JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS, Vol. 6, No. 3, 2000, 431-451
REMARKS ON PERIODIC SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS V. V. FILIPPOV
Abstract. The paper is written in comparison of the power of the Leray–Schauder method and the method of translation along trajectories in the boundary values problems theory for ordinary differential equations. The author suggests a new version of the continuation principle for the method of translation along trajectories. Then he shows how to use the new approach to obtain in a simpler way a reinforced analog of a thaeorem of J. Mawhin.
Introduction This paper is written in comparison of the power of the Leray–Schauder method and the method of translation along trajectories in the boundaryvalue problems theory for ordinary differential equations. The article [1] contains a new version of the continuation principle for the method of translation along trajectories. This new version implies both my previous results in this direction, see in particular [2], and the wellknown Leray–Schauder continuation principle for integral operators, constantly used in the theory of existence of periodic solutions for ordinary differential equations, see Sec. 4 below. In this paper we repeat the mentioned result from [1] and supplement it by the discussion of appearing relations. Then we show how to use this theorem to obtain a reinforced analog of Theorem 4.2 from [3] (see also [4]). Here Theorem 4.2 from [3] is not our main goal. We consider it as a specimen of a long list of similar results. We give here a new its proof, which does not use the Leray–Schauder method and which is based on translation along trajectories. 1991 Mathematics Subject Classification. 34B15, 34C25. ˇ Key words and phrases. Periodic solutions, solution spaces, Cech homologies. This work is supported by the Russian Foundation for Basic Research, Project No. 9801-00061. 431 c 2000 Plenum Publishing Corporation 1079-2724/00/0700-0431$18.00/0
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We leave to the reader the direct comparison of our method of the proof and of the one in [3], [4]. Note that we use only the theory of homoloˇ gies (although those are Cech homologies by Vietoris–Begle theorem) and the Leray–Schauder theory is based on a superstructure of the theory of homologies. Moreover, the geometric condition of the statement of the theorem implies directly the required estimate of the degree of the translation and when we apply the Leray–Schauder theory, the computation of the degree needs additional efforts (this is incident to all similar results). A new approach to the theory of ordinary differential equations is accounted in [5]–[9] and in my other papers. It is based on detailed studies of topological structure of spaces of solutions. In particular, in the framework of the approach [5]–[9] we have the theory of the Cauchy problems for equations with very weak restrictions on properties of right-hand sides and for equations with multivalued right-hand sides (differential inclusions). So the method of translation along trajectories turns out to be applicable to equations and inclusions with very weak restrictions on right-hand sides. On the background of this Cauchy problem theory restrictions on right-hand sides turn out to be unsurmountable obstacle to the direct application of the Leray–Schauder method in so large suppositions. Even the equation with the right-hand side being continuous everywhere except one point need not lie in the sphere of application of the Leray–Schauder theory due to absence of the corresponding integral operator. Theory developed in [5]–[9] investigates topological object, whose properties are close to those of solution spaces of ordinary differential equations. The spaces of the class Rceu (U ) in [5]–[9] correspond to local dynamical systems. The spaces of the class Rce (U ) correspond to local multivalued dynamical systems. Our descriptions of such systems are useful for investigation and applications. The monograph [8] contains the most complete account of this theory except homological properties of solution spaces. In the framework of the discussions of this paper the most helpful tool are spaces of the class Rceh (U ), corresponding to local multivalued dynamical systems in which set of solutions starting at a fixed point is homologicaly trivial. For solution spaces of ordinary differential equations with continuous right-hand sides the verification of this condition is the contents of Aronszajn theorem. Results of this paper go in the direction of simplification and of the widening of the theory of boundary-value problems for ordinary differential equations. In order to simplify understanding we do not aim at widest generality and we do not discuss details which are not essential for our goals. But note that our reasons here work in many other cases.
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ˇ 1. Remarks on Cech homologies In order to do not break the account below by small references we start with recalling some simplest well-known, or easily verifiable facts, related ˇ to Cech homologies. Here and in Sec. 2 m = 0, 1, . . . . Let (1.1) V1 be an open and L2 be a closed subset of a compactum L1 , a set V2 ⊆ L2 is open in the compactum L1 and V2 ⊇ V1 , see Fig. 1.1. Remark 1.1. With the notation (1.1) the couple L2 \ V2 ⊆ L2 lies in the couple L1 \ V1 ⊆ L1 . It means that we have inclusions of corresponding elements of the couples: L2 ⊆ L1 and L2 \ V2 ⊆ L1 \ V1 . So under suppositions (1.1) we have the homomorphism ˇ m+1 (L2 , L2 \ V2 ; Q) → H ˇ m+1 (L1 , L1 \ V1 ; Q), ¨ıL2 ,V2 ;L1 ,V1 : H which is generated by the inclusion.
Fig. 1.1 Remark 1.2. Let, in addition to (1.1), (1.2) L3 be a closed subset of the compactum L2 , a set V3 ⊆ L3 is open in the compactum L1 and V3 ⊇ V2 . Then the homomorphism ˇ m+1 (L3 , L3 \ V3 ; Q) → H ˇ m+1 (L1 , L1 \ V1 ; Q) ¨ıL3 ,V3 ;L1 ,V1 : H is equal to the composition of the homomorphisms ˇ m+1 (L3 , L3 \ V3 ; Q) → H ˇ m+1 (L2 , L2 \ V2 ; Q) ¨ıL3 ,V3 ;L2 ,V2 : H and ˇ m+1 (L2 , L2 \ V2 ; Q) → H ˇ m+1 (L1 , L1 \ V1 ; Q). ¨ıL2 ,V2 ;L1 ,V1 : H This means that ¨ıL3 ,V3 ;L1 ,V1 = ¨ıL2 ,V2 ;L1 ,V1 ·¨ıL3 ,V3 ;L2 ,V2 . We deduce this from the fact that the composition of generated homomorphisms of groups of homologies corresponds to compositions of continuous mappings of compacta.
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Remark 1.3. By the excision theorem if with notation (1.1), (1.3)
V2 = V1 ,
then the homomorphism ¨ıL2 ,V2 ;L1 ,V1 from Remark 1.1 is an isomorphism. Let now (1.4) U1 be an open and K2 be a closed subset of a compactum K1 , a set U2 ⊆ K2 ∩ U1 be open in K1 , see Fig. 1.2.
