one
or a
few ’s,
one can then tell where each is,
relative to these boundary curves.
Of course, in
general, is not
defined on the whole , and its domain is the union of several
non-overlapping open intervals. To do any numerical computation, it is
necessary to choose a computation range; to detect the changes of the
monotonicity of , one needs to decide how often the values of are to
be sampled; and to catch the possible finite limits of at , we have to tell the codes what a finite limit
means. So, our codes have several parameters to be used for specifying such details.
4.2. Geometry. For nonlinear continuous functions ,
consider the BVP’s
the simplest BVP’s of the form (2.1).
We now discuss what
their regions look
like, using numerical experiments. For these problems, one can always achieve by
applying a translation on the -axis
of the DE in (4.1). So, we only need
to handle
By Example 3.3, each is symmetric with respect to the line ;
if is odd, then Example
3.6 implies that every is also symmetric with respect to the origin.
We start from the
mathematically most natural cases, i.e., the cases where
are nonlinear polynomials. So, the degree of is greater than one. Using a translation of
the -variable, one can always get rid of the term
;
if ,
then by rescaling the -axis, we can always make for any given constant ;
via rescaling the -variable, one can always assume that the
leading coefficient ,
and hence either or .
Moreover, if is even, then after a reflection of the -axis when necessary, we arrive at the situation
where ;
similarly, if is odd, then we can always reach that .
Example 4.1. is a degree-3 polynomial. When the leading
coefficient ,
numerical experiments suggest that the BVP (4.2)
always has infinitely many solutions, i.e., .
When ,
there are only 3 normalized forms:
where .
Numerical experiments also indicate that if is increasing, i.e., if or ,
then the BVP always has a unique solution, i.e., and ;
and if ,
then the BVP always has a solution, i.e., .
Now, we give further
discussions of the case where . For the subcase with , numerical experiments provide evidences for
the following: if , then the BVP always has a unique solution;
and if for
some , then is a
mouth-shaped region with the two end points of the mouth removed, equals
the interior of the mouth,
and the mouth approaches a single point, the origin,
as .
The last observation can be stated as follows: the origin and are, respectively, the point of emergence
and starting time of the mouth. Moreover, every mouth has two symmetry
axes, which are perpendicular to each other: one goes through its end points,
to be called its longer axis, and the other bisects the segment joining its end
points, to be named as its shorter axis. The shorter axes of the mouths for the
’s and ’s with odd ’s are the line ,
while the longer axes of the mouths for the ’s and ’s with even ’s are the line .
Every mouth grows bigger and bigger as increases (both its length and its width keep
going up, with its length increasing faster first, its width becoming bigger
quicker then, and both going up slowly afterwards), but always stays in the
square ;
and each mouth approaches the boundary of the square as .
Figure 2 indicates the boundary curves for and . Click the following link
and press Ctrl+A followed by Ctrl+Y, adjust animation speed via
Edit|Preferences …|Graphics Options|Animation. The animation shows how the boundary curves evolve when varies from to ,
animation.
Figure 2. Regions of (4.2) with and various ’s.
Without using the
concept of ’s,
the number of solutions of the problem can be obtained from such graphs as
follows: if the point is
outside all mouths, then the BVP has a unique solution; when moves
to any of the two end points of a mouth from the outside of the mouth, the
number of solutions stays the same; when moves
to the boundary of the mouth, except the end points, the number increases by 1;
at the movement when enters
the interior of the mouth, the number goes up by 2.
For example, by the
first graph in Figure 2, when is
outside the mouth or at any of its two end points, the BVP (4.2) with and has a
unique solution; when is on
the boundary of the mouth, except at its end points, the problem has exactly 2
solutions; and when is in
the interior of the mouth, the BVP has precisely 3 solutions. According to the
second graph in (the first row of) Figure 2, when is
outside the smaller mouth, the number of solutions of the BVP with is as
in the first graph; when is at
any of the two end points of the smaller mouth, on the boundary of the smaller
mouth with the end points removed, and in the interior of the smaller mouth,
the problem has exactly 3, 4 and 5 solutions, respectively.
To understand the
mouth-shaped regions and their relations with the ’s,
one may imagine that a table is fully covered by an unfolded tablecloth.
Somehow, the cloth has a small and simple wrinkle, pressed onto the table
together with the rest of the cloth. In the middle of the wrinkle area, the
table is covered by two more layers of cloth, even though small, and hence it
is covered by three layers in total; on the boundary of the wrinkle area,
except at the two end points of the wrinkle, the table is covered one
additional time, and hence it is covered two times in total; and at each of the
two end points, the table is just covered once. Here, the table stands for the -plane,
and the tablecloth for the collection of all pairs obtained from the solutions that
are defined on the whole .
