Existence and Uniqueness in Differential Equations
By the proof of [Olv, pp.30-31, Theorem 1.29], the (a,b)'s
in [Bir, p.165, l.-7-l.-6] should have been replaced by (a,c)'s,
where c is near b.
Existence, uniqueness, and counterexamples.
Types of existence.
(1). Inverse image of a rectifying mapping [Arn, p.50, Theorem 7.3].
(2). Abstraction of a numerical approach [Arn, p.221, Corollary].
(3). Closed form solutions [Arn, p.13, Theorem 2.3 3)].
Discussion. (1) originates from intuitive geometrical considerations. However,
it is difficult to specify a rectifying mapping effectively. There are no
convenient packages (closed form solutions) in (2) because (2) uses sequential
limits.
Types of uniqueness.
(1)'. Geometric method:
Rectification [Arn, p.50, Theorem 7.4].
(2)'. Analytic method: Contraction
[Arn, p.222, Solution 1].
(2.5)'. [Bir, p.25, Theorem 6'].
(3)'. v(x_{0})¹0:
Inverse function theorem [Arn, p.14, l.!12]; v(x_{0})=0:
Comparison theorem [Arn, p.17, Theorem 2.7; Arn1, p.37, l.!15-p.38,
l.8].
Discussion. [Arn, p.222, Solution 2] is basically the same as [Arn, p.222,
Solution 1] because the former repeats the argument in [Arn, p.212, l.!2].
(3)' is the most elaborate because it
excludes a lot of impossibilities for nonuniqueness. (2.5)'
also
shows the solution's continuous dependence on initial conditions. From (3)'
to (1)' we gradually lose effective
information about how to construct the counterexamples for uniqueness [Arn1,
pp.36-37, §2.2].
(Hard core) For the solution's continuous and differentiable dependence on initial
conditions and parameters, Pontryagin uses successive approximation to
separately prove [Pon,
pp.170-181, Theorem 13, 14 & 15]. In fact, these proofs are very
similar to the proof of the existence theorem in [Pon,
§20]. We have to repeatedly check the
cumbersome details because, by successive approximation alone, we have no way to
visualize the big picture. However, from the viewpoint of rectification, all the
above theorems are nothing but trivial corollaries of [Arn1, p.89, Theorem 1]
(see [Arn1, pp.93-98, Corollary 1-Corollary 6; p.98, Remark]). Thus [Arn1,
pp.89-98] reduces the whole theory to its hard core.
(Isolation from complexity) Without using rectification it would be very
complicated to construct first integrals [Pon, p.183, (C)] and the solutions for
Cauchy's problem [Pon, p.186,
l.12-l.13]. Arnold uses the standard equation to interpret [Arn1, p.127, Theorem
& p.131, Theorem] because he would like to reveal what is really involved
besides rectification.
Why uniqueness breaks down:
Failure to satisfy a Lipschitz condition (see [Inc, pp.17-18, Example
2.1]). Nonunique solutions may become unique in a sense [Bir, p.23, l.1].
Failure to satisfy a growth condition [Joh, p.217, l.12-l.15].
Outside the cone in which initial conditions determine a unique solution
[Pet, p.76, l.-9-p.77, l.8].
[Har, p.8, Theorem 1.1] shows that every solution z(t) is the standard
solution (i.e., the limit of y_{n}(t)). [Arn1, p.277, Solution 2] proves
that any two solutions must be the same. The latter proof is more effective
because the procedure can be finished in finite steps.
The rectification theorem [Arn, p.48, Theorem 7.1].
C^{1} case [Bir, p.165, Theorem 12] & the analytic case [Che,
p.89, Lemma 1].
c in [Bir, p.165, l.-11] means
ξ in [Arn1, p.284, Fig.230]. Birkhoff proves
[Bir, p.165, Theorem 12] using the solution's the differentiable dependence on initial conditions
[Bir, p.165, l.-10]. For the proof of this
differentiable dependence, [Arn1, p.280, l.6-l.12] is simpler than [Bir, p.163,
Theorem 11] because Arnold avoids repeating the convergence argument.
The key to changing from the analytic case to the C^{r}-case
is the differentiation under integral sign [Arn1, p.143, l.12 & p.282, l.3].
Uniqueness based on formal expressions [Joh, p.148, l.-3-l.-1].
The energy integral is used in [Joh, p.139, l.-3-p.140,
l.-14] to prove the uniqueness of the
solution of a mixed problem for the wave equation.
(Uniqueness).
In Dirichlet's problem, boundary
conditions Þ uniqueness. For the
heat equation [Joh, p.217, (1.36a)], initial and growth conditions
Þ uniqueness [Joh, p.217, l.5]. [Joh, pp.215-217, 3
theorems] show how these initial and growth conditions can be
developed from boundary conditions.
