Existence Theorems for Ordinary Differential Equations

By Francis J. Murray and Kenneth S. Miller

Theorems stating the existence of an object—such as the solution to a problem or equation—are known as existence theorems. This text examines fundamental and general existence theorems, along with the Picard iterants, and applies them to properties of solutions and linear differential equations.
The authors assume a basic knowledge of real function theory, and for certain specialized results, of elementary functions of a complex variable. They do not consider the elementary methods for solving certain special differential equations, nor advanced specialized topics; within these restrictions, they obtain a logically coherent discussion for students at a specific phase of their mathematical development. The treatment begins with a survey of fundamental existence theorems and advances to general existence and uniqueness theorems. Subsequent chapters explore the Picard iterants, properties of solutions, and linear differential equations.

• Language: English
• Publisher: Dover Publications
• Release date: Nov 7, 2013
• ISBN: 9780486154954

Book preview

EQUATIONS

CHAPTER 1

The Fundamental Existence Theorems

1. The basic existence theorem

1.1 Let y be an unknown function of x. Frequently our information on y is its initial value y0 for x = x0 and its rate of change dy/dx. If our information on dy/dx is that dy/dx is a continuous function f(x), then we know that there is a unique y, which may be obtained by integration. If, on the other hand, as often happens, our information specifies dy/dx in terms of y as well as x, that is, dy/dx = f(y, x)¹ then even if f depends continuously on y and x the situation is not clear. For this equation on dy/dx does not yield prima facie evidence of the existence of y, since to evaluate f, y must be known, that is, y must exist.

there is an interval (x0, x0 + h) such that y is well approximated by y0 + f(y0, x0). (x — x0). If x1 = x0 + h and y1 = y0 + f(y0, x0) h we can take a second interval (x1, x1 + h) in which y is approximated by y1 + f(y1, x1) (x — x1). Continuation of this process will yield a polygonal line with slope in the form f(yi, xi) except at the vertices where in general there is not a unique slope. Intuitively, we would expect that if h is taken smaller and smaller the corresponding polygonal lines would converge to a solution y of the differential equation dy/dx = f(y, x).

However, the difficulties of the situation appear as soon as we try to make this procedure precise. Question one is, how do we know these lines will converge ? Secondly, if they do, will the resulting function have a derivative ? (The approximants in general will fail to have derivatives at more and more points as we take h smaller and smaller.) Thirdly, if the limiting function exists and has a derivative, will it satisfy the given differential equation ?

Relative to the first question one can show that if f is continuous one can choose a sequence of h’s approaching zero such that the polygonal lines will converge for some x interval with lower end point x0. However, in general, the size of this interval may be very small and has to be investigated in each individual case.

For questions two and three one has available certain technical devices. One transforms the problem from one in differential equations to an equivalent integral equation

Then if our approximants converge uniformly we obtain the desired results.

Precisely formulated,² the fundamental existence theorem reads as follows:

Theorem 1. Let f(y, x) be a real valued function of the two real variables y, x defined and continuousb such that

.

We shall prove Theorem 1 in Section 1.8. Note that the theorem has been stated for the case of one unknown function y. This will be generalized in later sections to n functions, implicit forms and higher derivatives. However, in order to appreciate clearly the ideas underlying the theory, we shall carry through the initial discussion for the one unknown function y.

It is important to note that Theorem 1 is an existence theorem and not a uniqueness theorem. Under the hypothesis of the theorem, cases exist in which there exist two or more distinct functions satisfying the conclusions of the theorem. Examples will be given later (cf. Section 1.1 of Chapter 3) to illustrate this phenomenon.

1.2 We shall unfold the theory as a logical entity. As we attempt to prove it, it will appear that certain auxiliary lemmas are necessary. These we shall state and prove only after we have seen the necessity for such a digression.

We are concerned with the differential equation

where f(y, x) satisfies the conditions of of x = x0. Hence, since dy/dx = f(y(x), x), dy/dx is continuous, and consequently

. Equation (1) therefore implies

On the other hand, a function y that satisfies . Thus the derivative of the integral on the right hand side of Equation (2) exists and equals the integrand. Consequently, differentiating Equation (2) will yield Equation (1).

We have thus proved the following lemmas:

Lemma 1which satisfies Equation (1), then dφ/dx exists and is continuous.

Lemma 2. The solutions of Equation (1) which at x0 equal y0 are identical with the solutions of Equation (2).

The study of the differential equation, Equation (1), has thus been reduced to the study of the integral equation, Equation (2). As stated above, this will permit us eventually to answer questions two and three.

1.3 In the statement of b appeared in the conclusion. The construction of this neighborhood as well as a discussion of its significance will now be given.

. (Cf. . That is, there exists a constant M such that

FIGURE 1

Now let

′. Clearly,

′.

. To demonstrate (ii), we note that

As will be seen later, the reason for introducing b is to make (ii) a true statement.

Our next task is to construct a polygonal line function which is intended to approximate a solution to Equation (1). Let h > 0 be given. We shall suppose that h is small relative to b. Furthermore, it is convenient to assume that b is an integral multiple of h, that is,

where p is a positive integer.

We define inductively a sequence of pairs of points y1, x1; … ; yp, xp = x0 + b by means of the relations

M. Now suppose f(y0, x0), …, f(yj-1, xj-1) exist and are less than M in absolute value. Then since

we have

M.

We now define a function y(x, h) by the equation

for

and

The quantity y(x0, h) is defined as

Note that the x0 and y0 are the initial point of Theorem 1.

x — xj+1 < 0 where yj = y(xj, h), xj = x0 + jh and j = — 1, — 2, …, — p.

x0 + b define the function F(x, h) by the equation

x < x0 let F(x, h) be defined by

x — xj+1 < 0, j = — 1, — 2, …, — p.

From Equation (3) we conclude that

Also

while

From these equations and Equations (4) and (5) we see that

′ and consequently

But from this result and Equation (6),

which by virtue of (ii) implies

b.

Since F(t, h) is a step function with a finite number of jumps, Equation (6) defines a continuous function of x and we may write

where, by definition,

1.4 , f is uniformly continuous. Hence given an ε > 0 there exists a δ with | x′ — x″ | < δ and |y′ — y″ | < δ. We shall refer to this δ = δ(ε) as the "uniform δ .

Lemma 3. Let ε > 0 be assigned. Let δ > 0 be the uniform δ for ε/b and let h0 = min (δ, δ/M). Then if h < h0 and |