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#### Stability and Periodic Solutions of Ordinary and Functional Differential Equations

**By T. A. Burton**

**Dover Publications, Inc.**

**Copyright © 2005 T. A. Burton**

All rights reserved.

ISBN: 978-0-486-15045-1

All rights reserved.

ISBN: 978-0-486-15045-1

CHAPTER 1

**An Overview**

To a reader who is unacquainted with differential equations, an overview such as this can be quite meaningless. Thus, for such readers, we state that **Section 1.1** of **Chapter 1**, together with **Section 4.1** of **Chapter 4**, will provide a basic introduction to theory of ordinary differential equations and constitutes a very standard and self-contained one-semester beginning graduate course. This is essentially the material used for such courses taught by the author during the past 21 years. And the reader without background in differential equations may do well to ignore the rest of this overview until those two sections are well in hand.

Most authors choose to treat separately the subjects of ordinary differential equations, Volterra equations, equations with bounded delay, and functional differential equations with infinite delay. Much too frequently the reader encounters these as entirely different subjects requiring a different frame of mind, different notation, different vocabulary, different techniques, and having different goals. [An exception to this is found in the excellent introductory book by Driver (1977) who presents the elementary theory of differential and finite delay equations in a unified manner.] When one moves from ordinary differential equations to functional differential equations it seems that one moves into a different world. One of the goals here is to show that many aspects of these worlds are so similar that one may move freely through them so that the distinctions between them are scarcely noticed.

But this is not merely a book on point of view. During the last three years there have been very significant advances in stability theory of various types of functional differential equations so that they can be brought into line with classical ordinary differential equations. There have also been dramatic advances in periodic theory. One may now discern unifying themes through very different types of problems which we attempt to expose in this book.

Thus, the second group of readers we address consists of those mathematicians, engineers, physicists, and other scientists who are interested in the broad theory of differential equations both from the point of view of applications and further investigations. Much of the material in this book is taken from preprints of papers soon to be published in research journals. It provides a synthesis of recent results.

And the third group of readers we address consists of those who have completed an introductory graduate course in ordinary differential equations and are interested in broadening their base, as opposed to exploring fewer concepts at great depth such as is accomplished by a study of the excellent book of Hartman (1964). We recommend a second course of study as follows:

(a) A quick reading of **Chapter 2** allows one to focus on a broad range of applications pointing out the need for a variety of types of differential equations.

(b)Follow (a) with a careful study of **Sections 1.2, 1.3**, and **1.4**, a quick look at **Sections 1.4.1** and **1.5**, and a careful study of **Section 1.6**.

(c) **Chapter 3** contains fundamental fixed point theory and existence theorems.

(d) Finish with the balance of **Chapter 4**.

The remainder of this overview is aimed at the seasoned investigator of some area of ordinary or functional differential equations. It should be helpful for any reader to frequently review it in order to maintain an overall perspective. For this is not a collection of results discovered about the subject, but rather it is an exposition of the cohesion of a very large area of study. Thus, in no manner is this a comprehensive treatment. With each topic we give references for full discussion which the reader can consult for something more than an introduction.

**0.1 Survey of Chapter 1**

**Section 1.1** presents the standard introductory theory concerning stability and structure of solution spaces of linear ordinary differential equations. The following properties emerge as central. It is interesting to see the easy transition from the elementary to the quite general theory.

(a) For the system

(0.1.1) x' = Ax

with *A* an *n* × *n* constant matrix, the solution through (*t*0, *x*0) can be written as

(0.1.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and every solution of

(0.1.3) x' = Ax + f(t)

can be expressed by the variation of parameters (VP) formula

(0.1.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) For the system

(0.1.5) x' = B(t)x

with *B* continuous, *n* × *n,* and *B* (*t* + *T*) = *B* (*t*) for all *t* and some *T* > 0, there is a periodic matrix *P*(*t*) and a constant matrix *J* such that a solution of **(0.1.5)** through (0,*x*0) can be expressed as

(0.1.6) x(t, 0, x0) = P(t)eJtx0;

and every solution of

(0.1.7) x' = B(t)x + f(t)

through (0, *x*0) can be expressed by the VP formula

(0.1.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(c) For the system

(0.1.9) x' = C(t)x

with *C* continuous and *n* × *n,* there is an *n* × *n* matrix *Z* (*t*) satisfying **(0.1.9)**, and a solution of **(0.1.9)** through (*t*0, *x*0) is written

(0.1.10) x(t, t0, x0) = Z(t)Z-1(t0)x0;

and a solution of

(0.1.11) x' = C(t)x + f(t)

is expressed by

(0.1.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Moreover, if *xp* is any solution of **(0.1.11)**, then any solution of **(0.1.11)** can be expressed as

(0.1.13) x(t, t0, x0) = Z(t)Z-1(t0) [x0 - xp(t0)] + xp(t).

