Principles of affinity in nonlinear problems

Journal o.[ Mathematwal S,'iem'e.s. Vol. 83. No. 4. 1997

P R I N C I P L E S OF A F F I N I T Y IN N O N L I N E A R P R O B L E M S N. A. Bobylev and M. A. Krasnosel'sky

UDC 517.988.63

Introduction When we investigate specific problems of nonlinear analysis, we often use equations that are different in principle. For instance, when seeking the conditions for oscillation of a nonlinear system, we either try to find the fixed points of the Poincar&-Andronov operator that acts in the phase space of the system or seek the solutions of different equivalent integral equations considered in different functional spaces. A similar situation may occur in the theory of boundary-value problems, namely, we can reduce the problem of solvability and an approximate construction of the solution of a boundary-value problem for a differential equation to the analysis of operator equations with different operators that possess special properties. Problems from the calculus of variations can also serve as an example. For instance, we can either carry out the investigation of a degenerate extremal of an integral functional (say, by the methods of field theory) in the space on which the functional is defined or reduce the problem to the investigation of the critical point of a function of a finite number of variables that is defined on the subspace of the degeneration. We can give more examples of this kind. Every approach to the specific problem in question has its advantages and drawbacks, and therefore the picture we obtain when investigating a problem becomes more complete if we manage to find internal connections between different approaches and use the positive aspects of each of them. The theory of affinity that studies the relationships between the topological characteristics of solutions of various equations encountered in the process of investigating a problem is one of the instruments for discovering these connections. Below we give general theorems that serve as the basis for the affinity theory and indicate some of its applications. The article is supplied with a bibliography.

1.

R o t a t i o n of a V e c t o r Field

1.1. C o m p l e t e l y c o n t i n u o u s v e c t o r fields. This section contains all the necessary information concerning the theory of rotation of vector fields. The fields that we consider below are either finite-dimensional or completely continuous. Since every finite-dimensional field is completely continuous, we only consider the properties of completely continuous fields. Suppose that E is a real Banach space and M C E is a set in it. A completely continuous vector field is a mapping r M ~ E which has the form r = I - A, where A: M , E is a completely continuous operator. The point x0 is a zero of the field r if ~(z0) = 0. If the field ~ has no zeros on M , then it is said to be nondegenerate on M. The completely continuous vector fields r = I - A0 and r = I - A1, defined on the set M C E, are homotopicon M if there exists a one-parameter family of nondegenerate fields r A) = I - A ( . , A)(1 < A < 1) such that the operator A: M x [0,1] , E is completely continuous and r = r ~(-, 1) = r In what follows, we denote by ~Po r the superposition of the fields ~ and r so that (~P o ~)(z) = ~P[r 1.2. Rotation and its p r o p e r t i e s . Let f~ C E be a bounded domain on whose boundary 0f~ a nondegenerate completely continuous vector field r is defined. Then the rotation 7(r Of~) of the field r on 0f~ is defined. The rotation 3'(~, 011) is an integer-valued topological characteristic possessing the following properties. i

ii

i

i

i i

i

ii

i

i

i

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. T,'maticheskie Obzory, Vol. 22, Dynamical Systems-3, 1995. 1072-3374/97/8304-0485 $18.00 9

Plenum Publishing Corporation

485

1. The rotation of the completely continuous vector fields r

and r

homotopic on Off, is the same:

(r 0fl)= ~(,,,0fl). 2. Suppose that fi~ C fi are pairwise nonintersecting subdomains and the field r is nondegenerate on \ Uifii. Then

oa,).

oa) = 3. If the field ~ is defined and nondegenerate on f~, then

~(r o n ) = 0 . 4. Let Zo E fi be an isolated zero of the field ~. The topological indez ind (zo, ~) of the zero xo is the rotation of the field r on the spheres S(r, zo) of small radii r > 0. If the field r is nondegenerate on OF/and has a finite number of zeros z 1 , . . . , z~ in fl, then Irl,

0fl) - I ;

ind (xi, ~).

i=1

5. Suppose that ~ is a completely continuous field defined and nondegenerate on 0f~ and 9 is a completely continuous field that is defined on E and has a single zero zo = 0. Then the completely continuous vector field ~ o ~ is nondegenerate on 0fl and

"7(~ o r

~ , ( ~ , 0 a ) ind (0, ~).

6. Suppose that Eo is a subspace of the space E and the completely continuous operator A: F/ ; E0 does not have any fixed points on Off. Let A0 be a restriction of the operator A to the closure of the domain flo = fi gl E0. Then the vector fields = I - A and ~o = I - Ao are defined, respectively, on F / a n d flo and the equality

~(~, 0 f l ) = ~(~o,0ao) holds (if no = o, then 7(~o, 0flo) = 0). 7. Suppose that Eo and Ex are real Banach spaces and r: Eo ; Ex is an isomorphism of Eo onto El. Let the completely continuous vector field r be defined and nondegenerate on the boundary 0flo of the bounded domain flo C Eo. Then the completely continuous vector field q~l - r o r o r -1 is nondegenerate on the boundary 0f~: of the domain fl: = r(flo) C E: and

