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Periodic Solutions in a Given Set of Differential Systems
Journal of Mathematical Analysis and Applications 264, 495᎐509 Ž2001. doi:10.1006rjmaa.2001.7683, available online at http:rrwww.idealibrary.com on
Periodic Solutions in a Given Set of Differential Systems Jan Andres1 Department of Mathematical Analysis, Faculty of Science, Palacky ´ Uni¨ ersity, Tomko¨ a 40, 779 00 Olomouc-Hejcın, ˇ´ Czech Republic E-mail: [email protected]
and Bohumil Krajc 2 Department of Applied Mathematics, Faculty of Electrical Engineering and Computer Science, Technical Uni¨ ersity of Ostra¨ a, 17 Listopadu, 708 33 Ostra¨ a-Poruba, Czech Republic E-mail: [email protected] Submitted by Raul ´ Manase ´ ¨ ich Received November 28, 2000
The existence of a periodic solution in a given set of Caratheodory differential ´ systems is studied. The proof of the main statement is based on the Wazewski-type ˙ approach jointly with the degree arguments. An illustrating example is supplied, and a comparison with the analogous results of other authors is indicated. 䊚 2001 Elsevier Science
Key Words: periodic solutions; given sets; Caratheodory systems; degree argu´ ments; Wazewski-type result. ˙
1. INTRODUCTION The study of periodic solutions to differential systems which remain in a given set is mostly related to the Žpositive or negative. flow invariance of this set. M. Nagumo’s classical result, already formulated in 1942, is a typical example of a so-called weak flow invariance, when there is not necessarily uniqueness for Cauchy problems. This theorem has been 1
Supported by the Council of the Czech Government ŽJ 14r98:153100011. and by a grant ˇ ŽF 0158r2000.. from the FRVS 2 Supported by the Council of the Czech Government ŽCEZ MSM272400019.. 495 0022-247Xr01 $35.00 䊚 2001 Elsevier Science All rights reserved.
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redeveloped and extended several times, with the use of various types of tangent cones, which allows us to obtain the existence results for periodic solutions by means of degree arguments. For the related references, see, e.g., the handbook wHPx. On the other hand, the case where no invariance is assumed is still far from obvious. Usually, the transversality behavior on the boundary of a given domain is required in terms of so-called canonical domains, Wazew˙ ski sets, bound sets, etc., verified by means of guiding or bounding functions, etc. Žsee, e.g., wA1, A2, AGJ, AMT, B, BK, BS, C1, C2, FZ, GGL, GM, GS, Kr1, Kr2, LW, M, R, Sa, Sr1᎐Sr3, T, Zx.. In wAK, Kcx, we considered entirely bounded solutions in given sets, when using simultaneously compression- and expansion-type conditions, similar to typical Wazewaki pictures Žalthough not applying the Wazewski ˙ ˙ topological principle, as developed and improved in wW, BS, H, JKx, etc... These results can also be used, for planar systems with uniqueness for Cauchy problems, to guarantee periodic oscillations, on the basis of the well-known Massera transformation theorem, which, however, is no longer true in ⺢ n, where n ) 2. Here we investigate again the problem given in the title by means of the Wazewski-type approach combined with degree arguments. Although the ˙ main result seems to be practically covered by geometric methods of R. Srzednicki in wSr1, Sr3x Žsee Remark 3 here., the new ideas in the proof make it original and still rather general. We also add an illustrating example in the concluding part.
2. SOME PRELIMINARIES Consider the differential system xX s f Ž t , x . ,
Ž 1.
where f : w0, 1x = ⺢ n ª ⺢ n satisfies Ži. f Ž t, ⭈ . is continuous, for a.a. t g w0, 1x; Žii. f Ž⭈, x . is Lebesgue measurable, for every x g ⺢ n ; Žiii. for every r g ⺢q, there exists r g L Žw0, 1x, ⺢q. such that, for a.a. t g w0, 1x, the implication <
«
f Ž t , . F r Ž t . .
. solutions x g A C Žw0, 1x, ⺢ n . of holds. We are interested in ŽCaratheodory ´ Ž1., satisfying x Ž 0. s x Ž 1. , Ž 2. the values of which are located in a given set ⌰.
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DEFINITION 1. Given a bounded open set ⭋ / ⌰ ; ⺢ n with the boundary frŽ ⌰ ., Ži. We call a point g frŽ ⌰ . an ingress point Žinto ⌰ . if, for every tU g Ž0, 1x and every solution ␥ of Ž1. in w0, 1x, the implication
␥ Ž tU . s
« ᭚⑀ g ⺢q
᭙ t g Ž tU y ⑀ , tU . : ␥ Ž t . g ⺢ n _cl Ž ⌰ .
takes place. Žii. We call a point g frŽ ⌰ . an egress point Žfrom ⌰ . if, for every U t g w0, 1. and every solution ␥ of Ž1. in w0, 1x, the implication
␥ Ž tU . s
« ᭚⑀ g ⺢q
᭙ t g Ž tU , tU q ⑀ . : ␥ Ž t . g ⺢ n _cl Ž ⌰ .