Fig. 1.2
Remark 1.4. By Remark 1.1 under hypothesis (1.4) the homomorphisms ¨ıK2 ,U2 ;K1 ,U2 and ¨ıK1 ,U1 ;K1 ,U2 are defined. By Remark 1.3 the homomorphism ¨ıK2 ,U2 ;K1 ,U2 is an isomorphism. So we can define the homomorphism jK1 ,U1 ;K2 ,U2 = ¨ı−1 ıK1 ,U1 ;K1 ,U2 : K2 ,U2 ;K1 ,U2 ·¨ ˇ m+1 (K1 , K1 \ U1 ; Q) → H ˇ m+1 (K2 , K2 \ U2 ; Q). H Example 1. Let compacta K1 and K2 in (1.4) be closed balls of the space Rm+1 . Let U1 be the interior of the ball K1 , U2 be the interior of the ball K2 . In this case the homomorphism ¨ıK1 ,U1 ;K1 ,U2 is an isomorphism. Therefore, the homomorphism jK1 ,U1 ;K2 ,U2 is also an isomorphism. Remark 1.5. Let, in addition to (1.4), K3 be a closed subset of the compactum K2 and a set U3 ⊆ K3 ∩ U2 be open in K1 . In this case the equality (1.5) ¨ıK2 ,U3 ;K1 ,U3 · ¨ıK2 ,U2 ;K2 ,U3 =¨ıK2 ,U2 ;K1 ,U3 =¨ıK1 ,U2 ;K1 ,U3 · ¨ıK2 ,U2 ;K1 ,U2 implies the equality ¨ı−1 ıK1 ,U2 ;K1 ,U3 = ¨ıK2 ,U2 ;K2 ,U3 · ¨ı−1 K2 ,U3 ;K1 ,U3 · ¨ K2 ,U2 ;K1 ,U2
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(see Remark 1.2). Then by the definition of the homomorphisms j...... we obtain the equality ıK1 ,U1 ;K1 ,U3 = jK1 ,U1 ;K3 ,U3 = ¨ı−1 K3 ,U3 ;K1 ,U3 · ¨ = (¨ı−1 ı−1 ıK1 ,U2 ;K1 ,U3 · ¨ıK1 ,U1 ;K1 ,U2 ) = K3 ,U3 ;K2 ,U3 · ¨ K2 ,U3 ;K1 ,U3 ) · (¨ = ¨ı−1 ı−1 ıK1 ,U2 ;K1 ,U3 ) · ¨ıK1 ,U1 ;K1 ,U2 = K3 ,U3 ;K2 ,U3 · (¨ K2 ,U3 ;K1 ,U3 · ¨ = ¨ı−1 ıK2 ,U2 ;K2 ,U3 · ¨ı−1 ıK1 ,U1 ;K1 ,U2 = K3 ,U3 ;K2 ,U3 · (¨ K2 ,U2 ;K1 ,U2 ) · ¨ ıK2 ,U2 ;K2 ,U3 ) · (¨ı−1 ıK1 ,U1 ;K1 ,U2 ) = = (¨ı−1 K3 ,U3 ;K2 ,U3 · ¨ K2 ,U2 ;K1 ,U2 · ¨
= jK2 ,U2 ;K3 ,U3 · jK1 ,U1 ;K2 ,U2 . See also the diagram
ˇ m (K3 , K3 \ U3 ; Q) H = ˇ m (K2 , K2 \ U3 ; Q) ˇ m (K2 , K2 \ U2 ; Q) → H H ∗ = = ˇ m (K1 , K1 \ U1 ; Q) → H ˇ m (K1 , K1 \ U2 ; Q) → H ˇ m (K1 , K1 \ U3 ; Q). H Its arrows correspond to the homomorphisms ¨ı...... , which are used in the above reasoning. The commutativity of the square ∗ corresponds to equality (1.5). 2. On homomorphisms and degrees Let ˜ 0 be an open subset of a compactum X; ˜ (2.1) X ˜ be a continuous acyclic mapping of a compactum X onto (2.2) B : X → X ˜ 0 ); ˜ X 0 = B −1 (X the compactum X, (2.3) H be a compact subset of the space X, a set G ⊆ H ∩ X 0 be open in X 0. By Vietoris–Begle theorem the generated homomorphism ˇ m+1 (X, X \ X 0 ; Q) → H ˇ m+1 (X, ˜ X ˜ \X ˜ 0 ; Q) B∗ : H is an isomorphism. So we can consider the homomorphism ˇ m+1 (X, ˜ X ˜ \X ˜ 0 ; Q) → H ˇ m+1 (H, H \ G; Q), lH,G : H defined by the formula lH,G = jX,X 0 ;H,G · (B∗ )−1 .
