Fixed points of mapping of Banach manifolds and some applications

FIXED

POINTS

OF MAPPINGS OF BANACH SOME APPLICATIONS

MANIFOLDS

AND

G.rBo~~sovrC and Ju. E.‘GLIKLIH Voronezh State University. Voronezh. U.S.S.R. (Receid

4 February

1979)

INTRODUCTLON UP TO NOW a lot of more or less wide and typical classes of nonlinear operator equations have been investigated from the point of view of existence of solutions, extension of solution, bifurcations, etc. (equations with completely continuous, condensing, monotone, Fredholm operators, etc.). For determination of a solution there exist topological characteristics, which are rather analogous to Leray-Schauder degree. Equations of almost all of these classes are considered in vector (Banach) spaces, and the geometry of linear spaces is essentially exploited in the construction of topological characteristics. The only exception IS the degree theory of Fredholm mappings, which includes Fredholm mappings of Banach manifolds and therefore it needs more essential topological methods. (Naturally, we do not consider a great quantity of variants of the theory of contracting maps in metric spaces which have no topological substance). But in the fixed point problem in infinite dimensional spaces all of these theories assume the linearity of the spaces. During some recent years the authors have investigated several non-standard problems, in which the existence of solutions is connected with the fixed points of compact and non-compact maps in nonlinear spaces. For example, it is the existence problem of a periodical solution of various differential equations on smooth manifolds. In classical dynamical systems the following existence principle of periodical solutions for a non-autonomous differential equation on compact manifold M is known: if the right side of the equation is w-periodical and Euler-characteristic of the manifold M is not zero, then the equation has an o-periodical solution. But if the equation under consideration is a functional-differential one, the shift operator is defined and acts not on the manifold M, but on the functional manifold C ([0, 01, M) of the continuous curves on M, and is completely continuous (or condensing). The classical proof of the existence of the periodical solution is impossible in this situation. To investigate fixed points one must extend the notion of topological characteristic or Lefschetz number. Another example which leads to the recomprehension of the traditional methods is two-point boundary value problem for second order differential equations on a smooth manifold. Integral operators of a new type (compact and locally compact) arise in this problem. The investigation of these and some other problems has allowed to generalise Lefschetz number and Leray-Schauder theories to the nonlinear infinite dimensional case. In this paper both our investigations mentioned above and results of the other authors, which are close to the named problems, are considered. 165

166

Ju. G. BORISOVI~ mw

Ju. E. GLIKLIH

The paper consists of two chapters. The first chapter is a geometrical one and deals with the topological methods in various existence theorems. In the first section we give a brief survey of the fixed points theory in linear spaces. In 1.2 we cite the F. E. Browder’s idea of the homological construction of Lefshetz number for compact mappings (i.e. the image of the whole space is compact) of topological spaces, that can be imbedded in the Banach space as a neighbourhood retract [19]. But a lot of operators arising in investigations of various differential equations on nonlinear spaces are not compact (see Chapter 2), and it is impossible to apply Browder’s method to them. In the papers [Ill, 251 the authors have constructed Lefschetz number for more wide classes of mappings: locally compact with compact iteration, condensing etc. We also consider manifolds which can be imbedded in a Banach space as a neighbourhood retract. The basic method of our construction is a transition to an auxilliary vector field in embodying Banach space. The similar idea was exploited by Leray but for compact operators. It is shown that our invariant does not depend on a choice of a Banach space, on an imbedding, retraction and so on. In 1.3 we set a detailed presentation of our construction for the case of locally compact mappings with a compact iteration. The case of condensing mapping is to be mentioned especially. At present time it seems to be embarrasing to create the general theory of condensing maps in nonlinear spaces, because the definition of abstract measure of noncompactness [47] exploits the notion of the convex closure. We make use of the metrical definition of measure of noncompactness and consider condensing mappings of Finsler manifolds which can be isometrically imbedded in a Banach space as a neighbourhood retract. This construction is explained in 1.4 [25]. The same method of a transition to an auxiliary vector field in embodying Banach space permits a creation of an analogy of the rotation of vector field on nonlinear space. It is a homotopic characteristic that is equal to algebraic number of fixed points of the mapping inside the domain of the manifold. The homotopic characteristic is also defined for locally compact mappings with a compact iteration, condensing mappings, etc. The homotopic characteristic is introduced in 1.5 [13]. Section 1.6 is devoted to the algebraic number of eigen-vectors of a nonlinear operator. This topological characteristic is constructed by V. G. Zvjagin [50, 511 and is similar to the rotation of vector field: it is a homotopic invariant; its being not equal to zero causes the existance of an eigenvector in the domain of Banach space. In the finite-dimensional case this invariant is defined as an obstruction for extension of a pair of noncollinear vector fields from the boundary onto the domain of the manifold: in the infinite dimensional case this construction is combined with finitedimensional approximations. The fact that this invariant is independent on the choice of approximation is not trivial and is proved by a beautiful construction. The second chapter is devoted to the applications of the results of the first chapter to the investigation of various differential equations on smooth finite-dimensional manifolds. In 2.1 we study the existence problem of w-periodical solutions for cu-periodical functionaldifferential equations on compact manifolds. The investigation is carried out by means of the shift operator which is generally speaking non-compact. As in [25] we construct the measure of noncompactness (in metrical terms) on the corresponding functional manifold with respect to which the shift operator is a condensing one, and show that the Lefschetz number for the shift operator is well-defined. This Lefschetz number is shown to be equal to Euler characteristics of the manifold where the equation is defined. And a simple corollary is obtained such that if the Euler characteristic of a manifold is not equal to zero, every periodic functional-differential equation of the class investigated has a periodic solution.

Fixed points

of mappings

of Banach

manifolds

and some applications

167

In 2.2. the operators of integral type intended for studying of global behavior of solutions of various differential equations on Riemannian manifolds are constructed [23, 24, 26-J. Their construction is based on the use of the Riemannian parallel translation. The operators are analogous to classical integral operators, but the specific character of non-linear situation causes the difference of their properties from the classical ones. For example these operators are defined not on continuous but only on smooth curves and that is why they are not completely continuous. Applications to periodical and two-point boundary value problem are given. In 2.3 the equations of the type ti(t)llt(t - h, m(t - h)) are considered, where [(t, m) is a continuous vector field on a Riemannian manifold and the sign /I means the parallelism along the trajectory. Such equations arise in some mechanical problems. They are analogous to classical equations with discrete delay but at the same time they are equations of the neutral type because parallel transport is defined only along smooth curves and depends on the derivative of the curve. The periodical problem is investigated provided that the field {(t, m) is o-periodical one. The case when the delay h is equal to the period w is of a special interest. If the equation is defined on a Euclidean space, the periodical solutions of such an equation are periodical integral curves of the field ((t, m). Here the solutions are the so called pseudointegral curves [13, 14, 151, i.e. periodical curves r(t) such that the vector {(t - o, y(t - u)) = c(t, r(t)) of the field, after the parallel translation along y into the point y(t) = ~(t - o) becomes equal to the tangent vector *i)(t). On the basis of operators of integral type constructed in 2.2 we construct the operator such that its fixed points (and only they) are pseudointegral curves. This operator acts on the set of Clcurves, is locally compact and have the second compact operation on the set of Cl-curves which are bounded in CO-metric. The last property permits to calculate its Lefschetz number and homotopic characteristic by means of theorems of relatedness. We show that the pseudointegral curves exist for every continuous vector field on the manifold with non-zero Euler characteristic, that the pseudointegral curves arise near the structurally stable cycle of an autonomous vector field under the perturbation of flat Riemannian metric, and near the structurally stable singular point of the field under the perturbation of the field by non-autonomous periodical addition. In 2.4 the operators for investigation of the periodical problem for the equation riz(t)II x ((t - h, m(t - h)) in general case are constructed. If h -c w the operator is locally compact, if h > co, it is locally condensing. It is possible to define the Lefschetz number or the homotopic characteristic for these operators too. In 2.5. the operator for determination of pseudointegral curves of functionaldifferential equations is described. It acts on the special functional manifold and locally condenses. CHAPTER].

FIXED

POINTS

OF

MAPPINGS

OF

BANACH

MANIFOLDS

1.1. Preliminaries 1. In this section we give rather a short survey of the theory of rotation of compact and close to them vector fields in linear spaces. It is impossible to extend the methods of this theory onto mappings of nonlinear spaces directly, but the investigation of nonlinear case finally will be reduced to the theory of vector fields. Let E be a Banach space, D c E be an open set and F: i7 + E be a continuous mapping with a compact image. Such mappings and corresponding vector fields 4 = I - F we shall call completely continuous. Zeros of the mappings 4 are called singular points of the vector field 4; obviously they and only they are fixed points of F. ConsiderV a finite s-net of the set F(n) and its linear span E”.There exists a natural s-shift of the compact F(Q) into E, which is called Schauder projector. We denote it by P. The finite-dimensional

168

Ju. G. BOKISOVIC AND Ju. E. GLIKLIH

approximation PF:a + E of the operator F defines the finite-dimensionai vector field 4 = I - PF:anE-+E". If4:fi --, E\Oand 8 is sufficiently small, then $:(Q n _i?)' --f B\O. Kronecker characteristic of the mapping 6 is called the rotation (in the sense of M. A. Krasnoselskii) of the vector field 4; it is denoted by ~(6, b). The detailed description of the construction and the proof that the rotation is well-defined are contained for example in [31,33]. The rotation ~(4, Q) is a homotopic characteristic of the vector field 4, because the rotation remains constant under the completely continuous homotopy F* :Clx [0, 11 -+E of the operator such that +* = I - F*:hx[O, l] -+ &\O. If ~(4, h) # 0 then there exists a fixed point of the operator F in R: the rotation y additively depends on the domain Q If x* E Q is isolated fixed point, then r(~$,s,)--the rotation on spheres of radius 6 with the centre x*-is constant at small 6’s and is called the index of the fixed (singular) point x *. If there is finite number of singular points in Q, then the rotation is equal to the sum of their indices. It is possible to show that from the topological point of view the rotation of M. A. Krasnoselskii is equivalent to Leray-Schauder degree. This construction can be extended to the case of locally-convex space [38]. In modern topological theory of nonlinear mappings there is rather a useful notion of the relative rotation of a completely continuous vector field [4,9]. Let t c E be a convex closed set, L n Q # @ and F:a n L -+L be a completely continuous mapping such that C$= I - F: (Cl n L)'-+E\O,where (Q n L)'is relative boundary in L.The relative rotation ~(4, (Q n L)', L) of the vector field 4 relative to the sub-space L is defined via the method of finite dimensional approximations or using the total index of solutions in the sense of Leray [37]. It has the same properties like ‘absolute’ rotation. 2. One can realize a transition to non-compact mappings via two ways: to construct ‘good’ spectrum of finite dimensional approximations or to find an invariant convex subspace. Both approaches are useful in modern investigations (monotone and condensing mappings). For various classes of condensing mappings the second method is more preferable. It is possible to formulate a general principle of existence of an invariant subspace [ 161. Let X be a topological space, 2’ be the totality of all closed subsets of X with the exponential topology. If X were a metric space, one should consider Hausdorff metrics p(A, B) in 2’. Let E be a linear real topological space with the cone K of non-negative elements [32]. A totality {$,} of closed subsets 4i E 2’ we shall call &system if two conditions hold: (a) a nonempty intersection of any totality of sets &Abelongs to &system; (b) for every A E 2” there exists 4A 3 A. A mapping x: 2x --f K is called a distinguishing one if the following axioms hold :