takes place. The set of ingress or egress points of ⌰ will be denoted by ⌫q or ⌫y, respectively. DEFINITION 2. Given a bounded open set ⭋ / ⌰ ; ⺢ n with the boundary frŽ ⌰ ., we say that the quadruple Ž ⌰, h, ⍀, g . is admissible w.r.t. Ž1. if the following conditions are satisfied: Ži. every boundary point Ži.e., of frŽ ⌰ .. is either an ingress or an egress point of ⌰ Žw.r.t. solutions of Ž1..; Žii. h is a homeomorphism of ⺢ n into itself and hŽ ⍀ . s ⌰; Žiii. g: ⺢ n ª ⺢ n is a continuous mapping such that Ža. if hŽ . g ⌰ j ⌫q and g w0, 1., then hŽ y Ž1 y . g Ž .. f q ⌫ ; Žb. if hŽ . g ⌫y and g w0, 1., then hŽ y Ž1 y . g Ž .. g n ⺢ _clŽ ⌰ .; Žc. degŽ g, ⍀, 0. / 0, where deg stands for the Brouwer degree. Remark 1. Denote wy1, 1x 0 = wy1, 1x s wy1, 1x = wy1, 1x 0 s wy1, 1x and assume that n
hy1 Ž ⌰ . s ⍀ s Ž y1, 1 . , hy1 Ž ⌫q . s
s g 0, . . . , n4 ,
nys
D Ž wy1, 1x sqiy1 = y1, 14 = wy1, 1x nysyi . , is1
hy1 Ž ⌫y . s
sy1
D Ž wy1, 1x j = y1, 14 = wy1, 1x nyjy1 . . js0
Defining g: ⺢ ª ⺢ , n
n
g Ž 1 , . . . , n . s 12 Ž y 1 , . . . , y s , sq1 , . . . , n . ,
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one can easily check that g satisfies conditions Žiii.Ža., Žb. and, since g is odd, Žiii.Žc.. Another possibility for satisfying Žiii.Žc. is that ⍀ is a so-called AR-space Žin particular, a retract of a convex set. and g is a self-mapping Ži.e., into. Žcf., e.g., wGx..
3. MAIN THEOREM THEOREM. Let Ž ⌰, h, ⍀, g . be an admissible quadruple w.r.t. Ž1.. Then problem Ž1. ᎐ Ž2. admits a solution with ¨ alues located in ⌰. Proof. We proceed in eight steps. ⺢
n.
Ži. At first, define ᒏ: C Žw0, 1x, clŽ ⍀ .. = clŽ ⍀ . = w0, 1x ª C Žw0, 1x, by ᒏ Ž q, , . Ž t . s
½
hy1 Ž ␥ Ž t . . ,
for t g w 0, x ,
hy1 Ž ␥ Ž . . ,
for t g Ž , 1 x ,
where ␥ g A C Žw0, 1x, ⺢ n .,
␥ Ž t . s hŽ . q
t
H0 f Ž s, h Ž q Ž s . . . d s.
Žii. Furthermore, define ᒐ: C Žw0, 1x, clŽ ⍀ .. = clŽ ⍀ . = w0, 1x ª ⺢ n by ᒐ Ž q, , . s g Ž ᒏ Ž q, , . Ž . . . Žiii. At last, define ᑪ s Ž ᑪ 1 , ᑪ 2 .: C Žw0, 1x, clŽ ⍀ .. = clŽ ⍀ . = w0, 1x ª C Žw0, 1x, ⺢ n . = ⺢ n, by ᑪ 1 Ž q, , . s q y ᒏ Ž q, , . , ᑪ 2 Ž q, , . s y ᒏ Ž q, , . Ž . y Ž 1 y . ᒐ Ž q, , . . Živ. Let us fix g w0, 1x, q g C Žw0, 1x, clŽ ⍀ .., and g clŽ ⍀ . and denote ␥ s h( q. One can then observe that ᑪ Ž q, , . s 0
Ž 3.
X
␥ Ž t. s f Ž t, ␥ Ž t. . ,
«
for a.a. t g w 0, x , for every t g Ž , 1 x ,
Ž 4.
␥ Ž t . s ␥ Ž . ,
Ž 5.
q Ž 0. s s q Ž . y Ž 1 y . g Ž q Ž . . .
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499
In particular, for s 1, we have
␥X Ž t. s f Ž t, ␥ Ž t. . ,
for a.a. t g w 0, 1 x , ␥ Ž 0 . s ␥ Ž 1 . ,
i.e., the solution of Ž1. ᎐ Ž2.. To verify the solvability of ᑪ Ž q, , 1. s 0, we show that deg Ž ᑪ Ž ⭈, ⭈ , 1 . , C Ž w 0, 1 x , ⍀ . = ⍀ , 0 . / 0, where deg stands for the Leray᎐Schauder degree. Žv. Now we show that, for every g w0, 1x and every pair Ž q, . g frŽ C Žw0, 1x, ⍀ . = ⍀ ., we have ᑪ Ž q, , . / 0. Ža. At first, we prove by contradiction the implication
g w 0, 1 x ,
q g fr Ž C Ž w 0, 1 x , ⍀ . . ,
g⍀
«
ᑪ Ž q, , . / 0.