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Let now, in addition to (2.1)–(2.3), (2.4) f : H \ G → S1m be a continuous mapping of the compactum H \ G in the unit sphere S1m of the Euclidean space Rm+1 . The mapping f generates the homomorphism ˇ m (H \ G; Q) → H ˇ m (S1m ; Q) (= Q). f∗ : H We will consider the composition f∗ · ∂ of this homomorphism with the boundary operator ˇ m (H, H \ G; Q) → H ˇ m (H \ G; Q). ∂:H Lemma 2.1. If under hypotheses (2.1)–(2.4) the mapping f can be extended to a continuous mapping F : H → S1m , then the homomorphism f∗ ·∂ is trivial (≡ 0). Proof. The diagram ˇ m+1 (H, H \ G; Q) → H ˇ m+1 (H, H; Q) H ∂ ∂ ˇ ˇ Hm (H \ G; Q) → Hm (H; Q) ˇ ˇ F∗ f∗ m ˇ ˇ Hm (S1 ; Q) = Hm (S1m ; Q). is commutative. Its horizontal homomorphisms correspond to inclusions. ˇ m+1 (H, H; Q) is trivial (= 0). Therefore, the homomorphism The group H f∗ · ∂ = F∗ · ∂ · ¨ıH,G;H,∅ is also trivial. This gives what was required. The lemma is proved. Lemma 2.2. Let under hypotheses (2.1)–(2.4) a compactum H1 ⊆ X contain the compactum H, a set G1 ⊆ G be open in X, the mapping f can be extended to a continuous mapping F : H1 \ G1 → S1m . Then the homomorphisms f∗ · ∂ · lH,G and F∗ · ∂ · lH1 ,G1 coincide. Proof. The equality ¨ıH1 ,G1 ;X,G1 · ¨ıH,G;H1 ,G1 = ¨ıH,G;X,G1 = ¨ıX,G;X,G1 · ¨ıH,G;X,G implies the equality (2.5)
ıX,G;X,G1 , ¨ıH,G;H1 ,G1 · ¨ı−1 ı−1 H,G;X,G = ¨ H1 ,G1 ;X,G1 · ¨
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see Remark 1.2. By (2.5) the equality f∗ · ∂ = F∗ · ∂ ·¨ıH,G;H1 ,G1 implies that f∗ · ∂ · lH,G = (F∗ · ∂ · ¨ıH,G;H1 ,G1 ) · (¨ı−1 ıX,X 0 ;X,G · (B∗ )−1 ) = H,G;X,G · ¨ = F∗ · ∂ · (¨ıH,G;H1 ,G1 · ¨ı−1 ıX,X 0 ;X,G · (B∗ )−1 = H,G;X,G ) · ¨ = F∗ · ∂ · (¨ı−1 ıX,G;X,G1 ) · ¨ıX,X 0 ;X,G · (B∗ )−1 = H1 ,G1 ;X,G1 · ¨ = F∗ · ∂ · ¨ı−1 ıX,X 0 ;X,G1 · (B∗ )−1 = F∗ · ∂ · lH1 ,G1 . H1 ,G1 ;X,G1 · ¨ That is what was required. The same result can be obtained from the commutativity of the diagram ˜ X ˜ \X ˜ 0 ; Q) ˇ m+1 (X, H = B∗ ˇ Hm+1 (X, X \ X 0 ; Q) ¨ıX,X 0 ;X,G ˇ Hm+1 (X, X \ G; Q) = ¨ıH1 ,G;X,G ˇ Hm+1 (H, H \ G; Q) ∂ ˇ m (H \ G; Q) H ˇ f∗ ˇ m (S m ; Q) H 1
ˇ m+1 (X, ˜ X ˜ \X ˜ 0 ; Q) H = B∗ ˇ = Hm+1 (X, X \ X 0 ; Q) ¨ıX,X 0 ;X,G1 ˇ → Hm+1 (X, X \ G1 ; Q) = ¨ıH1 ,G1 ;X,G1 ˇ → Hm+1 (H1 , H1 \ G1 ; Q) ∂ ˇ m (H1 \ G1 ; Q) → H ˇ F∗ ˇ m (S m ; Q). = H 1 =
The lemma is proved. If now under hypotheses (2.1)–(2.3) (2.6) F : H → Rm+1 is a continuous mapping of the compactum H in the Euclidean space Rm+1 and the mapping f = F H\G : H \ G → Rm+1 does not assume the value 0, then to prove the existence of a zero of the mapping f (x) F we can consider the mapping fˇ : H \ G → S1m , fˇ(x) = . Let us f (x) denote by χf the homomorphism fˇ∗ · lH,G . Lemma 2.3. If under hypotheses (2.1)–(2.3), (2.6) the homomorphism χf is not trivial ( ≡ 0), then the mapping F assumes (in a point of the compactum H) the value 0.
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Proof. If the mapping F does not assume in points of the compactum H F (x) the value 0, then the continuous mapping Fˇ : H → S1m , Fˇ (x) = , F (x) extends the mapping fˇ. By Lemma 2.1 the homomorphism (fˇ)∗ is trivial. Therefore, the homomorphism χf is also trivial. But this contradicts the assumption made. The lemma is proved. Let now with notation (2.1) ˜ × I be a continuous (2.7) −∞ < a b < ∞, I = [a, b] and A : X → X ˜ × I. acyclic mapping of a compactum X onto the product X Under hypotheses (2.1) and (2.7) for λ ∈ I and for a subset M of the ˜ compactum X we denote Mλ = M ∩A−1 (X×{λ}). In particular, in this way −1 ˜ we define the symbol Xλ = A (X × {λ}). With supplementary notation (2.3) we define in this way the symbols Hλ = Xλ ∩ H, Gλ = Hλ ∩ G. Let ˜ ×I → X ˜ be the projection onto the factor. Let pλ = p ˜ p : X : X×{λ} ˜ ˜ ˜ X × {λ} → X, Aλ = A : Xλ → X × {λ} and Bλ = p · Aλ . Xλ
For every λ ∈ I condition (2.1) is verified with B = Bλ . So with notation (2.6) we have the possibility to speak about the homomorphism χfλ , where fλ = f H
λ \Gλ
: Hλ \ Gλ → Rm+1 .
By Vietoris–Begle theorem the mapping p·A is acyclic. So letting B = p·A, we get the verification of conditions (2.1)–(2.3) and the possibility to speak about the homomorphism χf .
Lemma 2.4. Under hypotheses (2.1), (2.7), (2.3), and (2.6) for any λ ∈ I the homomorphisms χfλ and χf coincide.
Proof. In the diagram (2.8) ˇ m+1 (X, ˜ X ˜ \X ˜ 0 ; Q) H = (pλ )∗ ˇ ˜ ˜ \X ˜ 0 ) × {λ}; Q) Hm+1 (X × {λ}, (X = (Aλ )∗
=
→
ˇ m+1 (X, ˜ X ˜ \X ˜ 0 ; Q) H = p∗ ˇ ˜ ˜ \X ˜ 0 ) × I; Q) Hm+1 (X × I, (X = A∗
REMARKS ON PERIODIC SOLUTIONS
= (Aλ )∗ ˇ m+1 (Xλ , Xλ \ X 0 ; Q) H λ ¨ıXλ ,Xλ0 ;Xλ ,Gλ ˇ m+1 (Xλ , Xλ \ Gλ ; Q) H = ¨ıHλ ,Gλ ;Xλ ,Gλ ˇ Hm+1 (Hλ , Hλ \ Gλ ; Q) ∂ ˇ Hm (Hλ \ Gλ ; Q) ˇ (fλ )∗ ˇ Hm (S1m ; Q)
439
= A∗ ˇ m+1 (X, X \ X 0 ; Q) → H ¨ıX,X 0 ;X,G ˇ m+1 (X, X \ G; Q) → H = ¨ıH,G;X,G ˇ → Hm+1 (H, H \ G; Q) ∂ ˇ → Hm (H \ G; Q) ˇ f∗ ˇ = Hm (S1m ; Q),
the arrows directed up correspond to isomorphisms (this follows from the Vietoris–Begle theorem or from the excision theorem). The horizontal arrows correspond to homomorphisms, generated by inclusions. Therefore, the assertion of the lemma follows easily from the commutativity of diagram (2.8). The lemma is proved. Corollary. Under hypotheses (2.1), (2.7), (2.3), and (2.6) for every λ, µ ∈ I the homomorphisms χfλ and χfµ coincide. (Because they both coincide with the homomorphism χf .) Let now, in addition to (2.1)–(2.3), and (2.6), ˜ 0 be open ˜ lie in Euclidean space Rm+1 and the set X (2.9) the compactum X m+1 . in the space R ˜ lies in the interior of a ball D. Let Do be the Then the compactum X interior of the ball D. Then domain of definition and range of values of the homomorphism m ˇ ˇ χ ˇf = χf · jD,Do ;X, ˜ X ˜ 0 : Hm+1 (D, D \ Do ; Q) → Hm (S1 ; Q)
ˇ m (S1m ; Q) with are isomorphic to Q. Moreover, we may identify the group H ˇ the group Hm+1 (D, D \ Do ; Q) = Q, because the boundary homomorphism ˇ m+1 (D, D \ Do ; Q) → H ˇ m (D \ Do ; Q) is an isomorphism. So the ∂ : H homomorphism χ ˇf can be considered as an homomorphism from Q to Q. It can be uniquely restored according to the image of the unity of the group Q, which will be called the degree of the mapping f , as usually is done in similar situations.