(1) x(A)d x(B) if A c B (2) x(Au R) d x(A) ifX(R) = 0 (3) for every A = 2’, x(+,Qq 4,) = X(A) (4) x(X) = 0 for every point x E X It follows from 1 and 2 that (2’)

x(Au R) = x(A) ifX(R) = 0

Let us denote by Ker x the totality guishing mapping). Let R c X be an open set.

of sets A E 2’ such that

x(A)= 0 (the kernel of the distin-

Fixed points

of mappings

of Banach

manifolds

A mapping F:Q + X is called the coordinated with the distinguishing mapping 1. Ker x is invariant with respect to F, i.e. x(m) = 0 if ~$4) = 0 for A c R; 2. for every A c Q x(m) + X(A) if x(A) # 0 Now we formulate the fundamental property of a coordinated with x operator. THEOREM 1 [16]. Every coordinated

with x mapping

169

and some applications

F:Q -+ X has a set 4 E (4,)

x if:

such that:

1. F(4 n a) c 4

3. C$1 R where R is a chosen set from Ker x. The set of fixed points of F belongs to Ker x, so (Theorem 1) we can construct a set 4 including all the fixed points of F. Below we give a brief description of the rotation of vector fields with condensing operator F via this way. Let x be a Banach space. If {#,} are convex closed sets and Ker x is a totality of compact sets then x is a measure of noncompactness under the definition of [47]. Mappings coordinated with a measure of noncompactness, are called condensing ones. Let a be a closure of an open set in X. For a condensing mapping F:Q + X there exists a compact convex set C$such that 0 n C$ # @, fl n 4 includes all fixed points of F and F(Q r? 4) c 4. So it is possible to define the rotation of vector field I - F on Q [47] as the relative rotation of F on h. The value of the rotation doesn’t depend on the choice of C$if 4 is rather ‘large’ [45]. Ju. I. Sapronov has shown in [44] that there exists a set C#Iwith the property of fundamentality [35]. From the Dugundji theorem a convex set is a retract of the whole space, so one can homotopy the mapping F on Q to the mapping with values in 4. This homotopy is condensing and has no fixed points on the boundary because C#J is fundamental. Thus there exists a completely continuous mapping in every homotopic class of a condensing mappings. One can consider this construction as a simpler method for the definition of a rotation. See details in [44]. It should be pointed out that the operators which decrease a measure of nonconvexity [21], are also included in the scheme of mappings which are coordinated with a distinguishing mapping. 3. It is impossible to apply the previous notions directly for determination of fixed points of mappings of nonlinear spaces. First of all, the specific character of nonlinear space does not allow to pass from the mapping to the correspondent vector field, i.e. the problems of determination of zeros of a vector field and fixed points of a mapping are essentially different. Theoretically it is possible to construct topological invariants for fixed points estimation using Theorem 1. For example if one could construct a &system and a distinguishing mapping on nonlinear manifold X such that Ker x consisted of compact ANR, then for a coordinated with x mapping F: X + X one should be able to define Lefschetz number A, as a Lefschetz number of the corresponding restriction. But the problem of constructing of appropriate +-systems (for example, substituting convex sets) in nonlinear spaces is an important unsolved problem even now. 1.2. Lefschetz number of F. E. Browder for compact mappings As it has been said the fixed points problem in nonlinear spaces cannot be reduced to the existence problem of zeros of vector fields, i.e. one cannot use directly the degree theory here. Another way for this problem is to construct topological invariants of Lefschetz number type. This invariant is constructed by F. E. Browder [19] for compact mappings of spaces which can

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Ju. G. BORISOVICAND Ju. E. GLIKLIH

be imbedded in a Banach space as a neighbourhood retract. The construction is based on homological methods. Here we describe it briefly. Let X be a topological space and H,(X) its s-dimensional homology group with the coeflicients in the field Q of rational numbers. Letf : X -+ Y be a continuous mapping and f,,: H,(X) --f H,( Yjthe induced homomorphism of homology groups. H,(X) and H,(Y) are linear spaces, f,, is linear mapping. We shall say that a linear mapping h: I/ + I’ (V is a linear space of possibly infinite dimension) has the trace if the image h(V) has a finite dimension. In this case the trace of h, tr(k), is defined to be equal to the trace of h restricted to any subspace of a finite dimension which contains h(V) (cf. also [37]). If f: X --f X is a continuous mapping, the Lefschetz number A(j) off is said to exist if f,, = 0 except for a finite number of S 3 0, while f,, has a trace for each s 3 0. Then the Lefschetz number is defined by the usual formula.

WI

= f (- 1)”WJ s=o

A space Y is said to be a Lefschetz space if for each continuous self-mapping f of E: A(f) is defined and if A(f) # 0 then f has a fixed point. The examples of Lefschetz spaces are the following ones: compact metric absolute neighbourhood retracts, convexoid spaces, a finite union of compact subsets of a Banach space. A mapping f : X + X is called compact if f(X) is compact. 2 [ 191. Let f : X -+ X be compact mapping and X can be imbedded E as a neighbourhood retract. Then A(,f) is well-defined.

THEOREM

in a Banach

space

Idea of proof. Let Y: U + X be a retraction of an open neighbourhood U of X in E onto X. We define a mapping F: U -+ U by F = f o r. The compact set F(U) can be covered by a finite number ofconvex sets N,in U. Let us denote by Ki the convex closure of the compact set Ni n f(X). Obviously Ki is convex and compact and f(X) = F(U) c Y = v Ki. Y is Lefschetz space, and it is possible to show that all spaces H,(Y) are finite dimensional while H,(Y) # 0 for finite number of indices only. It is possible to consider F as the superposition of the mapping of U in Y and of the imbedding, so all F,, have a trace and for F: U + U Lefschetz number is welldefined. One can represent f as the superposition of the imbedding and F therefore all f,, have a finite dimensional image and f,, = 0 for all s except for a finite number if s, i.e. A(f) is welldefined. q.e.d. It is not difficult to show that tr(f,,) = tr(FJ for all s and therefore A(f) = A(F). If A(f) # 0 there exists a fixed point of f in X. If X is contractible (or acyclic), A(f)

= 1.

1.3. Locally compact mappings. Lejhcket: number In this section we consider a more wide class of mappings that is locally compact mappings which have a compact iteration (spaces on which the mappings act, are the same as in the previous section). Here we construct Lefschetz number as in [ll] making use of the transition to the auxiliary vector field in embodying Banach space. The Lefschetz number has usual properties:

Fixed points of mappings of Banach manifolds and some applications

171

in particular its non-equality to zero is a sufficient condition of fixed point existence, for contractible spaces the Lefschetz number is equal to 1. We should point out that some concrete sufhcient conditions of fixed points existence for this class of mappings have been obtained by F. E. Browder in [19]. In particular the existence of a fixed point is proved there for some concrete contractible spaces. Let X be a topological space which can be imbedded in a Banach space as a neighbourhood retract, f : X --f X be a continuous locally compact mapping and also f “X is compact at certain n 2 1. Let us imbed X in a Banach space E and let r: U -+ X be a retraction of an open neighbourhood U of the set X in E onto X. We define the mapping F: U -+ U by the following formula F = f0 r. By the definition F is continuous and locally compact, while F”(U) c X is compact. Moreover all fixed points of F belong to F”(U). It follows from the compactness of F”(U) that there exists a finite numbar of open sets 0, Y in U such that R = u fl, 1 F”(U) and F(n,) is compact for each c(. Hence F(Q) is also compact. It should be remarked that there are no fixed points on the boundary of Q, thus we may consider the rotation y(Z - F, fl) of completely continuous vector field I - F on the boundary h (For locally compact mapping cf. also [ZO]). Definition 1. The number

~(1 - F, fi) is said to be the Lefschetz

number

A,, of the mappingf

on

X. Let us show that the Lefschetz LEMMA 1. A, does not depend Proof. Consider

number

is well-defined.

on the choice of U and r.

in E two neighbourhoods

U, and

U, and retractions

r1 : U, + X, r2: U, +

X. Let U = U, n U,, !2, be c-neighbourhood of f”X in E, U 3 R, such that both F,(QJ = f 0 r1 x (Cl,) and F2(Qe) = fOr,(!S,) are compact. There exists a number 6 > 0 such that we have (IFlU - F,UII < E at u E !2,, i.e. the interval y of a straight line between Flu and F,u belongs to Q2,.After the retraction of y onto X we obtain a curve in X which joins Flu and F,u. It is easy to

show that the obtained homotopy of F, and F, is compact on Q, x [0, l] and has no fixed points on the boundary h,. Thus the rotations of vector fields I - F, and I - F, on h, are equal, q.e.d. LEMMA 2. fIf does not depend

on the choice of E and imbedding.