Let there exist g w0, 1x, q g frŽ C Žw0, 1x,⍀ .., and g ⍀ such that ᑪ Ž q, , . s 0. Then q Ž . g frŽ ⍀ ., for some g w0, 1x, and, according to assumption Ži. in Definition 2, hŽ q Ž .. is an ingress or egress point of ⌰. For g Ž0, ., Ž3. implies ᭚⑀ g ⺢q such that either hŽ q Ž t .. g n ⺢ _clŽ ⌰ ., for all t g Ž y ⑀ , ., or hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž , q ⑀ ., takes place, respectively. This is, however, a contradiction of the choice of q g frŽ C Žw0, 1x, ⍀ ... Let s 0, g w0, 1.. If s 0, then Ž5. implies g Ž q Ž0.. s 0, which is a contradiction of condition Žiii.Žc. in Definition 2 Žactually, it is rather a contradiction of conditions Žiii.Ža. and Žiii.Žb., which guarantee that the degree in Žiii.Žc. is defined.. For g Ž0, 1. in Ž5. we know that q Ž0. s q Ž . y Ž1 y . g Ž q Ž .., where Ž q . g clŽ ⍀ .. If hŽ q Ž0.. g ⌫y, then ᭚⑀ g ⺢q : hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for every t g Ž0, ⑀ ., which is a contradiction of the choice of q g frŽ C Žw0, 1x, ⍀ ... Therefore, we can assume that hŽ q Ž0.. g ⌫q, which, however, contradicts conditions Žiii.Ža., Žb. in Definition 2. Let s 0, s 1. Since Ž5. implies q Ž0. s q Ž1., we can proceed quite analogously to the foregoing case. So, ᭚⑀ g ⺢q such that either hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž1 y ⑀ , 1., or hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž0, ⑀ ., take place, respectively. However, this is again a contradiction of the choice of q g frŽ C Žw0, 1x, ⍀ ... Let Ž0, 1x 2 G g w0, 1.. If s 0, then can be taken, according to Ž4., to be equal to 0 as well. Put g Ž0, 1.. Then Ž4. implies that q Ž . s q Ž . and Ž3. implies that hŽ q Ž .. is not an ingress point into ⌰. Otherwise, ᭚⑀ g ⺢q such that hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž y ⑀ , .. 䢇
䢇
䢇
䢇
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If hŽ q Ž .. s hŽ q Ž .. is an egress point from ⌰, then in view of Ž5. and condition Žiii.Žb. in Definition 2, the initial value hŽ q Ž0.. does not belong to clŽ ⌰ ., which is a contradiction of the choice of q g frŽ C Žw0, 1x, ⍀ ... Let s s 1. Then Ž5. implies q Ž0. s q Ž1. and Ž3. implies ᭚⑀ g ⺢q such that either hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž1 y ⑀ , 1., or hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž0, ⑀ ., takes place, respectively. This is, however, a contradiction of the choice of q g frŽ C Žw0, 1x, ⍀ ... Žb. Now we prove again by contradiction the implication 䢇
g w 0, 1 x ,
q g C Ž w 0, 1 x , cl Ž ⍀ . . ,
g fr Ž ⍀ .
«
ᑪ Ž q, , . / 0.
Hence, let there exist g w0, 1x, q g C Žw0, 1x, clŽ ⍀ .., and g frŽ ⍀ . such that ᑪ Ž q, , . s 0. Let us employ that s q Ž0.. Let g w0, 1.. If s 0, then Ž5. implies that g Ž q Ž0.. s 0, which contradicts Žiii.Žc. in Definition 2. For g Ž0, 1., we know according to Ž5. that s q Ž0. s q Ž . y Ž1 y . g Ž q Ž .., where q Ž . g clŽ ⍀ .. If hŽ q Ž0.. g ⌫y, then ᭚⑀ g ⺢q : hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž0, ⑀ ., which is a contradiction of the choice of q g C Žw0, 1x, clŽ ⍀ ... Therefore, we can assume that hŽ q Ž0.. g ⌫q, which is, however, a contradiction of conditions Žiii.Ža., Žb. in Definition 2. Let s 1. Then Ž5. implies q Ž0. s q Ž1.. Thus, ᭚⑀ g ⺢q such that either hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž1 y ⑀ , 1., or hŽ q Ž t .. g ⺢ n _ clŽ ⌰ ., for all t g Ž0, ⑀ ., takes place, respectively, which is a contradiction of the choice of q g C Žw0, 1x, clŽ ⍀ ... Žvi. Furthermore, we show that the image of the operator 䢇
䢇
ᑯ : C Ž w 0, 1 x , cl Ž ⍀ . . = cl Ž ⍀ . = w 0, 1 x ª C Ž w 0, 1 x , ⺢ n . = ⺢ n ,
Ž 6.