The solution's uniqueness
for Cauchy's problem is determined by
the coefficients of its power series [Pet, p.18,l.-2-p.20,l.20].
How to strengthen the argument for uniqueness.
Allowing more solutions to be included.
Analytic functions [Pet, p.20, l.18-l.20] ®
nonanalytic functions [Pet, p.34, §4].
Weakening the condition that ensures uniqueness.
Growth condition [Joh, p.217, (1.36c)] ®
unilateral condition [p.222, (1.57c)] (see [Joh, p.222, l.9-l.14]).
Using general properties instead of special ones.
[Pet, p.75, §11] shows that we may
use Green's theorem to prove the
uniqueness for wave equations. It is not necessary to go that far [Joh, p.129,
(1.14)].
In practice, the uniqueness theorem of P.D.E. allows short cuts to obtain
a solution [Cor, p.230, l.10; p.231, Example].
(Baire's category theorem [Roy, p.139, Corollary 16; Dug, p.251, Ex. 6])
The existence given in [Dug, p.300, Theorem 4.2] is logical
[Wan3, p.109]. This existence is not as effective as the constructive existence
[Wan3, p.109] given in [Gel, p.38, l.-6-p.39, l.3].
It is a setback that modern mathematicians rashly adopt short proofs [Roy, p.141,
Exercise 30.d; Ru2, p.121,
Exercise 14] but neglect the most precious part of mathematics, effectiveness. [Mun,
§7-8] tries to improve the effectiveness of
the existence given in [Dug, p.300, Theorem 4.2]. The attempt is futile because Munkres' use of Baire's category theorem has ruined effectiveness in the first
place.¬
A mathematical problem can uniquely define a solution set [Dug, p.411,
Definition 1.1]. First, we must ask whether the problem has solutions. If the answer is yes, then
we must ask what they are. In order to find the elements of the solution set
[Dug, p.411, Proposition 1.2], we must solve the problem first.
(Fundamental theorem of algebra)
We may prove the fundamental theorem of algebra using Liouville's theorem
(see [Con,
p.77, Proposition 3.5]), the argument principle (see [Con, p.121, l.-3-p.122,
l.5]), [Mun00, p.349, Lemma 55.3] (see [Mun00, p.354, Theorem 56.1]), or [Usp,
p.102, Theorem] (see [Usp, Appendix I]). It is an incorrect approach to list the fundamental theorem's various proofs without
discriminating their effectiveness [Mun00, p.353, l.-2-p.354,
l.5]. The second and the third proofs use the same idea but different settings.
The second proof uses analytic functions, winding numbers, and the logarithmic
function [Con, p.121, l.-9] while the third proof uses
the corresponding but less effective concepts such as continuous functions, the
fundamental group of path-homotopy classes, and covering spaces. As a result,
there is a difference in effectiveness between the second proof and the third
one. Given a specific polynomial of degree n. Based on the second or third proof, we may find an R > 0 such that all the n roots
lie in |z|
< R [Mun00, p.356, Exercise 1]. However, the second does not use reduction to
absurdity. In order to achieve the same result, the third proof employs reduction
to absurdity three times due to its abstract setting. Based on the
first proof, there is no way to gain any information about any root's location.
The fourth proof given in [Usp, pp. 293-297], based on Gauss' proof, is more effective
than the above three. In fact, it is the only proof that can be translated into
an effective algorithm and implemented in a computer program. This is because the the
following steps of the proof
enable us to further narrow down our search:
(a). Find the n roots of T on |z| = r > R. See [Usp, p.295, l.18]. We search for roots on a 1-dim arc rather than
in a finite area.
(b). Start from |z|
= R and draw the curve T = 0 outwardly by using (a).
(c). Start from (1) on |z|
= R and proceed inwardly along T = 0 to find a root of U. See [Usp, p.297, Fig.].
Again, we search for roots on a finite 1-dim curve rather than in a finite area.
Remark. The proofs of [Dug, p.342, Proposition 3.1] and [Arn1, p.311, Example 2] are essentially the same as
that of [Mun00, p.354, Theorem 56.1] except that instead of a covering space Dugundji uses the degree of f: S^{1}
® S^{1} and Arnold uses the indexes of a curve. All these
concepts are equivalent. Note that it is unnecessary to assume that f(z)¹0
for all finite z in [Dug, p.342, l.6]. All we need to assume is that f(z)¹0
for |z|
£ t_{0}. Note also that [Dug, p.342,
Proposition 3.1] applies to continuous functions other than polynomials.
An idea vs. its execution on constructing an external free product of groups
[Mun00, p.416, l.13-l.23]
The idea of constructing an external free product of groups is simple and intuitive: juxtapose words and then reduce the result
[Mun00, p.418, l.14-l.15]. However, for an simple and elegant proof of its
existence, we must make a detour by considering P(W) instead of W [Mun00 p.416,
l.4-l.7].