~(~0, 0flo) - ~(~,, 0fl,). 8. Suppose that 1 is not an eigenvalue of the linear completely continuous operator A: E the zero Zo = 0 of the completely continuous vector field ~ = I - A is isolated and ind(0,~) =

, E. Then

(-i)~(a),

where fl(A) is the sum of multiplicities of the real eigenvalues of the operator A which are greater than 1. 9. Suppose that the space E is the direct sum of the subspaces Eo and E1 and Po: E ~ Eo, 1='1: E ~ El are projection operators for which Pox = 0 for z E E1 and P l z = 0 for z E Eo. Suppose, furthermore, that fl0 C Eo and fll C E1 are bounded domains in the subspaces Eo and E:. The product fl = flo x fll of the domains flo and f~l is a domain in E which ~:onsists of points of the form z = Zo + x l , Xo ~_ flo, z: E ill. The direct sum ~ = ~o @ ~1 of the fields ~o and ~1 defined, respectively, on fl and fil is the vector field

r

= ~o(Po~)+ ~,(P,~),

9 e ~o x fl, = h,

defined on ~. If the fields r and q~l are nondegenerate on 0flo and 0ill, respectively, then the field 9 = ~o@~1 is nondegenerate on c3fl and 7 ( r Off) - 7(r 0flo). ~(~,, 0fl,). 486

0

T h e P r o b l e m on Forced Oscillations of N o n l i n e a r S y s t e m s

2.1. T h e shift o p e r a t o r . Let us consider the differential equation dx

= f(t,

e

a ")

(2.1)

with a continuous right-hand side which is T-periodic with respect to t. We are interested in the problem of finding T-periodic solutions of Eq. (2.1), i.e., solutions z(t) such that z(t + T) ___-z(t) and, in particular, z(0) = z(T).

(2.2)

Let us assume that every initial condition z E RN defines the unique solution p(t,z) (p(O,z) = 0) of Eq. (2.1) that can be extended to the interval [0, T]. Then, in the phase space Rtr of Eq. (2.1), the operator

(2.3) uCz) = pCT, z) is defined, which is known as the Poincarg-Andronov operator or shift operator along the trajectories of Eq. (2.1). Under these conditions, the fixed points of the shift operator (and only these points) are the initial conditions for the T-periodic solutions of Eq. (2.1). Therefore, the search for T-periodic solutions of Eq. (2.1) reduces to the search for the zeros of the vector field ~o = I - U considered in the phase space ~N. 2.2. An e q u i v a l e n t i n t e g r a l e q u a t i o n . Another approach to the problem on T-periodic solutions of Eq. (2.1) consists in the reduction of this problem to finding the solutions of equivalent integral (or integrofunctional) equations. The following equation is the simplest one:

(2.4) 0

The solutions of Eq. (2.4) coincide with the T-periodic solutions of Eq. (2.1), or, what is the same, with the zeros of the vector field (I) = I - A, where A is an operator defined by the right-hand side of Eq. (2.4). Equation (2.4) and, together with it, the field (I), are studied, in the sequel, in the space C[O, T] of the vector functions z(t), with a uniform norm, continuous on [0, T]. In this space the field (I) is completely continuous. The essential advantages of the methods of investigation of the T-periodic problem for (2.1), based on the study of the vector field ~ = I - U with the operator (2.3), consist in the finite dimensionality of the problem, in the geometric visuality, and in the possibility of carrying out effective computations of the field qo. The drawback of the method is that the operator U, in explicit form, is, as a rule, unknown. It exists only under restrictive requirements placed on the right-hand side of Eq. (2.1). The advantages of the methods that use the integral equation (2.4) are that the corresponding operator A and the field (I) are defined by explicit formulas which do not need any assumptions of the uniqueness of the solutions of the Cauchy problem or of the local continui,y of the~. oolutions for their construction. But it is difficult to carry out a direct computation or estimation of the rotation of the corresponding fields in infinite-dimensional spaces, which is the main difficulty of these methods. The theory of affinity, which establishes the relationship between the rotations of the fields ~ and (I), makes it possible to compensate for the drawbacks of every one of the indicated methods by the advantages of some other method. Already the first applications of this theory have led to new theorems on the existence of periodic oscillations in distributed argument systems (and, in particular, in time-lag systems), to new effective tests for the convergence of various computation procedures in the problem of approximate construction of periodic solutions and in the investigation of a number of problems concerning systems with hysteresis nonlinearities, etc. 2.3. T h e f u n d a m e n t a l t h e o r e m . Now we shall give the main result. We say that the bounded domains G C n~N and fl C C[0, T] have the same core if the fields tp and are nondegenerate, respectively, on OG and 0fl and if the set of initial values of the T-periodic solutions of Eq. (2.1) that lie in G coincide with the set of values for t = 0 of all T-periodic solutions of this equation lying in f/. 487

Theorem

2.1.

Let the d o m a i n s G C R Iv and fl C C[O, T] have the same core. Then the relation

~(~, 0a) = ~(r 0a)

(2.5)

holds. Proof.

We set I

Im

(2.6)

\I

~E~, O