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Lemma 2.5. If under hypotheses (2.1)–(2.3), (2.6), and (2.9) the mapping F does assume the value 0, then the degree of the mapping f is equal to zero. Proof reduces to the reference to Lemma 2.1. Lemma 2.6. If under hypotheses (2.1)–(2.3), (2.6), and (2.9) the degree of the mapping f is different from zero, then the mapping F assumes (at a point of the compactum H) the value 0. Proof reduces to the reference to Lemma 2.5. Lemma 2.7. Let under hypotheses (2.1), (2.7), (2.3), (2.6), and (2.9) fλ = f H \G : Hλ \ Gλ → Rm+1 for λ ∈ I. Then for all λ, µ ∈ I the λ λ degrees of the mappings fλ and fµ coincide. Proof reduces to the reference to Lemma 2.4. Lemma 2.8. Let, with the notation (2.1), (2.7), (2.3), and (2.9), f : W → Rm+1 be a continuous mapping of an open subset W ⊇ Ha of the and compactum H in Euclidean space Rm+1 , the set f −1 (0) be compact do not intersect the set H \ G. Let the degree of the mapping f Ha \Ga be different from zero. Then for all λ ∈ I the intersection Hλ ∩ f −1 (0) is not empty. Proof. By the normality of the compactum H there exits an open subset V ⊇ f −1 (0), whose closure (in the compactum H) lies in the set W . Lemma The mapping f is defined on the whole set Ha . Therefore, 2.2 implies the coincidence of the degrees of the mappings f H \G , f H \V , a a a a and f [Va ]\Va . The mapping f is defined on the whole set [V ]\V . Therefore, Lemma 2.7 implies the coincidence of the degrees of the mappings f [V ]\V λ λ for all λ ∈ I. So for every λ ∈ I the degree of the mapping f [V ]\V is λ λ different from zero. This fact and Lemma 2.6 imply the nonemptity of the intersection Hλ ∩ f −1 (0). The lemma is proved. Remark 2.1. If under hypotheses of Lemma 2.8 for some λ ∈ I the set W contains Hλ , then the degree of the mapping f H \G coincides with λ λ the degree of the mapping f . This follows from the fact that the Ha \Ga
Lemma mapping f is defined on the whole set (H \ V )λ and, therefore, 2.1 implies the coincidence of the degrees of the mappings f H \G , f H \V λ λ λ λ and f . [Vλ ]\Vλ
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Let now under hypotheses (2.1)–(2.3), and (2.9) (2.10) L = {(x1 , . . . , xm , 0) : x1 , . . . , xm ∈ R} (m-dimensional subspace ˜L = X ˜ ∩ L, X ˜0 = X ˜ 0 ∩ L, XL = B −1 (X), ˜ X0 = of the space Rm+1 ), X L L m −1 ˜ 0 B (X ), HL = H ∩ XL , GL = G ∩ XL , f : H \ G → R be a continuous mapping of the compactum H \ G in the m-dimensional space Rm and let the mapping f do not assume the value 0 on the set HL \ GL , the mapping F : H \ G → Rm+1 be defined by the formula F (x) = (f (x), Bm+1 (x)), where Bm+1 (x) is (m + 1)th coordinate of the point B(x), fL = f HL \GL : HL \ GL → Rm . The degree of the mapping fL is defined in the same way with the change of m by m − 1. The homomorphism δ of the cutting of cycles (see [10]; here we speak about the cutting by the subspace L and by the subspaces which are related to L) is permutable with the action of all homomorphisms, used in the definition of the degree of a mapping. So the diagram (2.11) δ ˇ m+1 (D, D \ Do ; Q) →= H ¨ıD,Do ;D,X˜ 0 δ ˇ ˜ 0 ; Q) →→ Hm+1 (D, D \ X = ¨ıX, ˜ X ˜ 0 ;D,X ˜0
ˇ m+1 (X, ˜ X ˜ \X ˜ 0 ; Q) H = B∗
δ
→→
δ ˇ m+1 (X, X \ X 0 ; Q) →→ H ¨ıX,X 0 ;X,G δ ˇ m+1 (X, X \ G; Q) →→ H = ¨ıH,G;X,G
ˇ m+1 (H, H \ G; Q) H ∂
δ
→→
ˇ m+1 (DL , DL \ Do,L ; Q) H ¨ıDL ,Do,L ;DL ,Do,L ˇ ˜ 0 ; Q) Hm+1 (DL , DL \ X L = ¨ıX˜ L ,X˜ 0 ;DL ,X˜ 0 L L ˇ m+1 (X ˜ \X ˜ 0 ; Q) ˜L, X H L L = B∗ ˇ m+1 (XL , XL \ X 0 ; Q) H L ¨ıXL ,XL0 ;XL ,GL ˇ m+1 (XL , XL \ GL ; Q) H = ¨ıHL ,GL ;XL ,GL ˇ m+1 (HL , HL \ GL ; Q) H ∂
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∂ ˇ m (H \ G; Q) H ˇ F∗ ˇ m (S m ; Q) H 1
∂ δ
→→
δ
→=
ˇ m (HL \ GL ; Q) H ˇ f∗ ˇ m−1 (S m−1 ; Q) H 1
is commutative. Under the cutting of (m + 1)-dimensional ball D we reˇ m+1 (DL , DL \ Do,L ; Q) of ˇ m+1 (D, D \ Do ; Q) → H ceive the isomorphism H the first line of this diagram. (Here DL = D ∩ L and Do,L = Do ∩ L.) Under the cutting of m-dimensional sphere S1m we receive the isomorphism ˇ m (S m ; Q) → H ˇ m−1 (S m−1 ; Q) of the last line of the diagram. H 1 1 The composition of the homomorphisms in the left column of diagram (2.11) coincides with the homomorphism χF and the composition of the homomorphisms in the right column of diagram (2.11) coincides with the homomorphism χfL . So we obtain Lemma 2.9. With the notation (2.1)–(2.3), (2.9), and (2.10) the degrees of the mappings fL and F coincide. Due to the obvious induction on n > m Lemma 2.8 implies Lemma 2.10. Let with notation (2.1)–(2.3), and (2.9), (2.12) m < n, L = {(x1 , . . . , xm , 0, . . . , 0) : x1 , . . . , xm ∈ R} ⊆ Rn (m˜L = X ˜ ∩ L, X ˜0 = X ˜ 0 ∩ L, XL = dimensional subspace of the space Rn ), X L 0 −1 ˜ −1 ˜ 0 B (X), XL = B (X ), HL = H ∩ XL , GL = G ∩ XL , f : H \ G → Rm be a continuous mapping of the compactum H \ G in m-dimensional space Rm , and the mapping f do not assume the value 0 on the set HL \ GL , (2.13) the