In **Section 1.2** several periodic results are discussed. It is noted that if all solutions of **(0.1.1)** tend to zero as *t* -> ∞ and if *f* (*t* + *T*) = *f* (*t*) for all *t* and some *T* > 0, then all solutions of **(0.1.3)** approach

(0.1.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a *T*-periodic solution of **(0.1.3)**. And, in the same way, if all solutions of **(0.1.5)** tend to zero as *t* -> ∞, then all solutions of **(0.1.7)** approach

(0.1.15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a *T*-periodic solution of **(0.1.7)**.

In **Section 1.3** we examine a variety of Volterra integrodifferential equations and note a complete correspondence with the preceding.

(d) For the system

(0.1.16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with *A* constant and *n* × *n* and with *D* continuous and *n* × *n,* there is an *n* × *n* matrix *Z* (*t*) satisfying **(0.1.16)** on [0, ∞) such that each solution of **(0.1.16)** through (0, *x*0) can be expressed as

(0.1.17) x(t, 0, x0) = Z(t)x0;

and a solution of

(0.1.18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

though (0, *x*0) is expressed by the VP formula

(0.1.19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(e) For the system

(0.1.20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with *B* and *E* continuous, there is an *n* × *n* matrix *Z* (*t*) satisfying **(0.1.20)** and each solution through (0, *x*0) can be expressed as

(0.1.21) x(t, 0, x0) = Z(t)x0;

and if *xp* (*t*) is any solution of

(0.1.22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then every solution through (0, *x*0) can be written as

(0.1.23) x(t, 0, x0) = Z(t)[x0 - xp(0)] + xp(t).

**Equation (0.1.22)** also has a VP formula given by **(0.1.26)**, below.

In **Section 1.4** we note for **(0.1.18)** that if *D* and *Z* are *L***1**[0, ∞] and if *f* (*t* + *T*) = *f* (*t*), then every solution of **(0.1.18)** converges to

(0.1.24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a *T*-periodic solution of the *limiting equation*

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Considerable stability theory is introduced to verify *Z* [member of] *L***1** [0, ∞). There is also a subsection on the existence of periodic solutions when *Z* [not member of] *L***1** [0, ∞).

In **Section 1.5** we consider periodic theory for **(0.1.22)** with *B* (*t* + *T*) = *B* (*t*), *E* (*t* + *T, s* + *T*) = *E* (*t, s*), and *f* (*t* + *T*) = *f* (*t*). There is an *n* × *n* matrix *R* (*t, s*) satisfying

(0.1.25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so that a solution of **(0.1.22)** through (0, *x*0) can be expressed by the VP formula

(0.1.26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Moreover,

(0.1.27) R(t + T, z + T) = R(t, s).

If *R* (*t,* 0) -> 0 as *t* -> ∞ and if both ∫t0 |R(t, s)|ds and ∫t0 |E(t, s)|ds are bounded, then each solution of **(0.1.22)** approaches

(0.1.28) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

a *T*-periodic solution of the limiting equation

(0.1.29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The classical result of Perron for ordinary differential equations holds for Volterra equations also and allows one to verify that ∫t0 |R(t, s)|ds is bounded.

*Remark* 0.1.1. The solution spaces of the various equations are seen to be similar because of the variation of parameters formulas and especially by **(0.1.13)** and **(0.1.23)**. But the property which brings unity to the periodic theory for these linear systems is isolated in **(0.1.27)**; this property is valid for all kernels of the VP formulas in the periodic case. And in each of those cases we are able to consider *x* (*t* + *nT,* 0, *X*0) in the VP formula, change the variable of integration, and let *n* -> ∞ obtaining the integral representation for the periodic solution, all because of **(0.1.27)**.