ᑯ Ž q, , . s Ž q, . y ᑪ Ž q, , .
is relatively compact. q⬁ Let us consider the sequence Ž z n , n .4ns1 of points from the image of q⬁ ᑯ. Let Ž qn , n , n .4ns1 be a sequence such that Ž qn , n . y ᑪ Ž qn , n , n . s Ž z n , n ., for all positive integers n. q⬁ Ža. At first, let us consider the sequence ␥n4ns1 of absolutely continuous solutions of related problems, i.e., the function sequence defined by
␥n Ž t . s h Ž n . q
t
H0 f Ž s, h Ž q Ž s . . . d s, n
We prove that ␥n : n g ⺞4 is relatively compact.
t g w 0, 1 x .
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PERIODIC SOLUTIONS FOR DIFFERENTIAL SYSTEMS
According to the well-known Ascoli lemma, it is sufficient to show that all functions from our set are uniformly bounded and equicontinuous. For a suitable r g ⺢q, it obviously holds that ᭙ t g w 0, 1 x : ␥n Ž t . s h Ž n . q
F h Ž n . s F K⌰ q
1
t
H0 f Ž s, h Ž q Ž s . . . d s n
1
H0
f Ž s, h Ž qn Ž s . . . d s
r Ž s . d s,
H0
where K ⌰ s max< < : g clŽ ⌰ .4 - q⬁ and, due to the Caratheodory ´ conditions, the Lebesgue integral on the right-hand side is finite. The equicontinuity follows easily as well; because of a finiteness of the integral of r , for every g ⺢q there is ␦ g ⺢q such that, for all n g ⺞ and t, p from w0, 1x with < t y p < - ␦ , we arrive at
␥n Ž t . y ␥n Ž p . F
t
Hp
f Ž s, h Ž qn Ž s . . . d s F
t
HP Ž s . d s r
- .
q⬁ Žb. Let n k 4ks1 be a sequence of positive integers such that
lim ␥n k s ␥ ,
lim n k s .
kªq⬁
kªq⬁
Then
¡h Ž ␥ Ž t . s~ y1
znk
¢h
y1
nk
for t g 0, n k ,
Ž t..,
Ž ␥n Ž n . . , k
for t g Ž n k , 1 .
k
Putting zŽ t. s
½
hy1 Ž ␥ Ž t . . ,
for t g w 0, x ,
hy1 Ž ␥ Ž . . ,
for t g Ž , 1 x ,
we show that lim z n k s z.
kªq⬁
Choose g ⺢q. The function hy1 (␥ is uniformly continuous on w0, 1x, by which there exists ␦ g ⺢q such that < hy1 Ž␥ Ž t .. y hy1 Ž␥ Ž p ..< - 2 ,
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whenever < t y p < - ␦ , t g w0, 1x, p g w0, 1x. Let k 0 g ⺞ be such that < n y < - ␦ , k
sup tg w0, 1 x
hy1 Ž ␥n kŽ t . . y hy1 Ž ␥ Ž t . . -
2
,
for all ⺞ 2 k ) k 0 . Observe that the latter inequality is correct, because the set ␥n k : k g ⺞4 is bounded. If t g w0, min n k, 4x, then z n kŽ t . y z Ž t . s hy1 Ž ␥n kŽ t . . y hy1 Ž ␥ Ž t . . -
2
-,
for ⺞ 2 k ) k 0 . If t g Žmax n k, 4 , 1x, then z n kŽ t . y z Ž t . s hy1 Ž ␥n k Ž n k . . y hy1 Ž ␥ Ž . . F hy1 Ž ␥n k Ž n k . . y hy1 Ž ␥ Ž n k . . q hy1 Ž ␥ Ž n k . . y hy1 Ž ␥ Ž . . -
2
q
2
s,
for ⺞ 2 k ) k 0 .
If t g w n k, x, then < n k y t < - ␦ , and, subsequently, z n kŽ t . y z Ž t . s hy1 Ž ␥n k Ž n k . . y hy1 Ž ␥ Ž t . . F hy1 Ž ␥n k Ž n k . . y hy1 Ž ␥ Ž n k . . q hy1 Ž ␥ Ž n k . . y hy1 Ž ␥ Ž t . . -
2
q
2
s,
for ⺞ 2 k ) k 0 .