mapping F : H \ G → Rn be defined by the formula F (x) = (f (x), Bm+1,...,n (x)), where Bm+1,...,n (x) is (n − m)-dimensional vector, composed by (m + 1)th, . . . , nth coordinates of the point B(x), fL = f H \G : HL \ GL → Rm . L L Then the degree of the mappings fL and F coincide. Lemma 2.11. Let, with notation (2.1)–(2.3), (2.9), and (2.12), θ : Rn−m → Rn−m be a nondegenerate linear mapping. Let the mapping F : H \ G → Rn be defined by the formula F (x) = f (x, θ(Bm+1,...,n (x)) and fL (x) = f HL \GL . Then the absolute values of the degrees of the mappings f and fL coincide. Proof. I. If the linear mapping θ preserves the orientation, then it is homotopic with preserving the nondegenaracy to the identity mapping and we obtain what was required from Lemma 2.10.
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II. If the linear mapping θ changes the orientation, then we obtain what was required from item I and from the consideration of the composition of the mapping F with the linear mapping (x1 , . . . , xn+1 , xn ) → (x1 , . . . , xn+1 , −xn ), which changes the orientation in the space Rn and in its subspace Rn−m . The lemma is proved. 3. Homotopy to a degenerated linear system In this section we obtain an analog of Theorem 4.2 in [3]. Our version ˇ has the similar properties in applications to equations. But it uses Cech homologies. Due to remarks of the next section Theorem 4.2 in [3] follows from our result. In order to simplify the comparison with the mentioned theorem we do not tend to the widest generality and we remain at a similar level of assumptions. But we weaken a little restrictions on the right-hand side and we consider differential equations y = f (t, y), where the right-hand side f satisfies the conditions: (3.1) the function f (t, y) is defined on the set U = R × Rn , is continuous in y ∈ Rn , is measurable in t ∈ R and is locally majorized by integrable functions in t, see [5]–[9]. Let us suppose that (3.2) for (t, y) ∈ U and λ ∈ [0, 1] h(t, y, λ) = λf (t, y) + (1 − λ)A(t), where (3.3) A(t) is a square matrix of the order n, whose elements are integrable on the segment [t0 , t1 ], the dimension of the space P of periodic solutions of the equation y = A(t)y on the segment [t0 , t1 ] is equal to m, L = {y(t0 ): y ∈ P }. Let us start with recalling some facts, related to periodic solutions of linear equations. Let us fix m linearly independent periodic solutions ϕ1 , . . . , ϕm of the equation y = A(t)y in such a manner that their initial values ϕ1 (t0 ), . . . , ϕm (t0 ) compose an orthonormalized system. Let us extend the system to a basis ϕ1 , . . . , ϕn of the space of all solutions of the equation y = A(t)y, but preserving the condition that the vectors ϕ1 (t0 ), . . . , ϕn (t0 ) compose an orthonormalized system. We will consider the last system as a basis of the space Rn . Let us denote by C(t) the matrix, in columns of which we have the vectors ϕ1 (t), . . . , ϕn (t). Then (3.4) the columns of the matrix (C ∗ )−1 compose a fundamental system of solutions of the conjugate equation y = −A∗ (t)y. The dimension of the space of periodic solutions (on the same segment [t0 , t1 ]) of the conjugate equation y = −A∗ (t)y is also equal to m, see,
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e.g., [11] (although they deal there with equations with continuous righthand side, but we have not differences in the question under consideration). Let us fix m linearly independent periodic solutions ψ1 , . . . , ψm of the equation y = −A∗ (t)y and extend them to a basis ψ1 , . . . , ψn of the space of all solutions of the equation y = −A∗ (t)y. Note two facts: (3.5) if y1 is an arbitrary solution of the equation y = A(t)y and y2 is an arbitrary solution of the equation y = −A∗ (t)y, then the inner product (y1 (t), y2 (t)) is constant and (3.6) the (unique) solution ∆ of the Cauchy problem y = A(t)y + a(t), t −1 C (s)a(s)ds. y(t0 ) = 0 can be written as ∆(t) = C(t) t0
By (3.6), for every function ψj , j = 1, . . . , n, we have t ∗ (∆(t), ψj (t)) = C ∗ (t)ψj (t), C −1 (s)a(s)ds = ∗
= ψj (t0 ),
t
t0
C −1 (s)a(s)ds =
t0
t =
t
ψj (t0 ), C −1 (s)a(s) ds =
t0
∗ −1 ∗ (C ) (s)ψj (t0 ), a(s) ds =
t0
t (ψj (s), a(s)) ds t0
(in ∗ we take (3.4) into account). In particular, t1 (∆(t1 ), ψj (t1 )) =
(3.7)
(ψj (s), a(s)) ds. t0
Let us note that (3.8) for every solution δ of the equation y = A(t)y and for every function ψj , j = 1, . . . , m, (δ(t1 ) − δ(t0 ), ψj (t1 )) = 0, because due to equality ψj (t0 ) = ψj (t1 ) and (3.5) we have (δ(t0 ), ψj (t1 )) = (δ(t0 ), ψj (t0 )) = (δ(t1 ), ψj (t1 )). Remark 3.1. The mapping Ψ : Rn → Rn , Ψ(u) = yu (t1 ) − u, where yu is the (unique) solution of the Cauchy problem y = A(t)y, y(t0 ) = u, is linear. Its kernel coincides with the set L of initial values of periodic solutions of the equation y = A(t)y. Its dimension was denoted by m. This implies that the dimension of the image Q of the operator Ψ is equal to n − m. By (3.8), if u ∈ Q, then (Ψ(u), ψj (t1 )) = 0 for every function ψj , j = 1, . . . , m. Due to