ProoJ: Let E,, E, be Banach spaces; i,:X + E,, i,:X --) E, be imbeddings; r,:U, + i,X, r1 : U, + i,X be retraction of corresponding neighbourhoods on X; F, : U, + i,X F, : U, --t i,X lifts of J Let us consider the Banach space E = E, 0 E, with the norm lie11= are natural projections of E onto Ei) and the natural imbedding of X in E llPlelll + i = (iI, iJ. Obviously iX is r-retract of U = U, 0 U, where r = (rl, r2). The mapping F: U + X, induced by the tetraction r, is defined on U. Let a, and Q2 be the neighbourhoods of the set f”(X) in U, and U, which define the Lefschetz numbers A,f and A, f as the rotations y(Z - F,, h,) and y(l - F,, !b,). It is sufficient to show that these numbers are equal. We shall show that both are equal to ~(1 - F, h2, where Q = f12,0 Q,. Let us consider the homeomorphism q(u) = u - i, 0 i- ’ 0 r(u) of U onto v(U). The image q(U) is retracted onto i,X with the same libres as U onto ix. We should point out that q(U) n E, = U,, #2) n E, = 0,. The mapping cp induces the mapping F*:(p(U) -+ q(U) by the rule be corresponding

IIP2el12 (Pi

172

Ju. G. BORISOVI? ANDJu. E. GUKLIH

F*(U - i, 0 i-i 0 r(u)) = F(u) - i, o i- ’ o rF(u). Hence ~(1 - F, 01 = y(Z - F*, (p(h)). The image of F* belongs to i, X thus we have ~(1 - F*, cp(Q)) = y(I - F*, Q,) by the theorem on the product of rotations [31, p. 1341. But F* is equal to P, on a,, therefore ~(1 - F*,cp(h,)) = y(Z - F,, h,) and ~(1 - F, ti) = ~(1 - F,, a,). In complete analogy with this result we may prove that y(l - F, h) = ~(1 - F,, h,) q.e.d. Let f have only isolated fixed points on X (it follows from the compactness off” that there is a finite number of fixed points in this case). The general properties of the rotation given the following preposition :

THEOREM3. Let f have only isolated a fixed point xi. Then A,, = 1 ji. I

fixed points on X,j, be the index of F in a neighbourhood

Let us clarify the relation of the introduced If X is a compact manifold, the classical

notions and the common known ones. Lefschetz number A; is well-defined.

THEOREM 4. Let X be a compact

Then A2, = AT.

manifold.

of

Prook It is possible to approximate the mapping f : X -+ X by a smooth mapping with isolated fixed points such that the smooth mapping is homotopic to f: The numbers A and A* are constant under the homotopy, thus without loss of generality it is possible to assume f as a smooth mapping. Then A,: coincides with the sum of indices with respect to X of fixed points xi. On the other hand, A,r is equal to the sum of indices ji of F in xi, so we make use of the theorem on the product of rotation and restrict F onto the maps of the manifold X which include the points xi. Hence by the Theorem 3 we have Al = AT. If .f : X + X is a compact (see 1.2).

mapping,

THEOREM 5. If ,f : X --f X is compact

the Lefschetz number

mapping,

A(j’) of F. E. Browder

is well-defined

A, = A(.f).

Proof: In the paper [19] (see 1.2) A(S) and A(F) are defined and it is shown that A(f) = A(F). Let E” be an approximating space, P,: F(U) -+ E” be Schauder projector. Since P,F is homotopic to F, A(F) = A(P,F). Let us cover the image P,F(U) by a finite number of convex closed sets in E”, their union Y is a convexoid space in the sense of Leray [36]. A(P,F) is equal to the ordinary Lefschetz number A;,F,u (see [19]). In the papers [6,9] it is shown that APmF,,, = y(Z - P,F, (U n E”)‘). Therefore

A,, = ~(1 - F, ii) = A(f).

q.e.d.

Now we shall study the properties of the Lefschetz number. Definition 2. We shall say that f. is homotopic to f,(& _ f,) if there exists a mapping w h.ICh’is continuous, locally compact, has a compact Ith iteration, (1 > max(n,, [O,l]-*X cplxxioi = f o>

401XXII!

=

.fl.

cp: X x n,)) and

173

Fixed points of mappings of Banach manifolds and some applications THEOREM 6. If f,

- f,, A.,<]= Af,.

Proof: Consider the compact set ‘p’(X x [0, l]), cover it by a finite family of neighbourhoods Q, such that for every CIthe image cp(Q, x [0, 11) is compact. Let Q be the union of the neighbourhoods. Then+ = qQr:r-‘(Qx[O,l])-+r-l X is a homotopy of F, and F, without fixed points on the boundary fi. So it follows from the properties of the rotation that A,,0 = A/,. THEOREM 7. If A,, # 0, f has a fixed point.

Proof Since ~(1 - F, fi) # 0 there exists a fixed point X* of the mapping tion x* E X, so fx* = x*. THEOREM 8. If X is contractible,

F. By the construc-

f has a fixed point in X.

ProoJ: Let ‘pr be a contraction, i.e., q,(X) is continuous on X x [O, 1) and (pO = id, q,(X) = X* E X. The image of X x [0, 1) under the mapping ‘p!0j” is compact. Let R be a neighbourhood of the image, such that f(sl) is compact. Obviously the homotopy qt o f has no fixed points on !$ and the index of the only fixed point X* of the mapping vi 0 f is equal to 1. So f has a fixed point by the Theorem 7. THEOREM 9. (Restriction

theorem). Let us suppose in addition, that X is a smooth Banach manifold. Let the image j:X --+ X belong to a closed submanifold M c X. Then A, = ASIM(flM is the restriction off on M).

Proof Let 0 be a tubular neighbourhood of M in X. It is common knowledge that M is a deformation retract of X. Let the deformation be denoted by r,: 0 x [0, l] -+ 0, i.e. rt is a homotopy, r,, = id, rl is a retraction. Then we denote by rx a retraction rx: U, + X of an open neighbour hood of X in E, and consider the open neighbourhood V, = ril 0 of M in E with the retraction rM = rI o rx. We have the mappings: F,: U, + ZJ,, F,(u) = f 0 r,.&) and F,,,,: U, + V,, FJu) = f o r,Ju). Let Q =f”(X), R = U,, be an open set such that F(n) is compact. It is easy to see that the homotopy F,u = f o rr o TX(u)satisfies definition 2, has no fixed points on !$ F, = F,, F, = F,. q.e.d. 1.4. Condensing and locally condensing mappings, Lejkhetz number This section is devoted to the construction of Lefschetz number for condensing and locally condensing mappings. There is extensive literature devoted to these mappings in linear spaces (see, for example, the bibliography in [47]). The transference of general theory of condensing mappings to the case of nonlinear spaces seems to be embarrassing now because the definition of abstract measure of noncompactness is based on the notion of convex closure. Maybe it would be possible to define analogies of abstract measures of noncompactness with the help of the notions &system and distinguishing mapping [16] (see 1.1). We make use of measures of noncompactness defined in metrical terms. As in the previous section we shall construct the Lefschetz number via a transition to an auxiliary vector field [25]. Let X be a Finsler manifold which can be isometrically imbedded in a Banach space E as a neighbourhood retract. Finsler metric defines the distance p on X as the intimum of lengths of curves which connect points. So X is a metric space. Definition 3. The Hausdorfl has a finite &-net in X>.

measure

of noncompactness

of a set SL is a number

x(Q) = inf(@

Ju. G. BORISOVI? ANDJu. E. GLIKLIH

174

Definition 4. The Kuratowski measure of noncompactness of a set Q is a number CC(Q)= inf{dl one can decompose the set R in a finite number of subsets with diameters less than d). x and a have the important property: they are equal to zero if and only if R is compact. If fir = Q,> x(fi 1) < x(Q,) and a@,) < a(Q,). Later in this section the words ‘the measure of noncompactness @ mean x or a. Definition 5. A continuous operator f :X + X condenses with respect to $ ($-condenses) the constant q < 1 if the inequality I(l(fQ) < q $(Cl) holds for every bounded set Q c X.

with

DeJinition 6. A continuous operator f : X + X is called the locally condensing one with respect to 9 (locally $-condensing) if for every point x E X there exists a neighbourhood U such that for every set Q c U the inequality $(fQ) < $(Q) holds. Let f$-condense with the constant q < 1 and X has a finite diameter. Let us consider the set

_f”X = \

f”X (fk is kth iteration off).

k=l

LEMMA 3. f”X

is

compact.

Proof. In the contrary, let $cf”X) = a > 0. We can choose a number k such that qk$(X) < u. So $(fkX) < $(f”X) that contradicts to the inclusion f”X c f”X. q.e.d.

It should be pointed out that f”X includes all fixed points off: Let us imbed X isometrically in a Banach space E. and define Hausdorff and Kuratowski measures of noncompactness in E via the natural metric pE, which is induced by the norm in E. Let I/J”be the measure of noncompactness in E, corresponding to tj. Generally speaking the operator f on X can be noncondensing with respect to ijE, but the following statement is true. LEMMA 4. For every point x E X there exists a neighbourhood Q c V, one has $E(fQ) < II/E(Q),i.e. f is locally +E-condensing.

V, c X such that for every set

This statement follows from the fact that in a small neighbourhood of a point, the metrics p and pE differ by an infinitesimal of the order more than 1. Let r: U + X be a retraction of an open neighbourhood U of the manifold X in E onto X. Let us define the operator F: U -+ X by the equality F = f 0 r. It is easy to see that F is continuous and locally II/” condensing, the set f’“X includes all fixed points of F. As in the previous section we can construct an open neighbourhood $2 of the compact set f”X such that F is t+!s condensing on Q and there are no fixed points of F on the boundary h. Definition 7. The rotation fi is called the Lefschetz

y(l - Fb)

number

of the condensing

A/ of the operator

LEMMA 5. Ar does not depend on the choice space E and an isometrical imbedding.

vector

field I -

f on the manifold

of a neighbourhood

F on the boundary

X.

U and a retraction,

a Banach

Here the arguments are analogous to the proofs of Lemmas 1 and 2. The reader can easily find a simple modification of the proof by himself. As usual from the properties of the rotation one can obtain the following statements.

Fixed points of mappings of Banach manifolds and some applications THEOREM

175

10. If A, # 0, the operator f has a fixed point in X.

THEOREM 11. If f, and fr are homotopic in the class of condensing with a constant q < 1 operators, then A,IO= Afl.