If t g w , n k x, then < y t < - ␦ , and, subsequently, z n kŽ t . y z Ž t . s hy1 Ž ␥n kŽ t . . y hy1 Ž ␥ Ž . . F hy1 Ž ␥n kŽ t . . y hy1 Ž ␥ Ž t . . q hy1 Ž ␥ Ž t . . y hy1 Ž ␥ Ž . . - q s, for ⺞ 2 k ) k 0 . 2 2
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503
Thus, we demonstrated that the set z n : n g ⺞4 is relatively compact. It just remains to realize that Ž< ␥nŽ n .< : n g ⺞4 is bounded. the set
n : n g ⺞ 4 s hy1 Ž ␥n Ž n . . y Ž 1 y n . g Ž hy1 Ž ␥n Ž n . . . : n g ⺞ 4 is relatively compact as well. Žvii. Here we show the continuity of the operator ᑯ defined by q⬁ means of Ž6.. Let Ž qn , n , n .4ns1 be a sequence, in the domain of ᑪ, such Ž . Ž that lim nªq⬁ qn , n , n s q, , .. We should prove that lim
nªq⬁
Ž Ž qn , n . y ᑪ Ž qn , n , n . . s Ž q, . y ᑪ Ž q, , . .
q⬁ Ža. At first, let us consider the sequence ␥n4ns1 of absolutely continuous solutions of the related problems, i.e., the function sequence defined by
␥n Ž t . s h Ž n . q
t
H0 f Ž s, h Ž q Ž s . . . d s, n
t g w 0, 1 x .
$ $
Observe that the functions f n, f nŽ t . s f Ž t, hŽ qnŽ t ..., t g w0, 1x, are measurable and have the common Lebesgue integrable majorant r . Moreover, $
lim f n Ž t . s f Ž t , h Ž q Ž t . . . ,
nªq⬁
for almost all t g w0, 1x. Thus, it follows, according to the Lebesgue dominant convergence theorem, that lim ␥n Ž t . s ␥ Ž t . ,
nªq⬁
for every t g w 0, 1 x ,
whenever ␥ Ž t . s hŽ . q H0t f Ž s, hŽ q Ž s ... d s, t g w0, 1x. Since ␥n : n g ⺞4 is Žsee part Žvi.Ža.. a bounded set of equicontinuous functions, our sequence tends to ␥ uniformly, on a compact interval w0, 1x. Žb. The remaining part of the proof can be performed quite analogously to the part Žvi.Žb.. Žviii. At last, we will show the non-triviality of the degree for the operator ᑪ Ž⭈, ⭈ , 0.. Ža. At first, observe that ᑯ Ž⭈, ⭈ , 0.: C Žw0, 1x, clŽ ⍀ .. = clŽ ⍀ . ª ⺢ n n = ⺢ and denote ⌬ s Ž C Ž w 0, 1 x , ⍀ . = ⍀ . l Ž ⺢ n = ⺢ n . , ᑫ s ᑪ Ž ⭈, ⭈ , 0 . < clŽ⌬ . .
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By the reduction property of the degree Žsee, e.g., wD, p. 59x., we have deg Ž ᑪ Ž ⭈, ⭈ , 0 . , ⍀ , 0 . s deg Ž ᑫ , ⌬ , 0 . . Žb. Following step by step the proof of the uniqueness of the degree in a finite-dimensional space Žsee again, e.g., wD, pp. 5᎐12x., we see that we can assume, without any loss of generality, that ᑫ g C Ž1. Ž ⌬, ⺢ 2 n . l C ŽclŽ ⌬ ., ⺢ 2 n .. For the same reasons, we can assume that 0 is a regular point of ᑫ, i.e., detŽ ᑫX Ž x .. / 0, for every solution x of the equation ᑫ Ž x . s 0. Recall that ᑫ Ž , . s Ž y , g Ž . . ,
for Ž , . g cl Ž ⌬ . .
So, if Ž , . is a solution of our equation, then det Ž ᑫX Ž , . . s det
ž
I Ž 0.
M g Ž. X
/
s det Ž g X Ž . . ,
where I is a unit matrix of the type Ž n, n., and Ž0. is a zero matrix of the same type as M. This, jointly with the assumption Žiii.Žc. in Definition 2, already implies that deg Ž ᑫ , ⌬ , 0 . s deg Ž g , ⍀ , 0 . / 0, as claimed.
4. APPLICATION AND CONCLUDING REMARKS To describe an appropriate given set for differential system Ž1., the Liapunov-like functions method can be used. For the sake of simplicity, we restrict our demonstration to the systems with continuous right-hand sides only. Thus, the following approach can be based on the following simple lemma, which we state without the proof. LEMMA. Consider system Ž1., where f is, additionally, continuous. Let be a solution of Ž1. in an inter¨ al T ; w0, 1x. Assume that y: ⺢ n ª ⺢ is a continuously differentiable function. Let y Ž Ž tU .. s 0, for fixed tU g T. Then there exists ⑀ g ⺢q such that Ži. tU ) inf T , Ž grad y . Ž Ž tU . . ⭈ f Ž tU , Ž tU . . - 0 U
«
U
᭙ t g Ž t y ⑀ , t . : y Ž Ž t . . ) 0, Žii. tU - sup T , Ž grad y . Ž Ž tU . . ⭈ f Ž tU , Ž tU . . ) 0 U
U
᭙ t g Ž t , t q ⑀ . : y Ž Ž t . . ) 0.