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the linear independence of the vectors ψj (t1 ), j = 1, . . . , m, the dimension of the space of solutions of the system (u, ψj (t1 )) = 0, j = 1, . . . , m,
(3.9)
is also equal to n − m. Therefore, the space Q coincides with the space of solutions of system (3.9). Remark 3.2. With notation (3.6), every solution of the equation y = A(t)y + a(t) can be written as y + ∆, where y is an arbitrary solution of the equation y = A(t)y. So with the notation of Remark 3.1 if the equation y = A(t)y +a(t) has a periodic solution y0 +∆, then ∆(t1 ) = y0 (t1 )−y0 (t0 ) ∈ Q. Thus, t1 (3.10)
(∆(t1 ), ψj (t1 )) =
(ψj (s), a(s)) ds = 0 t0
for all j = 1, . . . , m. But in the last case the set of periodic solutions of the equation y = A(t)y + a(t) can be written as y0 + P . So its dimension is equal to m (“Fredholm alternative”). Let, with the notation (3.1)–(3.3), (3.11) for i = 1, . . . , m, the mapping fi∗ : R → R be defined by the formula fi∗ (u)
t1 (f (t, yu (t)) − A(t)yu (t), ψi (t)) dt,
= t0
where yu is the (unique) solution of the Cauchy problem y = A(t)y, y(t0 ) = u, and for i = m + 1, . . . , n the mapping fi∗ : R → R be defined by the formula fi∗ (u) = (yu (t1 ) − yu (t0 ), ψi (t1 )). Theorem 3.1. Let under hypotheses (3.1)–(3.3) and (3.11) (3.12) W be an open subset of the space Cs+ (U, t0 ) and the set W ∗ = {z(t0 ) : z ∈ W } be a bounded subset of the space Rn , (3.13) for every λ ∈ (0, 1), every solution y ∈ ∂W of the equation y = h(t, y, λ) (defined on the segment [t0 , t1 ]) satisfy the condition y(t0 ) = y(t1 ). Let us suppose that (3.14) there exists a number r > 0 such that for every λ ∈ (0, 1) every solution y of the problem y = h(t, y, λ), y(t0 ) = y(t1 ) ∈ [W ] satisfies the condition y r. Let (3.15) the mapping F ∗ : Γ ∩ L → Rm , defined by the formula F ∗ (u) = ∗ (f1∗ (u), . . . , fm (u)), where Γ = {z(t0 ): z ∈ ∂W }, do not assume the value 0 and the degree of the mapping F ∗ be different from zero. Then the absolute
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value of the degree of the mapping F : Γ → Rn , defined by the formula F (u) = (f1∗ (u), . . . , fn∗ (u)), coincides with the absolute value of the degree of the mapping F ∗ (and, therefore, it is also different from zero) and the equation y = f (t, y) has a solution y on the segment [t0 , t1 ], for which y(t0 ) = y(t1 ) ∈ [W ]. The symbol Cs+ (U, t0 ) in condition (3.12) denotes the set of all continuous functions z : [t0 , a) → Rn (for all a > t0 ), whose graphs are closed subsets of the set U , see details in [2], where the used topological structure on the set Cs+ (U, t0 ) is described. Before starting the proof, let us note first that the mapping of the set D of solutions of the equation y = A(t)y, belonging to C + (U, t0 ) (they all are defined on the half-open interval [t0 , ∞)), which associates with a solution z its initial value z(t0 ), is an homeomorphism. So the set {z(t0 ): z ∈ W ∩ D} is open in the space Rn , and its boundary lies in Γ. Proof of Theorem 3.1. I. Let us fix arbitrarily ε ∈ (0, 1). ˜ ⊆ Rn , Due to the boundedness of the set W ∗ , there exists a compactum X 0 ∗ ˜ ˜ also which contains the set W . The interior X of the compactum X ∗ contains the set W . Let X be the set of couples (y, λ), where λ ∈ [0, ε] and y is a solution of the Cauchy problem y = h(t, y, λ), y(t0 ) ∈ [W ]. ˜ × [0, ε] by the formula A(y, λ) = We define the mapping A : X → X (y(t0 ), λ). We denote the set of couples (y, λ), where λ ∈ [0, ε] and y ∈ [W ] is a solution of the equation y = h(t, y, λ), by H. The set G of couples (y, λ) ∈ H, in which y ∈ W , is open in the space X. II. We define the mapping F˜ : H → Rn in the following way. Let its ith coordinate function for λ > 0 assume at the point (y, λ) ∈ H the value (y(t1 ) − y(t0 ), ψi (t1 )) for i = 1, . . . , m and the value (y(t1 ) − y(t0 ), ψi (t1 )) λ for i = m + 1, . . . , n. Let its ith coordinate function assume at the point (y, 0) ∈ H the value F˜ (y, 0) = F (y(t0 )) = (f1∗ (y(t0 )), . . . , fn∗ (y(t0 ))) . The continuity of (m + 1), . . . , nth coordinate function of the mapping F˜ is obvious everywhere. The continuity of first, . . . , mth coordinate function of the mapping F˜ is obvious everywhere, except for λ = 0, and for λ ≡ 0. In order to prove its continuity in the whole volume we need to check that for every sequence (yj , λj ) → (y0 , 0), where λj ∈ (0, ε], for every i = 1, . . . , m we (y(t1 ) − y(t0 ), ψi (t1 )) have: → fi∗ (y0 (t0 )). λ
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447
Consider the (unique) solution xj of the Cauchy problem y = A(t)y, y(t0 ) = yj (t0 ). yj (t) − xj (t) For ∆j (t) = we have: λj ∆ j − A(t)∆j (t) = f (t, yj (t)) − A(t)yj (t). Therefore, by (3.7), we have t1 (ψi (s), f (s, yj (s)) − A(s)yj (s)) ds →
(∆j (t1 ), ψi (t1 )) = t0
(3.16)
t1 →
(ψi (s), f (s, y0 (s)) − A(s)y0 (s)) ds = fi∗ (y0 (t0 )).
