Existence and Uniqueness Theorem. Let f(x, y) be continuous in a domain D C and satisfy a local Lipschitz condition with respect to y

Chapter VI: Boundary Value and Eigenvalue Problems

W. Integrating Factors (or Euler Multipliers). The differential equa- tion

VI. Existence and Uniqueness Theorem. Let f(x, y) be continuous in a domain D C and satisfy a local Lipschitz condition with respect to y

in D (this hypothesis is satisfied, for instance, if af/ay e C(D)). If E D, then the initial value problem

y'=f(x,y),

⁽⁵⁾

has exactly one solution. The solution can be extended to the left and right up to the boundary of D.

The line of reasoning used for the case n = 1 in §6 carries over. The following special case is proved first (compare with Theorem 6.1).

Vii. Theorem.

Let f(x, y) be continuous and satisfy the Lipschitz con- dition (4) in J x RTh, J = + a]. Then there is exactly one solution to the initial value problem

y'=f(x,y),

(6)

The solution exists in of J.

Proof. The initial value problem (6) is equivalent to the integral equation

y(x) = + f f(t,y(t)) dt

ⁱⁿ J, (7)

which can be written, using the more concise operator notation, in the form

y =

Ty where (Tz)(x) = ij

+ f f(t, z(t)) dt.

(7') The set of continuous vector-valued functions defined on J with the norm

lizil = max lz(x)Ie_2L1

§10. The Initial Value Problem for a System of First Order 109

is a Banach space (I I isthe norm in that appears in (4)). The operator T defined by (7') maps this space into itself. The proof that T satisfies a Lipschitz condition with Lipschitz constant is the same as the one given in 6.1. In this proof, y(x) is now a vector function, and the two simple facts in the following lemma are needed.

VIII.

Lemma. If z(x) ^is continuous in an interval [a, b] and I is a norm in then the scalar function = Iz(x)I is continuous in [a, b] and the inequality

b b

f

^z(x)dx

holds.

Proof. It follows from inequality (5.2) that

as xk—'x;

therefore is continuous.

Let us denote the integrals in the lemma by Ii and 12. Then the inequality

I

Iii

12 must be proved. We consider a partition P : a = x0

< x1 <

= ^b and its measure of fineness Fl = max{(x2 —

i =

1,...,p}.

Corresponding Riemann sums are given by

o(P; z) ^— for

=

f

^{z(x) dx,}

a(P; IzI)

= ^P

— for 12 = Iz(x)I dx.

Thetriangle inequality implies

z)I o(P; Izi). (*)

Now consider a sequence (Pk) of partitions with lim lPk I = 0. Then, by the Riemann definition of the integral,

o(Pk; z) and o(Pk;IzI) 12

as k oo. The inequality Iii 12 now follows from *).

I

The solutions of (6) are the fixed points of T. Since T is a contraction, Theorem VII follows from the Contraction Principle 5.IX. The solution is the limit of a uniformly convergent sequence of successive approximations

Yk+1

(Tyk)(x) = + f f(t, yk(t)) dt

^{(k = 0,1,2, .}..). ⁽⁸⁾ The first term yo(x) E C(J) can be arbitrarily chosen.

I

The general theorem is now derived from this special case in a series of steps that are completely analogous to those in §6 for the one-dimensional case.

In §7 and §8 two additional existence theorems were proved. The extension of the proofs of these theorems to the n-dimensional case is also straightforward.

Consequently, we will state these theorems without proof.

IX.

Peano Existence Theorem.

If f(x, y) is continuous in the do-

main D and

E D, then the initial value problem (5) has at least one solution. Every solution can be extended to the left and right up to the boundary of D.

Remark. The results on upper and lower solutions and on maximal and minimal solutions obtained in §9 do not extend to general systems, but only to systems that have a certain monotonicity property. We shall treat this impor- tant question in Supplement I below.

X.

Existence Theorem for Complex Differential Equations.

Let the vector function f(z, w) of n + 1 complex variables (z, w) =(z,

...,

with values in _C (i.e., each component

is continuously differentiable with respect to all n + 1 complex variables), and let (zo,wo) ED.

Then the initial value problem

w'=f(z,w), w(zo)=wo

(9)

has exactly one holomorphic solution w(z). The solution exists (at least) in the disk K : lz— zoI < a, where a> 0 is determined as in 8.11.

The solution w (i.e., each of the components can be expanded in a power series about the point zo with a radius of convergence a.

XI.

Autonomous Systems.

In this section, we develop a general frame- work for problems of the type introduced in 3.V. We call the system of differ- ential equations (1) autonomous if the right-hand side f(x, y) does not depend explicitly on x. Thus an autonomous equation has the form

y' =

f(y). (10)

Autonomous equations frequently arise in applications where the independent variable is time. With these problems in mind, we denote the independent variable by t and write y = y(t). In the results that follow,is assumed to be locally Lipschitz continuous in an open set G C Thus Theorem VI applies to (10) and it follows that initial value problems are uniquely solvable and the solution can be extended to the boundary of D = x G (cf. the definition in 6.VII). This leads us to a number of conclusions about solutions to (10):

(a) A solution y exists in a maximal open interval J = (a,b).

§10. The Initial Value Problem for a System of First Order 111

(b) If y is a solution of (10) in the interval J (a, /3), then z(t) := y(t + c) is a solution in the interval = (a^— c,/3 — c).

(c) Phase Space and Trajectories (Orbits). For autonomous systems, the space is called the phase space. The curve C

y(J) :=

{y(t) t J} C C in the phase space generated by a solution y on a maximal interval of existence .1 = (a,b) is called the trajectory or the orbit of y; cf. A.I for the definition of curves. If z is another solution with z(to) E C and z(to) = y(tj) for some t1 E J, then =

y(ti

^— t0 + t) is also a solution by (b), and the relation z(to) = implies z = ^Therefore, the trajectories of y and z coincide.

If the "half trajectory" y( [c, b)) with c E J is contained in a compact subset of C, then b ^{00; a}corresponding statement holds for the interval (a, c}.

(d) The Phase Portrait. Two trajectories are either disjoint or identical.

Each point of C belongs to exactly one trajectory. The collection of all trajec- tories is called the phase portrait of the differential equation; cf. 3.V.

(e) The two differential equations y' = f(y) and y' Af(y) (with A 0)

generate the same trajectories, hence the same phase portraits (with the same orientation if A > 0).

(f) Periodic Solutions. If y is a solution and y(to) =

y(ti)

for some to then y is periodic with period p = ^{to —}ti. This follows from (c), with z = y.

The maximal interval of existence is R. A nonconstant, continuous, periodic function has a smallest period T> 0, also known as the minimal period.

(g) Critical Points. A point a E C is called a critical point (also a stationary point or equilibrium point) of f if f(a) = 0. If a is a critical point of f, then y(t) a is a solution in IR. The corresponding orbit is the singleton {a}.

(h) If the solution y exists for t to and if a urn y(t) exists and belongs t-.oo

to C, then a is a critical point; i.e. f(a) = ^0.

Proof of(h). Suppose is a real-valued C'-function and Em = a 0.

t-+oo

Ifa> 0, then for large t, > a/2. It follows that lim =^oo. Similarly,

a <

⁰ implies = —00. Now the hypotheses imply that limy'(t) = f(a). The preceding argument applied to the components yk(t) of y shows that

]imy'(t)=O,i.e.,f(a)=O. I