«
PERIODIC SOLUTIONS FOR DIFFERENTIAL SYSTEMS
505
Now we are ready to give a non-trivial application of the theorem Žsee Remark 4.. EXAMPLE. Consider the system xX2 iy1 s x 2 i , xX2 i s f 2 i Ž t , x 1 , . . . , x 2 n . , where i s 1, . . . , n, and f s Ž x 2 , f 2 , x 4 , f 4 , . . . , x 2 n , f 2 n .: w0, 1x = ⺢ 2 n ª ⺢ 2 n is a continuous mapping. Denote ⌰ s Ž ␣ ,  . g ⺢ 2 : < ␣ < q <  < - 14 n. Assume that, for all t g w0, 1x, g clŽ ⌰ ., i s 1, . . . , n, the following inequalities take place:
Ž 7.
f 2 i Ž t , 1 , . . . , 2 iy1 , 1 y 2 iy1 , 2 iq1 , . . . , 2 n . ) 2 iy1 y 1, for 2 iy1 g w 0, 1 x ,
Ž 8.
f 2 i Ž t , 1 , . . . , 2 iy1 , 2 iy1 y 1, 2 iq1 , . . . , 2 n . ) 2 iy1 y 1, for 2 iy1 g w 0, 1 x ,
Ž 9.
f 2 i Ž t , 1 , . . . , 2 iy1 , y1 y 2 iy1 , 2 iq1 , . . . , 2 n . - 2 iy1 q 1, for 2 iy1 g w y1, 0 x ,
Ž 10 .
f 2 i Ž t , 1 , . . . , 2 iy1 , 1 q 2 iy1 , 2 iq1 , . . . , 2 n . - 2 iy1 q 1, for 2 iy1 g w y1, 0 x .
Then the above system admits a solution satisfying Ž2., the values of which are located in ⌰. At first, observe that ⌰ can be obtained as the intersection n
⌰s
F g ⺢ 2 n : yji Ž . - 0, for j s 1, . . . , 4 4 , is1
where, for i s 1, . . . , n, y 1 i : ⺢ 2 n ª ⺢,
y 1 i Ž . s 2 iy1 q 2 i y 1,
y 2 i : ⺢ 2 n ª ⺢,
y 2 i Ž . s 2 iy1 y 2 i y 1,
y 3 i : ⺢ 2 n ª ⺢,
y 3 i Ž . s y 2 iy1 y 2 i y 1,
y4 i : ⺢ 2 n ª ⺢,
y4 i Ž . s y 2 iy1 q 2 i y 1.
Hence, g frŽ ⌰ . if and only if there exist jU g 1, . . . , 44 and iU g 1, . . . , n4 such that y jU iU Ž . s 0 and y ji Ž . F 0, for all
Ž j, i . g Ž 1, 2, 3, 4 4 = 1, . . . n4 . _ Ž jU , iU . .
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Now observe that if g frŽ ⌰ . and y 1 i Ž . s 0, for some i g 1, . . . , n4 , then 2 i s 1 y 2 iy1 , 2 iy1 g w0, 1x, and, under the inequality Ž7.,
Ž grad y 1 i . Ž . ⭈ f Ž t , . s 2 i q f 2 i Ž t , . s 1 y 2 iy1 q f 2 i Ž t , . ) 0, for all t g w 0, 1 x . Analogously, applying inequalities Ž8. ᎐ Ž10., we obtain, in the sequel, for all i g 1, . . . , n4 ,
g fr Ž ⌰ . , y 2 i Ž . s 0
«
2 iy1 g w 0, 1 x , Ž grad y 2 i . Ž . ⭈ f Ž t , . - 0, g fr Ž ⌰ . , y 3 i Ž . s 0
«
2 iy1 g w y1, 0 x , Ž grad y 3 i . Ž . ⭈ f Ž t , . ) 0, g fr Ž ⌰ . , y4 i Ž . s 0
«
2 iy1 g w y1, 0 x , Ž grad y4 i . Ž . ⭈ f Ž t , . - 0. Thus, under the foregoing lemma, the sets ⌫q, ⌫y can be described as
g ⌫q
m
g cl Ž ⌰ . ,
and there exists i g 1, . . . , n4 such that
y 2 i Ž . s 0, 2 iy1 g w 0, 1 x or y4 i Ž . s 0, 2 iy1 g w y1, 0 x ,
g ⌫y
m
g cl Ž ⌰ . ,
and there exists i g 1, . . . , n4 such that
y 1 i Ž . s 0, 2 iy1 g w 0, 1 x or y 3 i Ž . s 0, 2 iy1 g w y1, 0 x , Observe that frŽ ⌰ . s ⌫qj ⌫y. Furthermore, denote ⍀ s Žy1, 1. 2 n. It is easy to verify that, under homeomorphism h: ⺢ 2 n ª ⺢ 2 n , h Ž . s 12 Ž 1 q 2 , y 1 q 2 , . . . , 2 iy1 q 2 i , y 2 iy1 q 2 i , . . . , y 2 ny1 q 2 n . , hŽ ⍀ . s ⌰ holds for ⍀. Hence, we obtain the following equivalences for g clŽ ⍀ .:
g hy1 Ž ⌫q . g hy1 Ž ⌫y .
m there exists i g 1, . . . , n4 such that 2 iy1 g y1, 1 4 , m there exists i g 1, . . . , n4 such that 2 i g y1, 1 4 .