t0
xj (t) is a solution of the equation y = A(t)y. λj Therefore, by (3.8) and (3.16), we have
yj (t1 ) − yj (t0 ) , ψi (t1 ) = λj
xj (t1 ) xj (t0 ) yj (t1 ) − xj (t1 ) , ψi (t1 ) + − , ψi (t1 ) = = λj λj λj
yj (t1 ) − xj (t1 ) , ψi (t1 ) = (∆j (t1 ), ψi (t1 )) → fi∗ (y0 (t0 )). = λj The function δ(t) =
This gives the required continuity. III. Now we are under hypotheses of Lemma 2.4. We will use the notation from Sec. 2, concerning this situation. The mapping H0 → Rn−m , which associates the point ((y(t1 ) − y(t0 ), ψm+1 (t1 )), . . . , (y(t1 ) − y(t0 ), ψn (t1 ))) with a point (y, 0) of the compactum H0 is equal to the composition of the continuous mapping (y, 0) → y(t0 ) and the nondegenerate linear mapping Ψ from Remark 3.1. Lemma 2.11 implies the coincidence of absolute values of the degrees of the mappings F ∗ and F . IV. Lemma 2.8 implies the existence of a solution yε ∈ [G] of the equation y = h(t, y, ε), for which y(t0 ) = y(t1 ). V. Condition (3.14) implies the compactness of the set M of couples (y, λ), where λ ∈ [0, 1] and y ∈ [W ] is a solution of the equation y = h(t, y, λ), for which π(y) = [t0 , t1 ] and y(t0 ) = y(t1 ). VI. Due to IV, for every λ ∈ (0, 1) there exists a solution yλ ∈ [W ] of the equation y = h(t, y, λ), for which π(yλ ) = [t0 , t1 ] and yλ (t0 ) = yλ (t1 ).
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V. V. FILIPPOV
VII. Due to the compactness of the set M (see IV) and due to V there exists a sequence of functions {yλi : i = 1, 2, . . . } of the kind mentioned in VI, for which λi → 1, and which converges to a solution y1 ∈ [W ] of the equation y = h(t, y, 1) (= f (t, y)). For it y1 (t0 ) = y1 (t1 ). The theorem is proved. Let us denote the space of all continuous functions [t0 , t1 ] → Rn by C([t0 , t1 ], Rn ). Remark 3.3. The set Cs+ (U ; t0 , t1 ) of functions from Cs+ (U ; t0 ), whose domains contain the segment [t0 , t1 ], is open in the space Cs+ (U, t0 ). The n + mapping α : Cs (U ; t0 , t1 ) → C([t0 , t1 ], R ), α(z) = z [t ,t ] is continuous. 0 1 ˜ = So if W is an open subset of the space Cs+ ([t0 ,t1 ], Rn ), then the set W + −1 α (W ) (= {z: z ∈ Cs (U, t0 ), π(z) ⊇ [t0 , t1 ], z [t ,t ] ∈ W }) is open in the 0 1 space Cs+ (U, t0 ). Another version of the proved fact can be stated as follows. Theorem 3.2. Let under hypotheses (3.1)–(3.3), and (3.11) W be an open bounded subset of the space C([t0 , t1 ], Rn ). Let conditions (3.13), (3.14), and (3.15) hold. Then for every ε ∈ (0, 1) the absolute value of the degree of the mapping F˜ Hε \Gε coincides with the absolute value of the degree of the mapping F ∗ (therefore, it is also different from zero) and the equation y = f (t, y) has a solution y on the segment [t0 , t1 ], for which y(t0 ) = y(t1 ) ∈ [W ]. Proof reduces to the reference to Theorem 3.1 (where we change the set ˜ , see Remark 3.3) and to Remark 2.1. W by the set W 4. On limit passages in the space of solution spaces We will discuss here the continuity of the dependence of the homomorphism χ... on parameters of the equation in question. More precisely, we will prove the local constantness of this homomorphism. Let (4.1) U = R × Rn , Z ∈ Rceh (U ), t0 , t1 ∈ R, t0 < t1 , a compactum ∆ lie in Rn , ∆0 be an open subset of the compactum ∆, {t0 } × ∆ ⊆ U and every solution of the Cauchy problem z ∈ Z + , inf π(z) = t0 , z(t0 ) ∈ ∆ be defined on the whole segment [t0 , t1 ], (4.2) W be a neighborhood of the set {z: z ∈ Z, π(z) = [t0 , t1 ], z(t0 ) ∈ ∆} in the space Cs ([t0 , t1 ], Rn ) and F : W → S be a continuous mapping of the set W into a compactum S. For Y1 , Y2 ∈ R(U ) and r ∈ [t0 , t1 ] we denote by Y1 ∗r Y2 the set of functions z ∈ Cs (U ), which satisfy at least one of the following three conditions: (1) z ∈ Y1 and sup π(z) r, (2) z ∈ Y2 and inf π(z) r, (3) r ∈ [t0 , t1 ],
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z
∈ Y ∈ Y2 , see Sec. VIII.7 in [6] or Sec. 11.5 in [8], where and z 1 [t0 ,r] [r,t1 ] a more general construction is considered. Now let (4.3) a sequence of spaces {Zj : j = 1, 2, . . . } ⊆ Rceh (U ) converge in U to a space Z. Let Yj = ∪{Z ∗r Zj : r ∈ [t0 , t1 ]}. Beginning with some j = j0 , every solution of the Cauchy problem z ∈ Yj+ , inf π(z) = t0 , z(t0 ) ∈ ∆ is defined on the whole segment [t0 , t1 ] and z [t ,t ] ∈ W . This follows from assertions 0 1 of Secs. VI.2 and VIII.7 in [6] or from assertions of Secs. 11.5 and 10.4 in [8] and from the closedness of the set {z: z ∈ Cs+ (U, t0 ), t1 < sup π(z)} in the + space Cs (U, t0 ), see [2]. We can define the mapping g : K+ → S, g(z, r) = F (z [t ,t ] ), where K = {(z, r): r ∈ [t0 , t1 ], z ∈ (Z ∗r Zj ) , inf π(z) = t0 , 0 1 z(t0 ) ∈ ∆}. Let U = {(z, r): r ∈ [t0 , t1 ], z ∈ (Z ∗r Zj )+ , inf π(z) = t0 , z(t0 ) ∈ ∆0 }. We have defined the acyclic mapping A : K → ∆ × [t0 , t1 ]. By the corollary of Lemma 2.4 the homomorphisms χgt0 = kgt0 ;Kt0 ,Ut0 ;Kt0 ,Ut0 · Bt−1 and χgt1 = kgt1 ;Kt1 ,Ut1 ;Kt1 ,Ut1 · Bt−1 0 1 coincide. (We define the mapping B in the same way as in Sec. 2.) But the homomorphisms χgt0 and χgt1 coincide with the homomorphisms χfj and χf , respectively, where fj : K → S, fj (z) = F (z [t ,t ] ) and 0
1