Note that the
regions shown in Figure 2 are symmetric with respect to both the line and the
origin, as predicted by the symmetry principles in Section 3. So, these
partially demonstrate the high accuracy of our codes (assuming, of course, a
high accuracy of the ODE solver used). We also remark that by the nature of the
design of the codes, the points plotted for very vertical parts of boundary
curves are sparse. This difficulty can be overcome using more focused runs of
the codes.
The effect of is as
follows. As increases from 0, the two end points of the biggest mouth are
dragged vertically up and horizontally to the right, respectively, so that all ’s
move in the upper right direction. (Hence, as decreases from 0, the motion is in the opposite ways, by the
relation between the situation with and
that with .) Note that now, each is
still symmetric with respect to the line . Figure 3 displays the regions with and 12
for (first
pair) and those with and for (second
pair). From these graphs, we see that the bigger the value of , the stronger the effect of a fixed . Animations show how the boundary curves evolve when increases, animation.
Figure 3.
Regions of (4.2) with and various ’s.
Example 4.2. is a polynomial of an odd degree .
When the leading coefficient ,
the BVP (4.2)
always has infinitely many solutions. When is increasing, the BVP always has a unique
solution.
When and has
(strictly) decreasing intervals, the BVP still always has a solution, i.e., . Assume that has exactly
decreasing intervals, where . Then, there are precisely areas
in the -plane
where mouths grow. These areas are rectangles when is odd.
Mouths in different areas can have different starting times; and hence for a
given , the numbers of mouths in different areas can
be different. Note that in general, each mouth has only one symmetry, even in
the situations where every has two
symmetries. Figure
4 presents the regions for , with two sequences of mouths growing out from
the points and , respectively, when for ; while Figure 5 shows the regions for , with one sequence of mouths coming out from
the point when for , and the other sequence from the point when for .
Figure 4.
Regions of (4.2) with and various ’s.
Figure 5. Regions of (4.2) with and various ’s.
For nonlinear
polynomials of odd degree ,
the following will be proved in Section 6: when the leading coefficient ,
the BVP (4.2)
always has a solution (see Example 6.4); when ( and is increasing), the problem always has a
unique solution (see Example 6.8); and when ,
the BVP always has infinitely many solutions (see Example
6.11).
Next, we discuss the
cases where nonlinear polynomials of an
even degree are used.
Example 4.3. is a degree-2 polynomial. We only have 3
normalized forms:
When ,
numerical experiments indicate the following: there is an and a strictly decreasing continuous function
such that ,
,
and the BVP (4.2)
with this has
(i) no solution if either ,
or and ,
(ii) a unique solution if
and ,
(iii) at least two solutions if and .
Using the ’s, these observations can be written as
Moreover, the existence region approaches the whole plane as and the first quadrant as .
For this reason, we call a region such as the one in (4.7) an infinite
fan. Figure 6 shows the boundary curves for and ,
with when and when .
Figure 6.
Regions of (4.2) with and various ’s.
When , numerical experiments suggest the following.
The regions and are
always similar to those in the case;
if , then (4.9) is true; and if for
some , then is a
mouth-shaped region with the two end points of the mouth taken away, equals
the interior of the mouth,
and the mouth approaches a single point, ,
as .
Moreover, approaches the whole plane as and the standard quarter plane with
vertex ,
i.e.,
as .
Each mouth grows bigger and bigger as increases, and eventually becomes the
boundary curve of the square .
Figure 7 presents the regions for and ,
with when ,
when ,
when and when .
An animation shows how the boundary curves
evolve when increases, animation.
Figure 7.
Regions of (4.2) with and various ’s.
Without using the
concept of ’s,
the number of solutions can be obtained from such graphs as follows: if the
point is
below the infinite fan, then the BVP does not have any solution; if is on
the boundary of the fan, then the BVP has a unique solution; if is
inside the fan, but outside all mouths, then the BVP has exactly two solutions;
and the increase of the number of solutions when crosses
the boundaries of mouths is as before.
To understand the
infinite fans together with mouth-shaped regions and their relations with the ’s,
one may imagine that a table is partially covered by a tablecloth, folded only
once. This folded cloth corresponds to the infinite fan. Somehow, the cloth has
small and simple wrinkles (only one layer of the cloth has wrinkles in this
example), pressed onto the table together with the rest of the cloth. These
wrinkles yield the mouth-shaped regions inside the fan.