Finally, define g : ⺢2n ª ⺢2n,
g Ž . s 12 Ž 1 , y 2 , . . . , 2 iy1 , y 2 i , . . . , 2 ny1 , y 2 n .
PERIODIC SOLUTIONS FOR DIFFERENTIAL SYSTEMS
507
and observe that all assumptions in Definition 2 are satisfied for our quadruple Ž ⌰, h, ⍀, g .. Remark 2. One can easily formulate similar examples for higher-order systems. Remark 3. Obviously, the transversality requirements imposed in the theorem on the boundary frŽ ⌰ . of ⌰ cannot be avoided here, even if clŽ ⌰ . is positively or negatively strongly flow invariant. Thus, the comparison with analogous results obtained by different techniques has meaning only when the transversality occurs. In wB, FZ, GS, Kr1, Kr2, and Sax, given sets are intersections of cones or convex sets and sets lying between several level surfaces of Liapunov-like functions. Sometimes, further additional restrictions are imposed that require the associated operators to be, e.g., convexrconcave, as, for example, in wBx. On the other hand, in wBx multiplicity results are obtained for differential inclusions. In wR, Sr1᎐Sr3x, the criteria for the existence of periodic solutions seem to be more similar to ours than to those quoted above; in particular, those in wSr1x an wSr2x seem to be even more general. More precisely, in wSr1, Sr2x, it is still assumed that clŽ ⌰ . and clŽ ⌫ ". are compact absolute neighborhood retracts, but this is practically never an additional restriction. Moreover, since conditions Žiii.Ža. and Žiii.Žb. roughly mean that the vector yg Ž . is directed outward ⍀ in the set hy1 Ž ⌫y. and inward in the set hy1 Ž ⌫q. , we can apply a formula due to R. Srzednicki Žsee wSr2, p. 728x. for calculating the degree, n
deg Ž g , ⍀ , 0 . s Ž y1 . deg Ž yg , ⍀ , 0 . s Ž cl Ž ⌰ . . y Ž ⌫y . , where stands for the Euler characteristic. Thus, condition Žiii.Žc. means that ŽclŽ ⌰ .. y Ž ⌫y. / 0. This inequality appears in wSr1, Theorems 1 and 2x as well as in wSr2, Corollary 7.4x, instead of our Žiii.Ža. ᎐ Žc.. Remark 4. In this light, it can also easily be seen that the existence of a solution in the example immediately follows from the mentioned results in wSr1, Sr2x, because ŽclŽ ⌰ .. y Ž ⌫y. s Žy1. n.
REFERENCES wA1x wA2x
J. Andres, On the multivalued Poincare ´ operators, Topol. Methods Nonlinear Anal. 10 Ž1997., 171᎐182. J. Andres, Using the integral manifolds to solvability of boundary value problems, in ‘‘Set Valued Mappings with Applications 1ᎏNonlinear Analysis’’ ŽR. P. Agarwal and D. O’Regan, Eds.., pp. 27᎐38, Gordon & Breach, New York.
508 wAGJx
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J. Andres, L. Gorniewicz, and J. Jezierski, Relative versions of the multivalued ´ Lefschetz and Nielsen theorems and their application to admissible semi-flows, Topol. Methods Nonlinear Anal. 16, No. 1 Ž2000., 76᎐92. wAKx J. Andres and B. Krajc, Bounded solutions in a given set of differential systems, J. Comput. Appl. Math. 113 Ž2000., 73᎐82. wAMTx J. Andres, L. Malaguti, and V. Taddei, Floquet boundary value problems for differential inclusions: a bound sets approach, Z. Anal. Anwendungen 20 Ž2001., 709᎐725. wBx R. Bader, A sufficient condition for the existence of multiple periodic solutions of differential inclusions, in ‘‘Topology in Nonlinear Analysis’’ ŽK. Ge ¸ba and L. Gorniewicz, Eds.., Banach Center Publ. 35 Ž1996., 129᎐138. ´ wBKx J. W. Bebernes and W. Kelley, Some bound value problems for generalized differential equations, SIAM J. Appl. Math. 25 No. 1 Ž1973., 16᎐23. wBSx J. W. Bebernes and J. D. Schuur, The Wazewski topological method for contingent ˙ equations, Ann. Mat. Pura Appl. Ž 4 . 