K = {z: r ∈ [t0 , t1 ], z ∈ Zj+ , inf π(z) = t0 , z(t0 ) ∈ ∆}, f : K 0 → S, f (z) = F (z [t ,t ] ) and K 0 = {z: r ∈ [t0 , t1 ], z ∈ Z + , inf π(z) = t0 , z(t0 ) ∈ ∆}. 0 1 Thus we have proved Assertion 4.1. Under hypotheses (4.1)–(4.3), beginning with some j = j0 , the homomorphisms χf and χfj coincide. This is just the property of the homomorphism χ... , mentioned at the beginning of this section. Remark 4.1. Let the right-hand side f of a differential equation y = f (t, y) be continuous on U = R × Rn and the condition of the uniqueness of solution of the Cauchy problem for the equation y = f (t, y) be satisfied. ˜ lie in the space Rn , the set X ˜ be open ˜0 ⊆ X Let t0 , t1 ∈ R, a compactum X n in the space R , and suppose that every solution of the Cauchy problem ˜ can be extended to the whole segment [t0 , t1 ]. With y = f (t, y), y(t0 ) ∈ X notation (2.1)–(2.4), let X ⊆ Cs+ (U, t0 ) denote the set of solutions of the ˜ and let the mapping B associate with Cauchy problem y = f (t, y), y(t0 ) ∈ X a function z ∈ X its value z(t0 ) at the point t0 . Let H = X, G = X 0 , and let the mapping F : H → Rn be defined by the formula F (u) = yu (t1 ) − u, where yu ∈ Cs+ (U, t0 ) denotes the unique solution of the Cauchy problem y = f (t, y), y(t0 ) = u. In this case the degree of the mapping F , defined as in Sec. 2, coincides with the degree of the translation along trajectories
450
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of the equation y = f (t, y) and we can use all results of the corresponding theory, see, e.g., [12]. Remark 4.2. Let, with notation (4.1), the space Z be the space of solutions of the equation y = f (t, y). When we use the Leray–Schauder theory to discuss the existence of periodic solutions for the equation, we consider, for instance, the integral operator t z = A1 (y),
f (s, y(s))ds.
z(t) = y(t1 ) + t0
It is shown in [12] (Theorem 28.5, named fundamental) that the degree of the translation (i.e., of our mappings Fλ , λ ∈ (0, 1), in Theorem 3.1) coincides with the Leray–Schauder degree of this integral operator under the assumptions of the continuity of the function f and of the uniqueness of the solution of the Cauchy problem for the equation y = f (t, y), see Remark 4.1. Assertion 4.1, the analogous property of preserving the degree for integral operators under limit passage (see the fourth step of the proof of Theorem 28.5 in [12]), and Weierstrass theorem on approximation of continuous functions by polynomials extend this theorem to an equation y = f (t, y) with continuous right-hand side f (therefore, we reject the condition of the uniqueness of the solution of the Cauchy problem). The same arguments when we use the approximation of the right-hand side which is mentioned in Remark 4.13.1 in [8] (see also Example 12.3.2 in [8] or Example 4.1 in [9]) give us an analogous result for the equation y = f (t, y), whose right-hand side f satisfies the Caratheodory conditions. Thus, Theorem 4.2 in [3] is a consequence of our Theorem 3.2 and Lemma 2.7.
References 1. V. V. Filippov, Mixture of Leray–Schauder and Poincar´e–Andronov methods in the problem on periodic solutions for ordinary differential equations. (Russian) Differ. Equat. 35 (1999), No. 12, 1709–1711. 2. , Homology properties of the sets of solutions to ordinary differential equations. (Russian) Mat. Sb. 188 (1997), No. 6, 139–160. English translation: Russ. Acad. Sci. Sb. Math. 188, No. 6, 933–953. 3. J. Mawhin, Continuation theorems and periodic solutions of ordinary differential equations. In: Topological methods in differential equations and inclusions, Kluwer Academic Publishers, Dordrecht–Boston–London, 1995, 291–376.
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´ 4. N. Rouche and J. Mawhin, Equations diff´erentielles ordinaires. Vol. 2, Masson, Paris, 1973. English translation: Ordinary differential equations, Stability and periodic solutions, Pitman, Boston, 1980. 5. V. V. Filippov, Topological structure of the solution spaces of ordinary differential equations. (Russian) Usp. Mat. Nauk 48 (1993), No. 1, 103–154. 6. , Solution spaces of ordinary differential equations. Moscow University Publishers, Moscow, 1993. , Basic topological structures of the theory of ordinary dif7. ferential equations. In: Topology in Nonlinear Analysis, Banach center publications, Vol. 35, 1996, 171–192. 8. , Basic topological structures of ordinary differential equations. Kluwer Academic Publishers, Dordrecht–Boston–London, 1998. , On the theory of the Cauchy problem for an ordinary differ9. ential equation with discontinuous right-hand side. Mat. Sb. 185 (1994), No. 11, 95–118. English translation: Russ. Acad. Sci. Sb. Math. 83 (1995), No. 2, 383–403. 10. P. S. Aleksandrov, Introduction to the homological dimension theory. Nauka, Moscow, 1975. 11. Ph. Hartman, Ordinary differential equations. John Wiley & sons, New York–London–Sydney, 1964. 12. M. A. Krasnosel’skii and P. P. Zabreiko, Geometrical methods of nonlinear analysis. (Russian) Nauka, Moscow, 1975. English translation: Springer, Berlin, 1984.
(Received 01.03.2000) Authors’ addresses: Department of Mechanics and Mathematics, Moscow State University 119899 Moscow, Russia E-mail: vvfi
[email protected]