When , numerical experiments give signs of the following:
there is a constant such
that for each , the regions are
similar to those in the case,
and for every , all ’s
are empty. The existence region approaches the whole plane as , moves in the upper right direction as increases,
and disappears at when . Moreover, as or , the shape of becomes
closer and closer to a half plane. Figure 8 displays the boundary curves for and , with when and when . One can give rigorous estimates of , but we omit the details.
An animation shows how
the boundary curves for evolve when increases, animation.
Figure 8.
Regions of (4.2) with and various ’s.
Example 4.4. is a polynomial of an even degree (and ). When is increasing (i.e., has only one local minimum point and no local
maximum point) and ,
all ’s are the same as those in the case. When is not increasing or has negative values, and are similar to those in the case, and the decreasing intervals of ,
except the first one (from the left) of them, give sequences of mouths for the ’s with .
If the values of on the first decreasing interval are all
non-negative, then this interval does not yield any mouth; and if has negative values on the first decreasing
interval, then this interval also gives a sequence of mouths, even though their
starting time may be very large. Moreover, when is sufficiently large, some mouths may touch
the boundary of the infinite fan (forcing the boundary of the infinite fan
makes sudden turns), and then parts of the boundaries of these mouths become
complicated (yielding extra triangular mouths near their end points). Furthermore,
when is very large, mouths from different
sequences may ‘overlap’. Here quotation marks are used to indicate that the
corresponding mouths are from different layers of the tablecloth.
Figure
9 presents the boundary curves for and ,
,
and ,
with when ,
when ,
when and when .
The first sequence of mouths have point of emergence and starting time for ,
while the second sequence of mouths come out of the point when for .
An animation shows how the boundary curves
for evolve when increases, animation.
Figure 9.
Regions of (4.2) with and various ’s.
For nonlinear
polynomials of an
even degree, the existence of an infinite fan yielding and , i.e., the existence of and , will be proved in Section 5 (see Example 6.13).
The above
discussions on the ’s
for polynomials actually apply to more general functions . For example, it seems to us that the
observations made for odd degree polynomials with a
positive leading coefficient hold for all functions that
are bounded from above on and
from below on ; and the numerical results above for even
degree polynomials are
true for all functions such
that as and are
bounded from below on . The next two examples demonstrate these
situations. Now, it is possible to have infinitely many areas where sequences
of mouths grow.
Example 4.5. Let ,
and consider the BVP (4.2).
Now the DE is ,
a form of the pendulum equation. For error prediction in solving BVP’s for
nonlinear DE’s such as the pendulum equation, see [6]. Numerical experiments indicate that there
are infinitely many sequences of mouths with points of emergence ,
;
and the mouths in each sequence have starting times ,
.
Figure 10 displays the regions for and .
Figure 10.
Regions of (4.2) with and various ’s.
Example 4.6. Consider the
BVP (4.2)
with
There is an infinite fan yielding and .
Numerical experiments show that there are infinitely many sequences of mouths
with points of emergence ,
;
and the mouths in each sequence have staring times ,
.
Figure 11 displays the regions for and ,
with for both choices of .
Figure 11.
Regions of (4.2) with given in (4.12) and various ’s.
In the above
examples, the solutions of the DE involved are either not explicitly known, or
too complicated to be used for a precise description of the ’s
when they are explicitly known. To overcome this difficulty, it is natural to
consider examples with piecewise linear functions . The last example of this section is of this
type.
We mention that
since every continuous function can be approximated by piecewise linear
functions, all BVP’s of the form (4.2), or even (2.1), are limits of
BVP’s of the piecewise linear type.
Example 4.7. Let ,
and consider the BVP (4.2).
The following are observations made from numerical experiments.
(i) When ,
the BVP always has a unique solution.
(ii) When ,
the BVP has:
no solution
if either and ,
or and ;
a solution
if ;
and
a unique
solution if .
See the first
graph in Figure 12.
(iii) When ,
there are two negative slopes and such that the BVP has:
no solution
if either and ,
or and ;
a unique
solution if either and ,
or and ;
and
exactly two
solutions if either and ,
or and .
See the second graph
in Figure 12 ( and when ).
Moreover, as , and ; while as , and . All these observations will be rigorously
verified in Example 5.1, with formulas for and derived.
Figure 12.
Regions for and various ’s.
In the first graph
in Figure 12, the
small circles suggest that the corresponding parts of the boundary of an are not
in . Note that at the origin in the graph, there
is a solid point on the boundary curve.