It is obvious that for a compact mapping f As is the same as in the previous section. The construction above allows to define Lefschetz number for locally condensing mappings with compact iteration. Indeed, instead of f”X we shall consider the compact set f”X. Since f is locally condensing, F is locally $E-condensing and condensing on a neighbourhood of f”X in E. 1.5. Homotopic characteristic The method of 1.3,1.4 permits to construct also an analogy of the rotation of vector field [13]. Let as in 1.3 X be a topological space, which can be imbedded in a Banach space E as a neighbourhood retract, 0 c X be an open domain. Let f:8 --) X be a continuous locally compact mapping such that j”“(G) is compact for certain n 2 1, and there are no fixed points off on the boundary 6. Let U be a neighbourhood of X in E, r: U --f X be a retraction: we obtain the mapping F by the equality F = f o r. Consider the set f”(o) n 0, it is compact, and it follows from the locally compactness of f that there exists a neighbourhood Q 2 f”(B) n a in U such that f(G) is compact. Naturally there are no fixed points on the boundary h. Without loss of generality we can assume that rI2 c 8. Definition 8. The rotation

y(Z - F, h) of the vector field I - F on the boundary h is called the homotopic characteristics S(f, 8) of the mapping f‘ on 8. If j”(G) n 0 = 0, we put by the definiton that S(f, 8) = 0. The following proposition

is a simple reformulation

of Lemmas 1 and 2.

does not depend on the choice of U and a retraction, E and an imbedding. Certainly, if X is a Banach space and f is a completely continuous operator, the homotopic characteristics off on 0 is equivalent to the classical rotation of vector field y(l - f, 0). This fact follows immediately from the theorem on the product of rotations [31]. Analogically if there are only isolated fixed points in 0, S(f; G) is equal to the sum of their indices. In the case under consideration the following delintion of homotopy is natural. LEMMA 6. S(f,8)

+ X and fi:o + X-are homotopic, if they are homotopic in the case of the definition 2 and for each t E [0,11,x E 0 there is cp(t,x) # x.

Definition 9. Mappings &:a

THEOREM 6.

q

If f0 N f,, S(f,, a) = S(f,, ai).

The proof is in complete analogy with the proof of Theorem 6. Let us formulate a proposition which can be proved like the Theorem 9. THEOREM 9. Let X be in addition a smooth manifold and imagef(@ fold M. Then S(f, @) = S(&, M,a n M).

belong to a closed submani-

176

Ju. G. Bomowi:

AND Ju. E. GLIKLIH

With the help of the analogous scheme of arguments it is easy to define the homotopic characteristics for mappings if Finsler manifolds, which are condensing with q < 1 on the closure of an open domain, and for locally condensing but having a compact iteration (see 1.4.). We leave the construction for a reader as a simple exercise. Remark. It seems to be natural to define the homotopic characteristic (Lefschetz number) via a construction of finite-dimensional approximations and the next transition to the limit. If X” is a finite-dimensional manifold, it is possible to construct homotopic characteristics via the homotopy theory [22,17]. Let U be a closed domain which is homeomorphic to the disk of the model space of the manifold X and S be its boundary. The homotopic characteristic is defined as an obstruction for extension off from S onto D without fixed points. This obstruction is an element of the relative homotopic group rc,(X”, X”\pt) = Z. In just the same way it is possible to consider the obstruction to the extension of a vector field from S onto U without zeros. This obstruction is an element of the group rc,_ ,(R”\pt) = Z and is a direct analogy of the rotation of vector field. But in the infinite dimensional case there are difficulties both in the construction of finite dimensional approximation and in the demonstration of the independence of the characteristic on the choice of the approximation. It leads to a sharp narrowing of admissible classes of mappings and manifolds. Nevertheless the approximation scheme above is realized in the problem of eigenvectors (see the next section). 1.6. Obstructions for extension of cross-sections and eigenvectors of nonlinear operators In this section we describe the construction of eigenvector index belonging to V. G. Zvjagin [50,51]. This index is constructed via homotopical methods and doesn’t reduce to the rotation of vector field. 1. Let Q be a domain of n-dimensional smooth oriented manifold. M such that K is CW-complex with the only cell of the highest dimension. Let (g, f) be a pair of vector fields on 0 which are linear independent on the boundary 0. Definition 10. The point x0 E 0 is named the singular one if there exists a real number one of the equalities f(x,) = Ag(x,) or Af(x,) = g(x,) holds.

A such that

Later we shall assume that n > 3. Consider the trivial fiber bundle 5 = (si x F,, ?. a, F,,,,, p) where Fn 2 is a set of pairs of linear independent vectors in the linear space E”. So S(s) = (x: g(x), .f(x)) is a cross-section of 5 over fi. We denote by 1~:B” -+ a the characteristical mapping of the closed n-ball B into the cell of the highest dimension. Since %\b”: S”- ’ + 52 the mapping (g, f)o+:S”-’ + Fn,2. equivalent to the Stiefel manifold The space F,, 2 is homotopically each integer m if n > 3. Thus the following definition is correct. Definition 11. The homotopical

class of the mapping

C,,, .$Y of the pair (%f). Ccg.,,,(a) is an element of the homotopy

(g, f) 0 xl,+: S”-

Vn z which is m-simple

’ ---f F,, 2 is named the index

group.

Definition 12. We shall say that (go, &) is homotopic

to (gr, fr) onh

for

if there exists a continuous

Fixed points of mappings of Banach manifolds and some applications

mapping

177

(h, k):!A x [0, l] -+ F,, 2 such that h(x, 0) = g,(x), h(x, 1) = g,(x), k(x, 0) = f,(x), k(x, 1)

= f,(x). Obviously

(g,,, &) is homotopic

to (gi, f,) if and only if C(,,,,,J@

= Cc41,,,,,(h).

THEOREM 12. If C,,,,,(h) # 0, there is a singular point of the pair (g, f‘) in R. If there are only isolated singular points in Q C(,, ,f)(Q) is equal to the sum of their indices.

The index is well-defined on the boundary of a multiply connected domain. See details of the construction in [50]. 2. Consider the following case: X = R”, g(x) = x, 0 4 a. Here a vector field is equivalent to the mapping f :W -+R” and a singular point x0 of the pair (I, f‘) is defined by the formula f‘(xJ = Ax,, i.e. f has an eigenvector. Later the index Ccr,S, (iz) will be named eigenindex of the mapping $ It is not difficult to show that the pair (I, f) on iz is homotopic to the pair (gi, f,) such that to e and i\f;(x)ll = 1. g,(x) = PER”, /lejI = 1, .f;( x 1 IS orthohonal Thus we obtain the mapping ,f, 0 x: s”- ’ -+ s” -’ which defines the elment of the homotopy group r~_~(s”-~) = Z2 (since n > 3) denoted by j,f; 1.. We should remind that nn_1(I;,2) = Z x Z, for even y1and TL,_i(V,. J = Z, for odd n. THEOREM 13. If n is even, C,,,,f, (h) = (0, ifi>).

If n is odd C,,,,f,(h) = {S, >.

3. Let E be an infinite dimensional real Banach space, !A be an open ball in E, F:a + E be a completely continuous mapping. Consider a finite s-net in the compact set F(Q). The linear span of the a-net is the finite dimensional subspace E” in E. We assume that 0 $0. (In general case one should also demand for a n E” to be CW-complex with the only cell ofthe highest dimension n and also the restriction of the characteristic mapping 31, of the cell a n E” on the ball Bk c Ek must coincide with the characteristic mapping xk of the cell Q n Ek for every n, k, n > k. The construction is true for this case too.). Consider the Schauder projector P,: F(a) + E”. LEMMA 7. Let F have no eigenvectors

has no eigenvectors The instead (I, I Let satisfy

on d and 0 $ F(h). There exists E > 0 such that P,,F also

on h.

character of the infinite dimensional situation requires to consider the pair (I, I - F) of the pair (1, F) as in finite dimensional case. Naturally a singular point x0 of the pair F) is an eigenvector of F. the pair (I, I - F) be linear independent on 0. Let S”-’ be the boundary of R n E” and E the condition of lemma 7.

Definition 13. Index C,,, I _p,F@‘-l) is called th e eigenindex of Z - F and denoted by C, _ ,(A). There appears the question if the eigenindex is well-defined. This question is not trivial for example because depending on the dimension of the approximating subspace the index C~I,r_pn,(S’-l) is an element of the group 2 x Z, or of the group Z,. THEOREM 14. Eigenindex

of the mapping

I - F is well-defined.

178

Ju. G. BORISOVIC

AND Ju. E. GLIKLIH

Proof. At first it is easy to show that for every projector P,’ such that Pi : F@) -+ E”, /IP,‘F, F,ll < E, we have C,, I_p F)(S”- ‘) = Cc,. r_rL,,(S”-‘) because P,F and PtF are naturally homotopic (P, is Schauder‘proiector, E is from Lemma 7). of Consider now a subspace En+’ 3 E” and choose the mapping P,F as an approximation F with the image in En+ ‘. The field I - P,E restricted to S” = Q n E”+l is of the form (cpl(X)? P,(X)? . . . 2cp,c4 x,, 1) where (q,(x), q,(x),

. . , q,(x)) = I - P,F:S”-

(here we assume that we have chosen a coordinate formula x,, I = 0).

I + E”.

system in En+’ such that E” is defined by the

LEMMA 8. C (I,I-P,F)V ‘) = CK-P,F, (Sn) where the equality the component Z, in the group n,_ i( V,, 2) and n,,( V,, I, J.

means the equality

of elements

of

Proof. Without loss of generality we can assume that the pair (I, I - P,F) on S”-’ is orthonormalized. By the construction (I, I - P,F) on S” is Freudenthal suspension over (I, I - P,F) on S” - I. Since 0 # n the identical mapping I is homotopic to a constant mapping : gt : S” - i x [0, l] -+ S”-‘,g, = I,g, z a w h ere a is a fixed vector of s”- ‘. There exists a homotopy (see for example to gr at each t. [3, Lemma 111) h,: S”-’ x [IO,l] + Sn-l, h, = I - P,F such that h, is orthogonal of Freudenthal suspension The homotopy gt: S”-’ x [0, l] 4 S ‘--l inducies the homotopy and the homotopy h,: S”-’ x [IO,l] -+ S”- ’ induces the homotopy of G,:S”x[O,l]-tS”, Freudenthal suspension H,: S” x [0, l] -+ S”. It follows from the properties of the finite dimen. . sional eigemndex that Cc,, I _ P,F)(Y-l) = c ,Y1,h,)(Sn-l) in E” and C,r,I-p,F)(Sn) = CI,,.,,,(S”) by the mappings in E”+l. By the Theorem 13 C,,I,,I,(S”-‘) and CtG,,H1)(Sn) are determined and belong to the component Z, of the group n,_ i( V,. 2) or h,:S”- LS”-2,H1:S”+S”-1 %(1/,+1 , 2) . Let us show that they coincide as elements of Z,. We have the diagram

where i,kare induced by the natural imbedding, pt are induced by the natural projection and C is Freudenthal suspension homomorphism. It is impossible to construct a homomorphism of but we are interested rr-l(l/n,2) into ~,(7/,+r,~ ) such that the diagram should be commutative, -+ CcG,,N,)(S”). By the construction of pairs (g,, h,) and only in the correspondence Cc4,, ,,,,(S-l) (G,, H,) we have Ccc1 HI)(Sn) = i:o X0 (i:-1)-1C~4,,h,)(Sn-1). It is easy to see that i:-’ and it are imbeddings and C is isomorphism. So C,,,, ,,,)(S- ‘) = CtG,,H,)(Sn) and therefore Co, r _p,,F)(Sn ‘) = Cc,, I_ p,n(S”). This completes the proof of the lemma. Then, as usual, of E” and E” are subspaces which do not include each other one most consider their linear span for obtaining of the previous situation. q.e.d. The following

propositions

THEOREM 15. If C,_.(h)

are naturally

true for the eigenindex.