*Remark* 0.1.2. In each case in which we conclude that a periodic solution exists there is a fundamental boundedness principle which can be deduced from the VP formula. It turns out that solutions are uniform bounded and uniform ultimate bounded for bound *B,* terms defined later. And this boundedness property yields periodic solutions for linear and nonlinear equations as well as for equations with finite and infinite delay. It is this boundedness property that unites the entire book by means of asymptotic fixed-point theorems. While the VP formula shows boundedness for linear systems, Liapunov's direct method is well suited to the nonlinear study. And it turns out that as we use the Liapunov theory to extend the periodic theory from the nonlinear ordinary differential equations to the delay equations, then the Liapunov theory itself extends to the delay equations in a very unified and cohesive fashion.

**Section 1.6** consists of an introduction to Liapunov's direct method for Volterra equations and construction of Liapunov functionals.

**0.2 Survey of Chapter 2**

In this chapter we look at a number of concrete applications ranging from ordinary differential equations models to models requiring functional differential equations with infinite delays. These include mechanical and electrical systems, biological problems, arms race models, and problems from economic theory.

**0.3 Survey of Chapter 3**

Many of the interesting problems in differential equations involve the use of fixed-point theory. [Cronin (1964) has a classical treatment of their uses.] And the fixed-point theorems frequently call for compact sets in the initial condition spaces which may be subsets of *R*** n**, subsets of continuous functions on closed bounded intervals into

*R*

**, or subsets of continuous functions from (-∞, 0] into**

*n**R*

**. Thus, one searches for a topology which will provide many compact sets, but is yet strong enough that solutions depend continuously on initial conditions in the new topology.**

*n*In **Section 3.1** we discuss compactness in metric spaces and a variety of examples. We also introduce the rudiments of Banach space theory. **Section 3.2** introduces contraction mappings, fixed-point theorems, asymptotic fixed-point theorems, and error bounds. **Section 3.3** applies the contraction mapping theory to obtain solutions of linear ordinary and Volterra differential equations.

In **Section 3.4** we discuss retracts, Brouwer's fixed-point theorem, and the sequence of results leading to the two common forms of Schauder's fixed-point theorem. We then discuss the asymptotic fixed-point theorems of Browder and Horn, as well as the Schauder-Tychonov theorem for locally convex spaces. These are the main results needed for existence of solutions and existence of periodic solutions of nonlinear ordinary and functional differential equations. **Section 3.5** contains existence theorems for nonlinear ordinary differential equations as well as functional differential equations with bounded or infinite delays.

**0.4 Survey of Chapter 4**

This chapter contains three main themes; limit sets, periodicity, and stability. **Section 4.1** is the basic introductory theory of nonlinear ordinary differential equations which, together with **Section 1.1**, constitutes a standard first graduate course in the subject. We introduce limit sets and arrive at the Poincaré-Bendixson theorem. This theorem concerns simple second-order problems, but its conclusion represents universal good advice: If we are interested in the behavior of bounded solutions, then we should concentrate on periodic solutions and equilibrium points.

**Theorem 0.4.1.** Poincaré-Bendixson *Consider the scalar equations*

(0.4.1) x' = P(x, y), y' = Q(x, y)

*with P and Q locally Lipschitz in (x, y). If there is a solution [] of***(0.4.1)***which is bounded for t* ≥ 0, *then either*

(a) **I** is periodic,

(b) * I* approaches a periodic solution, or

(c) **I** gets close to an equilibrium point infinitely often.

We are now ready to enlarge on **Remark 0.1.2**. Examine **(0.1.4)** with *f* bounded and all characteristic roots of *A* having negative real parts. It follows readily that

|eAt| ≤ Ke-αt

for *t* ≥ 0 and some positive constants *K* and α. One then easily shows that the following definition holds for **(0.1.3)** using **(0.1.4)**. In fact, the definition holds for all those linear systems for which we conclude that a periodic solution exists.

*(Continues...)*

Excerpted fromStability and Periodic Solutions of Ordinary and Functional Differential EquationsbyT. A. Burton. Copyright © 2005 T. A. Burton. Excerpted by permission of Dover Publications, Inc..

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