One can do
experiments for the BVP (4.2) with
(4.13) or
where .
Since the numerical results are similar to those in Example
4.7, we omit the details. However, an exact treatment of these two cases
omitted here will be given in Section
5, see Example 5.2 and Example
5.3.
It is also possible
to get either a sequence of mouths or an infinite fan together with a sequence
of mouths from a two piecewise linear . We will show such examples in the next paper
of this series.
5. Exact Examples
5.1. Overview. In this section, we work out a few examples
exactly. As partially mentioned in the introduction, they serve three purposes.
First, using these examples, we verify some numerical experiments of the
previous section. Second, the set-ups and conclusions in these examples motivate
the theoretical results to be presented in the next section. Third, these
examples can also be used later as references.
5.2. First example.
Example
5.1. Consider the BVP
The following statements give its precise number of
solutions.
(i) When ,
the BVP always has a unique solution.
(ii) When ,
the BVP has:
no solution
if either and ,
or and ;
uncountably
many solutions if ;
and
a unique
solution if .
(iii) When ,
set
where and are the unique solutions of
(5.3) and
on and ,
respectively, then the BVP has:
no
solution if either and ,
or and ;
a unique
solution if either and ,
or and ;
and
exactly
two solutions if either and ,
or and .
Moreover,
(5.4) ,
So, when :
as ,
the existence region goes to the half plane ;
and as ,
the existence region approaches the first quadrant. When ,
and ,
showing the high accuracy of the numerical approximations obtained in Example 4.7.
Next, proofs of
the claims (i)—(iii) are in order.
Fix an . Then,
So, the values of ,
,
cover the whole ,
and is strictly increasing in on .
Thus, if ,
then is strictly increasing in on ;
if ,
then is strictly increasing in on .
Assume further that .
Then, for ,
which has a zero only if ,
and the zeros, if exist, satisfy and are given by
However, for ,
.
See Figure 13. So, (5.10) does not have any root
in .
Hence, is also strictly increasing in on ,
since and .
Therefore, (5.1)
always has a unique solution.
Figure 13.
Intersections of and .
Let .
Then,
Thus, the values of ,
,
cover the whole one-to-one, i.e., (5.1) always
has a unique solution.
Pick an . Then,
So, the values of ,
,
cover the whole ,
and is strictly increasing in when .
For ,
whose zeros, if they exist, satisfy and are given by
However, for ,
.
See Figure 14. So, (5.14) does not have any root
in .
Hence, is also strictly increasing in on ,
since as .
Therefore, (5.1)
always has a unique solution.
Figure 14.
Intersections of and .
Fix an . Then,
So, is strictly increasing in when ,
and as . Direct calculations yield that
As above, we can show that does not have any negative zero. Hence, is also strictly increasing in when ,
since (5.16)
is true and .
Therefore, (5.1)
has no solution if and a unique solution if .
Let .
Then,
Thus, (5.1) has no solution if ,
uncountably many solutions if and a unique solution if .
Pick an . Then,
So, is strictly increasing in when ,
and as .
Direct calculations imply that (5.16) still holds. As above,
we can show that does not have any zero on .
Hence, is also strictly increasing in when ,
since (5.16)
is true and .
Therefore, (5.1)
has no solution if and a unique solution if .
Fix an .
Then,
So, as .
Since ,
the first equation in (5.3) has a unique solution in ;
as ,
;
and as ,
.
See Figure 13. Thus, from (5.9) and
(5.10),
has exactly one zero, i.e., .
This implies that is strictly decreasing in on and strictly increasing in on .
By (5.14),
(5.10)
and the definition of ,
where is given in (5.2).
Therefore, (5.1)
has no solution if ,
a unique solution if and exactly two solutions if .
Note that does not depend on ,
and
(5.21)
Let .
Then,
Thus, the values of ,
,
cover one-to-one; and the same claim is true for
the values of ,
.
Hence, (5.1)
has no solution if ,
a unique solution if and exactly two solutions if .
Pick an .
Set and ,
then ,
and
So, is strictly decreasing in on and strictly increasing in on ,
and as .
Thus, has its minimum point in ,
and hence has a zero in .
Since ,
the second equation in (5.3) has a unique solution in ;
as ,
;
and as ,
.
See Figure 14. Thus, from (5.13)
and (5.14),
has exactly one zero in ,
i.e., .
So, is strictly decreasing in on and strictly increasing in on .