# 0 there exists an eigenvector

of F in 0.

Fixed points of mappings

THEOREM

16. If F, and F, are homotopic

of Banach

manifolds

179

and some applications

on fi, then C,_,,(h)

= C,_,,(h).

In the last proposition one must consider a compact homotopy F*: b x [0,11 + E without eigenvectors on fi and such that 0 $ cl F*(h x [0, 11). It should be pointed out that for defining the eigenindex one may use pairs of the type (I, al - F) where c( is a scalar function such that (a(x)1 b LX,,> 0 in x E R. In this case it is natural to consider homotopies of fields 4, = a,Z - F* with the additional demand Ic+)I > a, > 0 for each t E [O, l), x E s1. If E is a Hilbert space, it is possible to prove the proposition which is an analogy of Rouche theorem. THEOREM

continuous

17. Let the inequality 1 mapping such that 0

Then C,_,(h)

= C, _F_,(fi);

3 a, > 0 hold on h and 40: G --f E be a completely

hence, if C, _ Jfi)

# 0, F + cp has in fi an eigenvector.

ProoJ The pair (I, Z - F) is homotopic to the pair (I, al - F) where a(x) = (Fx, x)([x~/-~ by means of the homotopy #t(x) = x - Fx - t(x - a(x)x). The pair (I, al - F) is homotopic to the pair (I, al - F - cp) by means of the homotopy $t(x) = cr(x)x - Fx - tcp(x). Finally the pair (I, al - F - cp)is homotopic to the pair (I, i - F - cp)by means of the homotopy 4:(x) = x - Fx - t(a(x)x - x) - q(x). The homotopies 4,, $,, 4: satisfy all conditions for conservation of the eigenindex. For example if 1x = $,(x) for some n E R, t E [0, I], x E h, then the scalar product of both sides of ilx = a(x)x - Fx - tq(x) with a(x)x - Fx would give IIF(x) - a(x)x)( d JJq~~p(x)l) on the contrary to (*). If ix = 4:(x) for some 1 E R, t E [0, 11, x E h, the mapping F + q would have an eigenvector on fi which would contradict to (*) via the same agruments. CHAPTER

2. SOLUTION

ANALYSIS OF DIFFERENTIAL EQUATIONS FINITE DIMENSIONAL MANIFOLDS

ON

SMOOTH

2.1. Periodical solutions offunctional-differential equations on compact manifolds Let M be a smooth manifold, J = C-T, 0] be a closed interval, T > 0. Consider the Banach manifold C”(J, M) of continuous maps of J into M. Functionaldifferential equation (F.D.E.) on M is a continuous map F: R x C”(J, M) --f TM such that for each t E R, q E C”(J, M) there holds zF(t, q) = ~(0) where TLis natural projection of TM onto M. A solution x(t) of F.D.E. on the interval [to - T, t, + A] with an initial condition (to, 9) is a continuous map of [to - T, t, + A] into M that is smooth in t 3 t, and such that i(t) = F(t, x,), t E [to, t, + A] where x,(e) = x(t + Q),0 E J, is a curve in the space C”(J, M) beginning at t = t, in the point q(B); i is a trangent vector to x(t). Autonomous F.D.E. on compact manifolds were investigated in [41]. That paper deals with the generic properties of bounded F.D.E. and also the existence and uniqueness theorem is proved under the assumption that F.D.E. is locally Lipschitzian; solutions of locally Lipschitzian bounded F.D.E. are defined on [to - T, co) and depend continuously upon t and initial conditions. We investigate locally Lipschitzian bounded nonautonomous F.D.E., naturally the properties mentioned above hold for them.

Ju. G. BORISOVI~‘ ANDJu. E. GLIKLIH

180

Let F.D.E. F be periodical upon t with the period o, i.e. F(t, cp) = F(t + o, cp). Let us consider a solution u(t, cp) of F with the initial condition (0, cp) defined on [ - 5, w], i.e. u(t, cp) = cp(t) in t E [-T, 0] and ;I(t, cp) = F(t, ut) in t E [0, w]. Definition

14. Shift operator

u, of F.D.E.

is an operator

that maps a curve cp(B) in the curve

(u,cp) (0) = u(o + 6, cp), 0 E J. Obviously u, is continuous, fixed points of U, and only they determine periodical solutions. It should be pointed out that the shift operator of F.D.E. in R” is widely exploited in investigations of stability, uniqueness, extendability, periodicity of solutions beginning from the papers of A. D. Myskis and N. N. Krasovskii. It is trivial that for bounded F.D.E. under z < o the operator U, is completely continuous. If z > o is it not so. Methods for F.D.E. in R” to overcome topological difficulties in fixed points problem here are: the compact reorganization of the shift operator [5] or use of the weak continuity of the operator under certain conditions [7]. For F.D.E. of neutral type the shift operator is not compact even if t < w [lo]. In the papers [46,30] it is shown that for these equations the shift operator condenses with respect to a special measure of noncompactness. Later, according to [25] we shall show that in our situation of F.D.E. on a compact manifold the shift operator is condensing with q < 1 with respect to Hausdorff measure of noncompactness in a natural metric on C’(J, M). The Lefschetz number A,, will be calculated. Take a Riemannian metric ( , ) on M. It defines a family of Riemannian metrics ( , ) (t) = e’( , ). Let p(t) be a metric on M which is induced by the tensor ( , ) (t). We define a metric on C”(J, M) by the formula Pt-,,&1? The metric p, _~, 01 is induced

(P2) = mEy p(r) (cPl(tj, (PJr)).

by the Finsler II~l(,-r,Ol = z?

metric e’llVr)ll

on

C”(J, M),

here V is a vector field along cpE C”(J, M), i.e. an element of T,C”(J, M). Let us consider an isometrical imbedding of M into a Eucledian space of enough high dimension (see [39,28]). The imbedding induces the isometrical imbedding of the Finsler manifold C”(J, M) in the Banach space C”(J, E) with the same norm. Obviously the retraction of a tubular neighbourhood U of M in E onto M induces the retraction of the open neighbourhood C”(J, U) of C”(J, M) in C”(J, E) onto C”(J, M). Consider Hausdorff measure of noncompactness (see 1.4) in the metric pt_ ~,0l on C”(J, M). Let 0 d I < T. We can restrict curves of C”(J, M) by [ - T, - Y] and [- r, 0] and consider the x,_~. ~_,] metrics P,-~, -rj and P,-r,Ol. Let us define two functions (measures of noncompactnesss) and xt_,., ,J in complete analogy with 1. Let Q c C”(J, M). LEMMA9. x(Q) 3 max{X,-,, THEOREM

-,(Q),

Xt-,,,,(a)).

18. x(u,Q) < e-“x(Q).

Proof F is bounded, M is compact, hence xt _ “~,,,,(u,R) = 0 (~$1 I _ w,ol is compact). it is easy to show that x(u,Q) = xt_,, _,,(u$) and by the definition of u, and x x(u,Q) = xt-,, -Ju$)

= e-UXrw-r,O,(Q)

If z > w

Fixed points

Then by Lemma 9 e-?(Q)

of mappings

of Banach

manifolds

2 e-Wmax{~I_,,,_,l(R),~I,_,,,l(n)}

and some applications

B e-wx,,_,.,,(Q)

181

= x(u,W. q.e.d.

So the operator u, is X-condensing with the constant q < 1. Together with the properties the manifold C”(.Z, M) it means that the Lefschetz number A,” is well-defined. THEOREM 19. A,_ is equal to the Euler characteristic

xw of the manifold

of

M.

Proof Let s E [0, 11. Let us define F.D.E. F” on the manifold Coct_sr, 01,Mj in the following way: fortZ(@ECO([-.sr,O],M)underthechangeofvariabletZ = s[,[E[-r,O],weobtain~](&C(J,M), then by the definition F”(t, ~(0)) = F(t, r(i)). Let cp”be restriction of a curve cp E C”(.Z, M) by the segment [ - ss, 01. We denote by zP(t, I$), t E [ - ST,01, the solution of F.D.E. F” with initial condition (0, t,P), and by U: : C”(.Z, M) + C”(.Z,M) the operator which is defined by the equality (u:cp) (0) = a”(& + o, cp”),6’E [ - 7, 01. It is easy to see that the homotopy u”, satisfies the condition of the Theorem 11 in 1.4 and U: = a,. The operator U: maps c”(J, M) into the isomorphic to M manifold of constant curves, where it coincides with the shift operator of the vector field F”. Hence U: is a compact operator, we can use Theorem 9 of 1.3 and restrict U: in M. Thus Au: = X~ since U: on M is homotopic to the identical mapping. COROLLARY. If X~ # 0 every F.D.E. of the investigated

class has an o-periodical

solution.