The ultimate goal of the general uniqueness theorem [Har, p.31, Theorem 6.1] concerning the solutions of ordinary differential equations
is to seek the
weakest hypothesis about f and an arbitrarily small
e>0 such that [Har, p.31, (6.4)] holds on [t_{0},t_{0}+e].
Both [Gon, p.750, Theorem 9.23] and [Gon, p.752, (9.16-3)] prove the
existence of the inverse function. However, there is a difference between the
two theorems. The proof of [Gon, p.750, Theorem 9.23] uses Rouché's
theorem which cannot give the exact location of zeros. In contrast, [Gon, p.752, (9.16-3)]
gives the exact values of the inverse function.¬
Sometimes, an existence that seems hypothetical can become constructive
existence, if we dig deeper and trace back to its origin. For example, based on
the definition of uniform convergence, the existence of b_{n}
in [Gon1, p.287, l.1] is hypothetical, However, if we make some effort and trace
back to its origin, it is not difficult to find a concrete value for
b_{n}. [1]
After constructing a mathematical object, we should list as many its
properties as possible.
However, in proving the object's uniqueness, we should use the minimum number of
required properties. For example, the positive measure m
constructed in [Ru2, p.42, Theorem 2.14] has properties (a), (b), (c), (d), and
(e). However, it suffices to use (a), (c), and (d) to prove the uniqueness of
such a measure.
Narrowing down the scope of an existence
If A is a proper subset of B and the solutions of a problem belong to A,
then we should say the solutions are in A rather than B [Sak,
p.221, l.-9-l.-7].
[Ahl, p.222, Theorem 1]
says that if G is a region and S^{2} - G is
connected, then G can be transformed conformally onto the open unit disk. In
contrast, [Ru2, p.292, Theorem 13.11] says that G is homeomorphic to the open
unit disk under the same hypothesis. Thus, the formulation of the Riemann
mapping theorem given in [Ahl, p.222, Theorem 1]
is better than that given in [Ru2, p.292, Theorem 13.11] because the set of the
conformal mappings is a proper subset of the set of the homeomorphisms. In
addition, it is easier to calculate B = sup _{fÎF}
f '(0) given in [Ahl, p.223, l.1] than
m = sup _{fÎM}
|f '(0)| given in
[Sak, p.228, l.16]. This is because F given in [Ahl, p.222, l.15] is a proper subset of M
given in [Sak, p.228, l.14] and because m requires the extra operation of taking
the absolute value. Remark. (Region whose complement does not have any
component that is a single point) We may use Perron's method [Ahl, p.240, l.6] to construct the
Riemann mapping [Ahl, p.243, l.-4-l.-1].
Perron's method is more effective than the method given in the proof of [Ahl,
p.222, Theorem 1] because it allows us to avoid using the diagonalizing process given in
[Ahl, p.215, l.8].
(Polygons) The Schwarz-Christoffel formula [Sak, p.233,
(8.4)] maps the upper half plane onto the inside of a polygon [1]. (Rectangles)
z = sn^{-1} w [Gon1, p.424, Fig. 5.19] maps
the upper half plane onto the inside of a rectangle.
If the convergence of Type I implies the convergence of Type II, then we say the convergence of Type I is stronger than the convergence of Type II. When we say an
improper integral exists, we mean that the integral
converges to a certain
limit. The strength of the existence of the integral value depends on the
strength of the convergence that we adopt as our method to evaluate the
integral [1].
The uniqueness theorem for the initial value problem x' = f (t, x), where f satisfies the Lipschitz
condition
The order of the following list of proofs is based on the effectiveness
of their arguments:
[Bir, p.142, Theorem 1] follows from [Bir, p.24, Lemma 2].
[Cod, p.10, Theorem 2.2] follows from [Cod, p.8, (2.2); case e = 0].
Note that [Cod, p.8, (2.2); case e = 0] is a special
case of [Bir, p.24, Lemma 2].
[Col, chap. I, §6, Sec.19] Advantage: it can be generalized to n-dimensional vector spaces.
Shortcoming: it uses the method of reduction to absurdity.
[Inc1, p.65, l.17-p.66, l.7; Har, p.9, l.-5-p.10, l.2]
Shortcoming: Both quoted passages unnecessarily use the method of mathematical induction.
[Pon, p.157, l.11-p.158, l.-7]
Criticism: If we compare it with A or B, it contains no new ideas except adding the
unnecessary topological mechanism to the argument. Topology is useful only in illuminating mathematical structures.
Although we have proved the uniqueness of the asymptotic expansion in [Wat1, p.154, l.8-l.9], sometimes we still have to verify the consistency of
two asymptotic expansions in the intersection of their domains. This is because the asymptotic expansions may not
be in the standard form.
Example [Wat1, p.349, l.1-l.2].