By (5.22),
(5.14)
and the choice of ,
where is given in (5.2).
Therefore, (5.1)
has no solution if ,
a unique solution if and exactly two solutions if .
Note that is independent of ,
and
(5.25)
Let ,
and set
Then, the discussions of the BVP
are similar to the above. Moreover, the case where can be transformed to the case where .
We omit the details.
5.3. Second example.
Example
5.2. Consider the BVP (5.27)
with
Then, the discussions of the number of solutions of the
problem are similar to those in Example 5.1, with
the only changes to the facts (i)—(iii) being the following: in (iii),
(5.29)
where and are the unique solutions of
(5.30)
and
on and ,
respectively. So, we omit the details.
5.4. Third example.
Example
5.3. Fix a ,
define
and consider the BVP (5.27).
All solutions of the DE in (5.27) are then periodic with a
period of .
So, we can assume that .
The exact number of solutions of the problem is given by the following
statements.
(i) When or ,
the BVP always has a unique solution.
(ii) When or ,
the BVP has:
no
solution if either and ,
or and ;
uncountably many solutions if ;
and
a
unique solution if .
(iii) When ,
set
(5.32)
where and are the unique solutions of
(5.33)
and
on and ,
respectively, then the BVP has:
no
solution if either and ,
or and ;
a
unique solution if either and ,
or and ;
and
exactly two solutions if either and ,
or and .
Moreover,
(iv) When ,
the BVP has:
no
solution if ,
and
uncountably many solutions if .
Remark 5.4. The negated
problems of the BVP's (5.27) with given by (5.26), (5.28)
and (5.31)are
the BVP's (5.27)
with defined by
(5.35)
and
respectively. So, one can get the number of solutions for
these problems from Example 5.1, Example 5.2 and Example 5.3.
We omit the details, but only mention that the existence region of the negated BVP's is below its boundary
when it is not the whole -plane.
6. Proofs of the Main Results
6.1. Overview. In this section, we prove a few theoretical results
about the BVP’s of the form (2.1). These results are
suggested by the numerical experiments of Section 4 and the exact examples of Section 5.
Recall that for each
and
every pair , denotes
the unique solution of the IVP (2.4), the
maximum domain of , and the
right-end point of ; when is
given and does not change, , and are abbreviated
as , and , respectively.
6.2. Proof of Theorem 2.1.
The following fact
reveals an advantage in omitting from
the DE.
Lemma
6.1. If is a solution of and is not in the maximum domain of ,
then is unbounded as ,
where is the right-end point of .
Proof. Note that by the definition of ,
since .
Let .
Then, for any ,
So, if stays bounded on ,
then so does by the assumption (2.2) on ,
and hence the solution can be extended beyond ,
contradicting to .
ÿ
Proof of Theorem 2.1. Note that by (2.2), (2.5)
and (2.6),
for each ,
there are functions such that
Fix .
Set
(6.4)
(6.5)
We can assume that on ,
,
and .
Then and .
Fact 6.2. Let .
If and ,
then on ,
is decreasing, and ;
if and ,
then on ,
is increasing, and .
To show the
first part of the fact, we note that there is a such that .
So, for each ,
This implies that is decreasing on .
In particular, .
Thus, any number in is a choice of ,
and hence the first part of the fact follows. The second part of the fact is
proved similarly.
Assume that for
each . Then, is
continuous on , since the solution depends continuously on
the initial values. By Fact 6.2, for ,
for ,
So, there is an such that .
To prepare for the
discussion of the remaining cases, let , and assume that . By Lemma 6.1, is
unbounded as . If does
not go or , then there are two values, either both in or both
in , such that for each sufficiently small , attains
both values infinitely many times on , and hence is
unbounded from both below and above on by the
Mean Value Theorem, which together with Fact 6.2 imply
that is
monotone on for
some , a contradiction. Therefore, either or . Moreover, if and , then ; if and , then .
Note that the set is
open. So, the set is
closed. If such
that and , then there is a such
that and , and hence by Fact 6.2, there exists a neighborhood of such
that for each , either , or and . There is a similar statement for the
situation where . Therefore,
are also closed.
If ,
we set
Then, ,
and hence ,
which implies that is closed; and there is a such that ,
and .
When is sufficiently close to ,
,
i.e., ,
and we have that and .
By Fact 6.2, for such an ,
is decreasing on ,
and hence .
Assume that ,
and .
Then, combining the above arguments, there is an such that .
Assume that , and . Then, after letting
similarly there is an such that .
Assume that , and . Then, after defining
there is an such that .