2.2. Operators of integral type on Riemannian manifold periodical and boundary-value problems In this section operators of integral type on Riemannian manifold are constructed and their properties are investigated. The construction exploits Riemannian parallel translation along smooth curves. See details in [23,24, 261. 1. Let M be a Riemannian manifold, Z = [a, b] be a closed interval, u: Z + TmoM be a continuous curve. There exists unique (?-curve Sv: Z -+ M such that Sv(a) = m, and the vector d/dtSu(t) is parallel along Sv to the vector v(t) for each t E I. Really, Su(t) = 6- ‘(&u(s) ds) where 6 is Cartan development. If M is complete manifold, S, is well-defined on the whole segment [a, b]. Leter we shall everywhere suppose that M is a complete manifold. So we have defined the operator S which maps the Banach space C’(Z, TmOM)of continuous mappings v:Z + TmoM into the Banach manifold C’(Z, M) of Cl-smooth mappings of Z into M. It is easily shown that S is continuous. Hence the parallel translation does not vary norm of a vector, the set SU, c C’(Z, M) (where U, c C”(Z, TmOM)is the ball of radius K and the centre 0) consists of curves such that in every point t the inequality Ili(t)ll d K holds. Let y E C’(Z, M) and X(t) be a continuous vector field along y(t). We denote by I,(y, X) a continuous curve in tangent space T,,,,M which is obtained by the parallel translation of vectors X(t) along y into the point y(s). If a continuous vector field <(t, m) is given on M, we denote by Is the operator that maps a curve y E C1(Z, M) into the curve I,(?, <(t, y(t)). It is easy to see that I, is continuous. The operator I, has an important property: LEMMAOF COMPACTNESS[26]. Let 0 be a set in C’(Z, M) and s be a set of continuous vector fields along the curves 0. If E is relatively compact in C”(Z, TM), the set I,(@, E) is relatively compact. Sketch ofthe proof The set 0 is relatively compact in Co-toppology. A limit point of 0 is, generally speaking, non-smooth curve. But it is not difficult to show that this continuous curve is Lipschitzian, i.e. the parallel translation is well-defined along it [18]. Now it is obvious that $

182

Ju. G. BOKISOVI(.AND Ju. E.

maps a convergent

sequence

the closure of I,(@, E) is compact. q.e.d. Let a vector field <(t, m) be defined on A4 and QK c C’(Z, M) be a set of curves y such that !i i(t) /i G K for every t E Z(K > 0 is a real number) and the set (&)(y E QK, t E Z} is bounded in M. THEOREM 20. The set I$&

into convergent

GLIKLIH

is compact

one and therefore

in C”(l, TM).

Proof: Q2, is compact in C”(Z, M), 5 is continuous, therefore the set of curves {<(t, ;(t))(y E Q,) is compact in C”(Z, TM). Thus we obtain the result by the lemma of compactness. COROLLARY. The operator Indeed, such that

I, is locally compact.

by the definition of Cl-topology there is a neighbourhood of a curve y E Ci(Z, M) for every curve y1 from the neighbourhood one has: l];/(t)]] 6 K + I-: where K =

m~~Mr)ll. Let us consider the superposition S 0 lYa:C’(Z, M) -+ C’(I, M). The operator So I, is a complete analogy of the Volterra operator of the theory of ordinary differential equations. Obviously fixed points of S 0 I, and only they are integral curves of the vector field <(t, m). But it is more convenient to make use of the superposition I, 0 S: C’(Z, TmOM)+ C’(I, T,,M). The operator I, 0 S acts on the (linear) Banach space and has the following property: a curve 1:E C’(Z, T_,M) is a fixed point of I, 0 S if and only if the curve Sv is an integral curve of the field ((t, m). It follows from the properties of S and I that the image ra @ S( U,) of a ball U, c C”(Z, T,,,M) is compact, i.e. the operator I, 0 S is relatively continuous. 2. If the manifold M is a Lie group G it is possible to define (on the basis of S and I’) an operator of integral type, whose fixed points and only they are periodical integral curves of a vector field on G. Let us define an arbitrary complete Riemannian metric on G (possibly non-coordinated with the group action). Let an o-periodical continuous vector field c(t, g) be given on G. We denote by Z the segment [0, 01 and consider an operator S”: C1(Z, G) -+ C’(Z, G) defined by the equality suY =

y(o)~p(o)~sOro.j’.

It is easily seen that fixed points of the operator and only they are o-periodical integral curves of t(t, g). As compared with the shift operator, S” has an advantage: one must demand neither uniqueness of integral curves nor their existence on the whole segment [0, 01 to define the operator S”. It follows from the local compactness of the operator I and from continuity of S and group action that S” is locally compact. Let Q c C’(Z, U) where Cr is a bounded domain in G. The continuous vector field <(t, g) is bounded on U x I by a constant K. It follows from the properties of the operators S, I and the continuity of group action that S”R is compact in C”(Z, G). Then we exploit the continuity of <(t,g), the lemma of compactness, the continuity of S and group action to obtain that (S”)2Q is compact in Ci(Z, G). Thus, if there are no fixed points of S” on the boundary h, the homotopic characteristic S(P, a) is well-defined.? 3. On the basis of S and I one can construct also an operator for studying of second order differential equations on manifolds. 7 The calculation of 6(S”, n) in terms of 5 on (i is described in the paper: Ju. E. Gliklih, integral curves of vector fields on Lie groups (Russian). Prikladnoi analiz, VGU, Voronezh

A. L. Haikin. (1979).

On periodical

Fixed points of mappings of Banach manifolds and some applications

183

Let us consider a continuous second order differential equation r(t, m, z) on a complete Riemannian manifold M, i.e. a continuous vector field on tangent bundle TM such that for each point (x, Z) E TM one has dzt(t, m, z) = z E T,M, where dn is tangent map of the natural projection 7~:TM -+ M. The representation of every tangent space ,7& =,TM as a direct sum of the connection H,,,, ;I allows to vertical subspace F$,,.z) (tangent to the fibre of TM) and Levi-C&t represent the field ((t, m, z) as the sum ((t, m, z) = &(t, m, z) + &(m, z) where &(t, m, z) E I/;,_) is called the vertical component, and the horizontal component t(,,,, is geodesic pulverisation of Levi-Civita connection. A curve 7: R -+ M is a solution of the equation t(t, m, z) if and only if the equality D/dt+(t) = p&(t, r(t), y(t)) holds in each point t; here D/dt is covariant derivative + T,M is a standard isomorphism of a tangent space to a vector space onto the [I, 4817 P:ym,z) the vector space. Now for a continuous curve 1~: R + TmoM and a vector C E T,,,Mwe can construct a Clcurve y: R -+ M such that y(O) - m,, d(o) = C and the vector D/dy j(t) is parallel along y to the vector MJ(~)in each t. This curve is unique and is described by the formula y(t) = S(J;+)ds

+ C)“S(C),,J

Along an arbitrary Cl-curve 7 the equation l(t, m, z) defines the vector field p<,(f, y(t), i(t)). We denote by rY: R -+ T,,,, M the curve such that the vector f,(t) is parallel along 7 to the vector &“k Y(L),y(t)). Denote by Z the segment [0, T] and consider the Banach space C”(Z, T,,M). The operator ?; 0 S(C) acts in C’(Z, T,,M). If w is a fixed point of the operator F 0 S(C), y = S(C) M’is a solution of g(t, m, z) with the initial condition y(O) = m,, y(O) = C. By the lemma of compactness and properties of parallel translation one easily obtains that f o S(C) is completely continuous. 4. Here we study the two-point boundary value problem for a continuous second order equation ((t, m, z). If t,,(t, m, z) s 0, i.e. 4 is geodesic pulverisation. by the Hopf-Rinow theorem there exists a geodesic joining arbitrary two points. Certainly, the case of general equation is not SO simple. There is the following proposition. THEOREM 21. [24,26].

Let points m. and m, be non-conjugate along some geodesic and I({, x (t, m. z)(( < K(K > 0) for all t, m, z. There exists a number L(m,, m,) > 0 such that for a segment [0, T], T < L(m,, m,), there exists a solution y of &t, m, z) with the condition y(O) = m,, Y(T) = ml. Sketch of the proof. The non-conjugacy of m,, m, along a geodesic means that there exists a vector C E T,,M such tha’t exp C = m,, d exp, is not degenerate, Since (S(C) 0) (1) = exp C it is not difficult to show that for each curve w from a suffkiently small neighbourhood of 0 in C’([O, 11, T,,M) there exists a vector C,,, such that (S(C,) w) (1) = m, and Cn: depends on w continuously. Let y = Su, where v E C’([O, 11, TmOM). The change of the variable t’ = Tt gives us the curves y* and v* defining on [0, T]. It is easily seen, that y* = S(T-Iv*). It is not diffkult to obtain from here that for small T every curve w from the ball U, c C’([O, T], TmoM) of radius K centred in the origin has the vector C, continuously depending on w and such that (S(C,),) w) (T) = m,. We denote by L the greatest upper bound of such ?: Now the completely continuous operator f o S(C,) maps U, c C’([O, T], T,$f) into itself and therefore it has a fixed point w* by Schauder fixed point theorem. The curve S(C,,) w* is a solution that we search.

Ju. G. BORISOW?ANDJu. E. GLIKLIH

184

The assumption of non-conjugacy of m0 and m, is essential. There are second order differential equations on the sphere S2 in R3, whose vertical component depends only on a point m of Sz and certainly is bounded, but every solution beginning in the South pole never comes in the North one: North and South poles are conjugate along every geodesic. It should be pointed out that for flat manifold M our construction gives the classical result: if the vertical component is bounded, the two-point boundary value problem is solvable for every two points and segment [0, T]. The explicit description of this item results is contained in [26]. There one also finds the details of the integral operators construction.