ÿ
6.3. Consequences and examples
related to Theorem
2.1.
As a direct
consequence of Theorem 2.1, we have the following
result.
Corollary
6.3. If is continuous on and bounded from above on and below on ,
then for every pair ,
the boundary value problem (2.1)
has a solution.
The following
example is a special case of Corollary 6.3.
Example 6.4. For any nonlinear polynomial of an odd degree and with a positive leading
coefficient, such as ,
the BVP
always has a solution.
Actually, the
proof of Theorem 2.1 implies the following facts.
Lemma 6.5. (i) If there are a constant and a function such that
then each solution of is bounded from above.
(ii) If there are a constant and a function such that
then each solution of is bounded from below.
(iii) If there are constants and functions such that (6.17) and (6.18) hold, then all solutions of are defined on the whole .
Example 6.6. By Lemma 6.5 (iii), all solutions of are defined on the whole .
Proof of Theorem 2.2.
Fix an ,
let satisfying ,
and set .
Then, there is a such that and on .
For each ,
Thus, on .
This implies that (2.1)
has at most one solution. However, by Theorem 2.1, (2.1)
has a solution. Therefore, (2.1) has a unique solution. ÿ
6.5. Consequences and examples
related to Theorem 2.1.
Corollary 6.7. If is continuous on and for each ,
is increasing on ,
then the boundary value problem (2.1) always has a unique solution.
Example 6.8. If an odd degree polynomial is increasing, such as with ,
then the BVP (6.16)
always has a unique solution.
In particular,
for each odd ,
the BVP
always has a unique solution. Of course, when ,
the solution is .
For each odd ,
the EP
does not have any non-positive eigenvalue, since when ,
the BVP (6.21)
only has the trivial solution. Actually, for such an ,
the set of real eigenvalues of the EP (6.21) is
with ;
and the remaining part of the proof of this fact will be given in one of the
next papers of this series.
Lemma 6.9. If for each ,
there are a constant and a zero measure set such that (2.7) holds, then for each ,
when is sufficiently negative, the solution of satisfying and has a critical point in ,
and the first such point fulfills
Proof. Let ,
and set .
Assume first that .
Define ,
and ,
where is given by (2.2).
Then .
When ,
(2.2)
implies that on the interval starting from such that on it, and there is a such that ,
from which one deduces that ,
where if does not have a critical point in .
For such an ,
we introduce the Prüfer angle for the solution by and ,
where .
Then,
and .
By (2.7)
and (6.23),
where .
Thus, .
This shows the claims under the assumption that .
If ,
then a part of the above proof yields the claims. ÿ
The introduction of
the Prüfer angle in the above proof of Lemma 6.9 can
be avoided if Theorem 5.2 in [2] is used.
Proof of Theorem 2.3. Note that each solution of is defined on the whole .
By assumption, there are and such that on ,
and for each ,
is increasing on ;
and on ,
and for every ,
is decreasing on .
Let ,
and .
Then .
First, fix an .
By (2.2)
with ,
there is an having the following property: for every ,
there is a such that ,
and on .
So, by Lemma 6.9, when is sufficiently negative, has a local minimum point in .
For such an ,
let be the first local minimum point of ,
then
,
and on .
Thus, if is sufficiently negative, then ,
which we assume from now on. Set ,
and denote by the inverse function of on .
Then, for each ,
Let be the maximum subinterval of such that on it, then on .
Set ,
and .
Then, ,
and .
Denote by the inverse function of on .
Then, for each ,
By (6.26), (6.27)
and the fact that increases in for ,
for each ,
which implies that
So, if ,
then there is a unique such that ,
and on .
Moreover, .
Note that
and as .
If ,
then
on each subinterval of such that on it. Hence, when is sufficiently negative, we have that ,
and there is a unique such that ,
and on .
Actually,
Thus,
and as .
So, by a result similar to Lemma 6.9, as long as is sufficiently negative, ,
and has a local maximum point such that on .
For such an ,
,
and
As above, when ,
there is a unique such that ,
and on ;
furthermore,
and hence as .
If ,
then
on each subinterval of such that on it. Thus, when is sufficiently negative, we have that ,
and there is a unique such that ,
and on ;
in this case,
and hence as .
This procedure can be repeated infinitely many times. Therefore, for each ,
only the first few terms of the sequence
in are defined, with the ’s used being the roots of in ,
the ’s the roots of ,
and the ’s the roots of ;
for every ,
each of ,
and is defined when is sufficiently negative; when defined,
By (6.41) and (2.1),
for any ,
for all sufficiently negative and every such that is defined. Note that each of the ’s, ’s and ’s depends continuously on on their domains of definition (with only at the ’s defined). Also, when is sufficiently large, there is an such that .