2.3. Equations with a discrete delay on Riemannian manifolds, Pseudointegral curves In this section we consider a special class of F.D.E. on manifolds which are analogous to disCrete-dealy equations in linear spaces, but the character of non-linear spaces leads to their being of neutral type. 1. Let a continuous vector field c(t, m) be given on a Riemannian manifold M. We consider C’-curves m(t) in M such that at each instant t the tangent vector ti(t) is parallel along the same curve m(t) to the vector @t - h, m(t - h)) of the field. The curve is described by the equation ti(t)[ /<(t - h, m(t - h)) where symbol /I means the parallel translation along m(t). Such equations arise in mechanics of non-flat configuration spaces. Let N be a configuration space of a mechanical system with a quadratic kinetic energy (see [27]), i.e. T is induced by a Riemannian metric ( , ) on N: T(X) = i(X, X). Now the equation ti(t)i(<(t - h, m(t - h)) means that the velocity ti(t) of the point depends on the configuration but, since there is a natural delay in a real situation, it is developed in the time interval h. Since the parallel translation is well-defined along smooth curves and depends on a derivative of a curve, the equations investigated are of the neutral type. (In flat linear spaces it is not so because the parallel translation does not depend on a curve.) Let y E C’(J, M), J = [-h, 01, be a C’-curve such that the vector j(O) is parallel along y to the vector <( -h, y( - h)). Then S 0 Toy (see the previous section) is a solution of the equation 2(t) x ll;“(r - h>Y(t - h)) on the segment [0, h]. Obviously the solution is unique. We are interested in periodical solutions. Let the field [(t, m) is o-periodical with respect to t. Methods of studying require to distinguish three cases: h > Q, h < LC).h = cc). At first we consider the case h = o. Here an o-periodical integral curve is a closed curve y(t) (with the period w) such that the vector j(t) at each t is obtained by the parallel translation of the vector <(t, -y(t)) along ‘J from the point l;(t - co) into the point y(t). These curves are called pseudointegral curves of a vector field [ 13,14,15]. If the metric on M is flat, pseudointegral curves are ordinary o-periodical integral one. Let us consider the Banach manifold C’(Z, M), I = [0, 01, and the operator S, = SO I(,,: C’(Z, M) -+ C’(Z, M) [23]. Obviously fixed points of S, and only they are pseudointegral curves. THEOREM 22.

The operator

S, is locally

compact.

This proposition follows from the corollary of Theorem 20 and the continuity of the operator. Consider the second iteration Si of S,. Let 0 be a bounded open set in M and C’(Z, 0) be subset of C’(Z, M) which consists of the mappings of Z into 8.

Fixed points of mappings of Banach manifolds and some applications THEOREM 23.

185

The set S;(C’(Z,G)) is compact.

Proof: Since 0 is bounded, B is compact and /l&t, m)(I IS b ounded on m x I by a constant K. Since the parallel translation does not change the norm of a vector, the curves T,C’(Z, 8) belong to mlM U,(m) (see 2.2) and the set S,C’(Z,G) consists of curves y such that (Ii(t)ll d K at each t. Thus the statement follows from theorem 20 of 2.2 and from the continuity of S. COROLLARY.

If M is a compact manifold, SiC’(Z, M) is compact.

Let us imbed the manifold M into a high-dimensional Euclidean space E, let r: U + M be a retraction ofa tubular neighbourhood U onto M. This retraction induces the retraction ofthe open neighbourhood C’(Z, U) of the manifold C’(Z, M) in the Banach space C’(Z, E) onto C’(Z, M). Thus it is possible to define the Lefschetz number or the homotopic characteristic in the sense of 1.2, 1.5, of Chapter 1. 2. Let M be a compact manifold. Hence SiC’(Z, M) is compact and the Lefschetz number hsw is well-defined. THEOREM 24

[ll].Asm

= XM-Euler characteristic of M.

Proof: Without loss of generality one may assume that integral curves of t(t, m) are locally unique. Really, let tr(t, m) be a locally Lipschitzian w-periodic approximafon of the field ((t, m). The linear homotopy which joins ti(t, m) and ((t, m) is bounded. Thus the induced homotopy S”,, s E [0,11, of the operator Sk and S, satisfies to the definition 2 (1.3), i.e. As_ = A,,. Denote by u(o) the shift operator of the field <(t, m). Let y E C’(Z, M). Consider the curve

qY(t)

= i

r(t)

O~ttsSU

m(t)

so$t
s E [0,l];m(t) is the integral curve of the field l(t, m) with the initial condition m(w) = y(so). Since T,y is piecewise smooth, the parallel translation is well-defined along it. Thus we can define the operator As: C1(Z,M) -+ C’(Z, M) by the equality Q(t) = So s. r,O T,y. Obviously A, = S,. The operator A, acts in the following manner. By the definition T,y is an integral curve of <(t, m) with the initial condition T,y(O) = y(O).The curve 0 . To o T,y: Z -P TToyfwjMis identically equal to the origin, therefore A&t) = Toy(w). So A, maps C’(Z, M) into the isomorphic to M manifold of constant mappings and coincides there with the shift operator V(o). Thus AS,,C1(I,M)= AUcUjjM = xM q.e.d. If xM # 0, every continuous o-periodic on M) has a pseudointegral curve.

COROLLARY.

vector field (for every Riemannian metric

3. Now we shall show that a pseudointegral curve arises in a neighbourhood of a structurallv stable singular point of an autonomous vector field under small non-autonomous periodical perturbation. Certainly, here one should exploit the homotopical characteristic for S,. But at first one must find the domain in the functional space such that there are no pseudointegral curves on its boundary.

Ju. G. BORISOVI~AND Ju. E.

186

GLIKLIH

Let dim M 3 3 and an autonomous vector field @I) on M be C’-smooth. Let m, be a structurally stable singular point (i.e. a hyperbolic one in terms of [40]) such that there are no periodic integral curves in its neighbourhood. LEMMA

integral

10. Under the conditions above there exists a neighbourhood curves (except of m,) of preassigned period o.

U of m, in M without

pseudo-

Proof We shall prove it by contradiction. Let there be an o-periodic pseudointegral curve in each neighbourhood of m,. It is easy to see that by decreasing the neighbourhood we can obtain an arbitrary Cl-closeness of the fields $(t) and ((t, y(t)) along every pseudointegral curve y in the neighbourhood. Really, by the definition the vector +(t) is a value at t + o of the solution of the parallel translation equation

with the initial conditions u(t) = t(y(t)), where yi are local coordinates of the pseudointegral curve. When we decrease the neighbourhood, \\t\\ tends to zero (i.e. dy’/dt also tends to zero), Cristoffel symbols I?; tend to constants FE(m,), hence u(t + o) tends to ((y(t)) by the smooth dependence of solutions on parameters and initial conditions (see, for example [29]). A pseudointegral curve y is C2-mapping of the circle S’ into M. Since dim M 3 3, one can arbitrarily C2-close approximate y by a C2-imbedding v (see, for example [48]). Via smooth partition of unity we construct Cl-vector field t such that it coincides with 7 on 7 and with 5 outside a small tubular neighbourhood of 7. Summarizing the previous arguments we can see that choosing y close to m, it is possible to obtain an arbitrarily Cl-closeness of the fields 5 and r. By the condition of the lemma the flow defined by the field in the neighbourhood of m,, is a Morse-Smale system, i.e. it is structurally stable (see [42]) and has no periodic integral curves. at the same time we have constructed the field 7J which is arbitrarily Cl-close to c and has the periodic integral curve 7. The contradiction proves the lemma. Surely the neighbourhood U from Lemma 10 may be considered as a bounded one. COROLLARY. of

the domain

For each w E (0, co) the homotopic characteristic Q = C’([O, w], U(o)) c C’([O, 01, M).

LEMMA 11. )6(S,,

a)\ = (6(m,)j

where

Prooj: Let us take a contractible In Euclidean metric SW is ordinary

S(Swsi) is well-defined

on the closure

6( m ,.j IS P omcare index of the singular point m,. U and define Euclidean metric and local coordinates integral operator A:

A?(r) = Y(W) +

' 5bW)

on U.

ds.

I 0

Consider the operator A on the set of continuous curves C’([O, 01, I’) where 1/ c U is such that AC’([O, w], I’) < C”( [0, w], U). It is well known that A is completely continuous on the space Co. Using Yu. I. Sapronov’s result [43] and the definition of the homotopic characteristic one can easily show that the rotation I of the vector field I - A on the boundary of the domain C’([O, 01, V) is equal to &A, 0) where 0 = C’([O, 01, V). It follows from [34] that III = (6(m,)l. Now we join the Euclidean metric and the original Riemannian metric on M by the linear

Fixed points of mappings of Banach manifolds and some applications

187

homotopy. By the same token we homotopy the operator A to So. The fact that the homotopy has no fixed points in O/m, is proved in complete analogy with Lemma 10. It is easy to see that the homotopy satisfies the definition 9 in 1.5. q.e.d. Let y~(t,m) be a continuous vector field on M which is o-periodical

with respect to t.

25. For sufficiently small E > 0 the field 5 + EVhas an o-periodic pseudointegral curve in a neighbourhood of m,.

THEOREM

Proof By Lemma 11 we have IS@,, a)/ = /s(m,,)( and also 6(m,) # 0 by the structural stability of m,. Let us join the fields t and 4 + EV,by the homotopy 5 + ilev, 1 E [0,11. The homotopy induces the locally compact and having compact the second iteration homotopy Sf) of the operator S, which has no fixed points on the boundary of Szif E is sufficiently small. Thus 6(Si, a) # 0 and there is a pseudointegral curve of 5 + E~Z in U. q.e.d. 4. In this item we show that pseudointegral curves may appear under the perturbation of Riemannian metric on the manifold [13]. From the mechanical point of view it means the perturbation of kinetic energy form. Let us consider an autonomous vector field t: on M with the uniqueness of the initial value problem’s solution Let y* be an isolated closed integral curve. The flow, induced by 5 defines the transformation of a local cross-section in a point m0 E y* which is called the point mapping. Definition

15. The Poincare index of point mapping’s isolated fixed point m, is called the index 6(y*) of the closed integral curve y*. Obviously 6(y*) does not depend on the choice of a point m, on y*.

Let it be possible to define a complete flat Riemannian metric ( , ). on a manifold M, let the field 5 have an isolated closed integral curve y* with a period o* and an index 6(~*) # 0. Then for each Riemannian metric on M, sufficiently close to ( , jo, there exists a pseudo-

THEOREM 26.

integral curve in a neighbourhood

of y* with a period close to o*.

Before proving the theorem we modify the method of functionalization of parameter (see for example [2,49]) to adapt it for our situation. Let m, E y* and o(m) be a smooth real-valued function on a neighbourhood of m, such that (50) (m,) > 0 and o(m& = o* (i.e. o(m) is functional of the first genus in terms of [2,49]). In the (m of a point m along the trajectory for the neighbourhood of m, there is well-defined shift Z_Jo(,,,) time w(m). It is clear that m, is an isolated fixed point of the operator Uo,*). Using the Theorem 9’ from 1.5 and the transversality of the cross-section it is easy to show that for sufficiently small neighbourhood I/ of the point m0 we have 6(u,(,), V) = 6(y*). Let d > w*. Denote by Z the segment [0, d] and consider the Banach manifold C’(Z, M). The mapping p: C’(Z, M) + M, py = y(O), defines on the domain p-l I/ the smooth function R by the formula R(y) = o(py). Now the operator S,:p-‘I/ -+ C’(Z, M), S,y = SO rac,,,y is well-defined in every complete Riemannian metric on M. Let us denote by Sg the operator in the metric ( , )o. Obviously the integral curve g(t) E C’(Z, M) of the vector field c with the initial condition g(0) = m, is an isolated fixed point of the operator S;.