Let be a constant such that if such an exists, otherwise. For ,
set equal to a number such that if such an exists, otherwise. Then, when is sufficiently large, .
For such an ,
define
From (6.41) and (6.42)
we obtain that
Given a ,
the continuity of in on then implies that is a value of in and in for every sufficiently large .
Therefore, (2.1)
has infinitely many solutions.
The case where can be
shown similarly, with only removed. ÿ
It is interesting to
find out if the monotonicity in assumptions on in Theorem 2.3 are necessary.
6.7. Consequences and examples
related to Theorem
2.3.
Among direct
consequences of Theorem 2.3 is the following fact.
Corollary
6.10. Assume that is continuous on ,
for each sufficiently negative ,
is increasing on ,
and for every sufficiently large ,
is decreasing on .
If
uniformly in on ,
then for every pair ,
the boundary value problem (2.1)
has infinitely many solutions.
The next example is
a particular case of Corollary 6.10.
Example 6.11. For each polynomial of an odd degree and with a negative leading coefficient, such
as ,
the BVP (6.16)
always has infinitely many solutions.
For every polynomial
of an
odd degree , with a negative leading coefficient and
without any constant term, the EP
has all the real numbers as eigenvalues: for each ,
the BVP (6.46)
has infinitely many solutions and hence a non-trivial one.
Proof of Theorem
2.4. Let be the
set of ’s
such that the DE has a solution defined on the whole whose
value at equals . Then, is open
and, by assumption, non-empty.
Now, fix an . Denote by the set
of ’s
such that . Then, is open
and non-empty. Let be the
set of connected components of , which are open intervals. By (2.6) and the proof of Theorem 2.1, for each pair , there is a constant such
that
In particular, if is an end point of some ,
then as .
From (2.7)
and Lemma
6.5 (ii) we see that if is an end point of some ,
then as in .
Fix a ,
let ,
denote by the in (6.47)
for ,
and set .
Then, by Lemma 6.9, there is an such that for each ,
the first local minimum point of is in .
Moreover, we can assume that for every ,
.
By applying an identity like (6.27) and estimates similar to
those in (6.28)
and (6.29),
from (2.7)
we deduce that the first root of in satisfies
Thus, (6.47) implies that
As a consequence of this fact, if is an end point of some ,
then as .
So, for each end point of every ,
as in .
Let be the
minimum value of on , and pick an such
that . If there are infinitely many ’s
which are , then the corresponding ’s
have an accumulation point, say , since these ’s
are bounded in view of (6.47) and (6.49). Note that . So, , and . By the latter, there is a such
that both and are
large, in particular, . When is
sufficiently close to , both and are
also large. This and Fact 6.2 yield that is
increasing on , contradicting . This shows that there are at most finitely
many ’s
in . So, has a
minimum value on , to be denoted by .
Then, (2.1) has
(i) no solution if either ,
or and ,
(ii) a solution if and ,
(iii) at least two solutions if and .
Moreover, is
continuous from above. Assume that is
not continuous from below at . Then, there are a constant and a
sequence in going
to such
that for all
. Pick an satisfying . Then, the ’s
are bounded, since (6.47) and (6.49) can be made
uniformly in in a
small neighborhood of in . So, we can assume further that the ’s
have a limit, say . Thus, : otherwise, the ’s
approach , impossible. Hence, , which also implies a contradiction as in the
last paragraph. This shows that is
continuous from below. Therefore, is
continuous.
The reverse problem
of (2.1) is
the DE has a solution defined on the whole ,
and satisfies the conditions corresponding to (2.7)
and (2.6).
So, there are a non-empty open subset and a continuous function such that (6.50)
has
(i) no solution if either ,
or and ,
(ii) a solution if and ,
(iii) at least two solutions if and .
Let .
Then, for all ,
(2.1)
with has a solution, and hence so does (6.50)
with by the Reversion Principle. This implies that
,
and for each .
Hence, ,
and for every .
Therefore, there is an such that ,
and is decreasing. Since is continuous, its graph does not have any
vertical segment (i.e., jump). So, is strictly decreasing. Moreover, since is strictly decreasing. ÿ
6.9. Consequences and examples
related to Theorem 2.4.