188

LEMMA 12. For sufficiently

Ju. G. BORISOVI~ ANDJu. E. GLIKLIH small neighbourhood

0 of the curve g in p-l I/ we have S(S&8)

=

&*). To prove the lemma one must homotopy the operator S, to the operator u,(,) in complete analogy with the homotopy of the second item of this section. Of course, the variable o(m) should be considered. ProofofTheorem 26. If a Riemannian metric ( , ) 1 is sufficiently close to ( , )o, the homotopy S& A E [0, I] induced by the homotopy ( , & = ,I( , l1 + (1 - A)( , )u of metrics has no fixed points on the boundary 0. Hence 6(S& 0) = 6(Sz, 0) = 6(y*) # 0. Thus the operator Sk has a fixed point in 0. q.e.d. If it is possible to see that S(S;,,@) # 0, the theorem is true without supposition of the uniqueness of initial value problem solutions (in this case the index 6(y*) is not defined). 5. It should be noted that it is possible to apply the methods given above of calculating of homotopic characteristic also to the operator S” (see 2.2) whose fixed points are periodic integral curves of vector fields on Lie groups. For example we give the proposition analogous to Theorem 25. THOEREM 27. Let 5 be a continuous vector field on a Lie group G, m, be a singular point with a non-zero PoincarC index such that there are no periodical integral curves in its neighbourhood, y(t, m) be a continuous o-periodical vector field on G. For sufficiently small e > 0 the vector field 4 + EY]has an w-periodical integral curve in the neighbourhood of m,. The proof is in complete analogy with the proof of Theorem 25. We must note that the analogy of Lemma 10 is not necessary here because there exists a neighbourhood of m, without periodical integral curves (fixed points of So) by the condition of the theorem. To prove the analogy of lemma 11 one must at first pass to a smooth vector field (I close to 5 on the neighbourhood of m,. The homotopic characteristic is not changed here (cf. the beginning of the Theorem 24).

2.4. The operators for investigation of the periodic solutions in the case of general equations with discrete delay Let us consider the periodical problem for the equation k(t) iit(t - h, m(t - h)) in a general case. In this section we construct appropriate operators and study their properties. Let a continuous vector field <(t, m) be w-periodic with respect to t and h < w. By I we denote the segment [-h, 0] and then we consider the Banach manifold C’(Z, M). Let :I’E (?(I, M). Let us apply the operator SO To to y, then we apply it once more to the curve S o r,?/ and so on N times, where Nh > o. The constructed family of the curves SO Toy, (S 0 To)‘;, , . . may be considered as a unique smooth curve U: [I-o, Nh - w] + M. By u. we denote the operator that maps y into the restriction of u by the segment [-h, 01. In fact U, is a shift operator along the trajectories of the equation rit(t)llt(t - h,m(t - h)) in the time o. Obviously, fixed points of U, and only they are w-periodic solutions of I”(t)ll<(t - h, m(t - h)). By the construction U, is locally compact and has a compact iteration, moreover if w > 2h, u, is compact. The proofs of these propositions are in complete analogy with corresponding ones for the operator S, in 2.3. Since the manifold C1(Z, M) can be imbedded into a Banach space as a neighbourhood retract, it is possible to define Lefschetz number or homotopic characteristic for u,.

Fixed points of mappings of Banach manifolds and some applications

189

If h > w, the construction of the operator is more complicated. We consider the case h < 20 and leave the general case for the reader as a simple exercise. Let C’(M) c C”([ - h, 01, M) be a set consisting of curves which are continuous on C-h, 0] and smooth on [ - h, -co] and [- w, 0] (possibly there is a discontinuity of the derivative at the point - 0)). THEOREM 28. C’(M) is a Banach

manifold and can be imbedded into a Banach space as a neighbourhood retract (cf. [12]). Proof. Let us consider the Banach manifold cl(M) = C’([ - h, -01, M) x C’([ -co, 01, M) and the mapping p: c’(M) + M x M, p(yl, y2) = ( y,( - o), y2( - co)). It is easy to see that the diagonal A in M x M belongs to the set of regulr values of p. Hence C’(M) = p- ‘A is a submanifold in c’(M). Let us imbed M into a high-dimensional Euclidean space E and construct C’(E) in analogy with C’(M). It is not difficult to show that c:‘(M) is a neighbourhood retract in c”(E) and C’(M) is a neighbourhood retract in C’(E). q.e.d. Let y E C’(M). Since ?/ is a piecewise smooth curve, the parallel translation is well-defined along it, and we can apply the operator SO I, to y. The curve SO Toy(B) is defined on the segment [-h, 01. Changing the variable t = 19+ h - w we define SO Toy on the segment C-0, h - co]. By the same manner we define y(B) on [-h - o, --co]. Now we denote by &,y E C’(M) the curve such that (z?,y) (t) = y(t), t E C-h, -co], (6~) (t) = SO r,?(t), t E [-co, 01. The operator 0, is in fact the shift operator in 3. Obviously zi, maps C’(M) into itself. Fixed points of 6, are smooth curves on [ -h, 01, moreover they and only they are o-periodic solutions of the equation rit(t)([<(t - h, m(t - h)). Using the properties of S and I (see 2.2) it is easy to show that for each domain R c C’(M) consisting of curves which belong to a bounded domain U < M, the set z?EQ is compact if (N - 1) w > h. But ii, is not locally compact. Riemannian metric on M induces the standard metric p* on the tangent boundle TM. Let us consider a family of metrics p, = e’p* on TM and the metric p on C’(M) which is defined by the formula

Let x be the Hausdorff metric p) set in C’(M).

measure

of noncompactness

in the metric p and Q be a bounded

(in the

THEOREM 29. The operator

ii, condenses with 4 < 1 on Q with respect to x. This theorem is proved in complete analogy with Theorem 18 in 2.1. Thus the operator 6, is locally condensing and has a compact iteration, i.e. one may define Lefschetz number of homotopic characteristic for it. 2.5. On pseudointegral curves 0ffunctionaMifferential equations Pseudointegral curves (see 2.3) arise in more general situations. In this section we briefly describe one of such generalizations: pseudointegral curves of functionaldifferential equations. The locally condensing operator for studying these curves is constructed, the existence theorem is proved. Consider a compact Riemannian manifold M with a given continuous u-periodic F.D.E. F of the class as in 2.1.

190

h. G. BORISOVIC AND Ju. E. GLIKLIH

Definition 16. An o-periodic CL-curve y in M is called a pseudointegral curve of the equation F if the vector F(t, yt) in each t is parallel along y on the segment [t, t + 01 to the vector ?(t). Pseudointegral curve of F.D.E. is a periodical motion of a complicated mechanical system: the velocity of the point depends on the previous history and since there is a natural delay in real situation, acts in a time interval o (cf. 2.3). Let us denote the segment [0, w] by I, the segment [ - T, 0] by J and consider the submanifold C(M) = {(4&Y)E CO(J, M) x C’K M)/q(O) = Y(O)) of the Banach manifold C”(J, M) x Ci(I, M). It is easy to show that the Banach manifold C(M) can be imbedded in the Banach space C(E), where E I M as a neighbourhood retract [12] (cf. Theorem 28 in 2.4). For the sake of convenience we shall later consider pairs (p, y) E C(M) as the mappings C(t): [ - z, o] + M which are continuous in t E [-~,a] and Cl-smooth in t E [0, w]. Let us consider an arbitrary point c = (cp, y) e C(M). F.D.E. F defines the vector field F(t, c,) along y. We parallelly translate this field along y into the point y(w) and then apply the operator S to the obtained curve in T,,,,M. As a result we obtain the curve Sac E C’(I, M) (cf. 2.2, 2.3). Now we can define the operator gu,:C(M) ---f C(M) by the formula (9,~) (t) = (ctlleo, SW(.). It is easy to see that the fixed points of the operator s, and only they are pseudointegral curves of F.D.E. F. It follows from the properties of the integral operators (see 2.2) and the shift operator (see 2.1) that 3, is continuous. Riemannian metric on hri induces the standard metric p* on the tangent boundle TM. Consider the metric p,_, o1on C”(J, M) (see 2.1). Let F.D.E. F be locally Lipschitzian with the constant p < 1, i.e. each curve cp E p(J, M) has a neighbourhood IJ, c C”(J, M) such that for every ‘pi, ‘Pi E U, and each r the inequality p*(F(t, cp,), F(t, cp,)) < PP,_,, ,,,(cp,, cp,) holds. Let us take an arbitrary real number 6 > 0. We define the metric pcOon C’(f, M) by the formula PO(Y1>YJ = (1 + P)- l (1 + V’

:,“I” P*(+,(a

?,V))

and denote by x, the Hausdorff measure of noncompactness in the metric pw on C’(I, M). Now we can define the measure of non-compactness $ on C(M) by the formula ti(((cp, Y)}) = max(x((cp)X x,({Y>)), i(q, Y)> c C(M), where the measure

of non-compactness

x on C”(J, M) is defined in 2.1.

THEOREM 30. For each curve c = (cp, 7) E C(M) there exists a neighbourhood that for every set Q c C(M) the inequality $(s^,Q) d y$(Q holds, where

U, c C(M) such

THEOREM 31. There exists an integer N such that the set StC(M) is compact. Thus SW is locally $-condensing and has a compact iteration. Taking the properties into account it means that the Lefschetz number A,“, is well delined for s,>.

of C(M)

of M. THEOREM 32. A,- = xMu,where xu is the Euler characteristic Corollary. If X~ # 0, every F.D.E. F of investigated class has a pseudointegral curve. See details of this Section in the paper: Ju. E. Gliklih, on pseudointegral curves of functional differential equations on Riemannian manifolds (Russian), Metody reSenija operatornyh uravnenii, VGU, Voronezh, 1978.

Fixed points of mappings

of Banach

manifolds

and some applications

191

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