Corollary
6.12. Assume that is continuous on ,
(i) the differential equation has a solution defined on the whole ,
(ii) as ,
uniformly in on ,
and
(iii) is bounded from below on ,
then the conclusions of Theorem
2.4 hold.
Example 6.13. Let be an even degree polynomial with a positive
leading coefficient. If has a real root, e.g., ,
,
etc, then the conclusions of Theorem 2.4 always hold
for (6.16),
since has a (real) constant solution; if does not have any real root, e.g., ,
etc, then there is a such that the conclusions of Theorem 2.4 hold for (6.16) if
,
and does not have any solution on the whole if .
For each even , the EP (6.21) has all the negative
numbers as eigenvalues: when , has a
negative root ; so, , and ; thus, (6.21) has at least two
solutions and hence a non-trivial one. Actually, the set of real eigenvalues of
the EP (6.21) is with ; and the remaining part of the proof of this
fact will be given in one of the future papers of this series.
6.10. Refinements of Theorem 2.1 and Theorem 2.4.
By relaxing (2.7),
we obtain the following improvement of Theorem
2.4. The proof of this improvement is
basically the same as that of Theorem 2.4 and hence is omitted.
Theorem 6.14. Assume that
(i) the differential equation has a solution defined on the whole ;
(ii) there are some constants and and a zero measure set satisfying (2.7);
(iii) there is a function fulfilling (2.6).
Then, the conclusions of Theorem
2.4 hold.
As a direct
consequence of Theorem 6.14, we have the following
result, which is an improvement of Corollary 6.12.
Corollary
6.15. Assume that is continuous on ,
(i) the differential equation has a solution defined on the whole ;
(ii) there is a constant such that
(iii) is bounded from below on .
Then, the conclusions of Theorem
2.4 hold.
Example 6.16. Consider a
function
where is bounded from below such that .
For example, ,
or .
If ,
then the conclusions of Theorem 2.4 hold for the BVP
(6.16).
The cases where and have been discussed in Example
5.1 and Example 5.2, respectively.
Remark 6.17. By Example 5.1 or Example 5.2,
the lower bound on the constant (or equivalently, the lower bound on ) in Theorem 6.14
and Corollary 6.15 is sharp. Moreover, Example 5.3 shows that the condition (iii) cannot be removed
in either of Theorem 6.14 and Corollary 6.15.
By replacing (2.5) and (2.6) by conditions on that
are weaker in the -direction,
we obtain the following parallel result of Theorem 2.1.
Theorem 6.18. If there are constants ,
and a zero measure set such that
then for every pair ,
the boundary value problem (2.1)
has a solution.
Among direct
consequences of Theorem 6.18 is the following fact,
which is an improvement of Corollary 6.3.
Corollary 6.19. Assume
that is continuous on .
If there is a constant such that
then for every pair ,
the boundary value problem (2.1)
has a solution.
Example 6.20. Consider a
function ,
where are continuous such that is bounded, and .
For example,
(6.56)
If ,
then the BVP (6.16)
always has a solution. The first example of in (6.56)
has been discussed in Example 5.3
Remark 6.21. By Example 5.3, the upper bound on (or equivalently, the upper bound on ) in Theorem 6.18
and Corollary 6.15 is sharp.
By replacing (2.6) by a condition on which
is weaker in the -direction,
we obtain the following parallel result of Theorem 6.14.
Theorem 6.22. Assume that
(i) the differential equation has a solution defined on the whole ;
(ii) there are constants ,
,
,
and zero measure sets such that (2.7) is satisfied and
Then, the conclusions of Theorem 2.4 hold.
As a direct
consequence of Theorem 6.22, we have the following
result, which is parallel to Corollary 6.15.
Corollary
6.23. Assume that is continuous on ,
(i) the differential equation has a solution defined on the whole ;
(ii) there are constants and such that
Then, the conclusions of Theorem
2.4 hold.
Example 6.24. Consider a
function ,
where are continuous such that is bounded, and .
Two such ’s are given in (6.56).
If ,
then the conclusions of Theorem 2.4 hold for the BVP
(6.16).
The case where is the first one in (6.56)
has been discussed in Example 5.3.
Remark 6.25. By Example 5.3,
the lower bound on and the upper bound on (or equivalently, the lower and upper bounds and on ) in Theorem 6.22
and Corollary 6.23 are sharp.
Acknowledgement: We are grateful to Lingju Kong, Qingkai Kong
and Anders Linnér for helpful discussions.
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Department of Mathematical Sciences, Northern Illinois University,
DeKalb, Illinois 60115, USA
E-mail address: wu@math.niu.edu