87 Ž1970., 271᎐279. wC1x P. Cardaliaguet, Conditions suffisantes de non-vacuite ´ du noyau de viabilite, ´ C. R. Acad. Sci. Paris Ser. I 1314, No. 11 Ž1992., 797᎐800. wC2x P. Cardaliaguet, Sufficient conditions of nonemptiness of the viability kernel, preprint no. 9363, CEREMADE, Universite ´ de Paris-Dauphine, Paris, 1993. wDx K. Deimling, ‘‘Nonlinear Functional Analysis,’’ Springer-Verlag, Berlin, 1985. wFZx M. L. C. Fernandez and F. Zanolin, On periodic solutions, in a given set, for differential systems, Ri¨ . Mat. Pura Appl. 8 Ž1991., 107᎐130. wGx L. Gorniewicz, ‘‘Topological Fixed Point Theory of Multivalued Mappings,’’ Kluwer ´ Academic, Dordrecht, 1999. wGGLx A. Granas, R. Guenther, and J. Lee, Nonlinear boundary value problems for ordinary differential equations, Dissertationes Math. Ž Rozprawy Mat.. 144 Ž1985., 1᎐128. wGMx R. E. Gaines and J. Mawhin, ‘‘Coincidence Degree and Nonlinear Differential Equations,’’ Lecture Notes in Mathematics, Vol. 568, Springer-Verlag, Berlin, 1987. wGSx R. E. Gaines and J. M. Santanilla, A coincidence theorem in convex sets with applications to periodic solutions of ordinary differential equations, Rocky Mountain J. Math. 12 Ž1982., 669᎐678. wHx P. Hartman, ‘‘Ordinary Differential Equations,’’ Wiley, New York, 1964. wHPx S. Hu and N. S. Papageorgiou, ‘‘Handbook of Multivalued Analysis,’’ Vol. 2, Kluwer Academic, Dordrecht, 2000. wJKx L. K. Jackson and G. Klaasen, A variation of the topological method of Wazewski, ˙ SIAM J. Appl. Math. 20, No. 1 Ž1971., 124᎐130. wKcx B. Krajc, On bounded solutions in a given set of systems of differential equations, J. Inequal. Appl. 6 Ž2001., 149᎐165. wKr1x M. A. Krasnosel’skii, ‘‘Positive Solutions of Operator Equations,’’ Noordhoff, Groningen, 1964. wKr2x M. A. Krasnosel’skii, ‘‘The Operator of Translation along the Trajectories of Differential Equations,’’ Am. Math. Soc., Providence, 1968. wLWx R. W. Legget and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Uni¨ . Math. J. 28 Ž1979., 673᎐689. wMx J. Mawhin, Bound sets and Floquet boundary value problems for nonlinear differential equations, Iagell. Uni¨ . Acta Math. 36 Ž1998., 41᎐53. wRx K. P. Rybakowski, Some remarks on periodic solutions of Caratheodory RFDES via ´ Wazewski principle, Ri¨ . Mat. Uni¨ . Parma Ž 4 . 8 Ž1982., 377᎐385. ˙ wSax J. Santanilla, Some coincidence theorems in wedges, cones and convex sets, J. Math. Anal. Appl. 105 Ž1985., 357᎐371.
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wSr1x wSr2x wSr3x wTx wWx wZx
509
R. Srzednicki, Periodic and constant solutions via topological principle of Wazewski, ˙ Acta Math. Uni¨ . Iag. 26 Ž1987., 183᎐190. R. Srzednicki, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations, Nonlinear Anal. 22, No. 6 Ž1994., 707᎐737. R. Srzednicki, On solutions of two-point boundary value problems inside isolating segments, Topol. Methods Nonlinear Anal. 13 Ž1999., 73᎐89. V. Taddei, Bound sets for Floquet boundary value problems: the nonsmooth case, Discrete Cont. Dynam. Syst. 6 Ž2000., 459᎐473. T. Wazewski, Sur un principle topologique pour l’examen de allure asymptotique des ˙ integrales des ` equations differentiales ordinaires, Ann. Soc. Polon. Math. 20 Ž1947., ´ ´ 279᎐313. F. Zanolin, Bound sets, periodic solutions and flow-invariance for ordinary differential equations in ⺢ n : some remarks, Rend. Istit. Mat. Uni¨ . Trieste 19 Ž1987., 76᎐92.
Periodic Solutions in a Given Set of Differential Systems Jan Andres1 Department of Mathematical Analysis, Faculty of Science, Palacky ´ Uni¨ ersity, Tomko¨ a 40, 779 00 Olomouc-Hejcın, ˇ´ Czech Republic E-mail: [email protected]
and Bohumil Krajc 2 Department of Applied Mathematics, Faculty of Electrical Engineering and Computer Science, Technical Uni¨ ersity of Ostra¨ a, 17 Listopadu, 708 33 Ostra¨ a-Poruba, Czech Republic E-mail: [email protected] Submitted by Raul ´ Manase ´ ¨ ich Received November 28, 2000
The existence of a periodic solution in a given set of Caratheodory differential ´ systems is studied. The proof of the main statement is based on the Wazewski-type ˙ approach jointly with the degree arguments. An illustrating example is supplied, and a comparison with the analogous results of other authors is indicated. 䊚 2001 Elsevier Science
Key Words: periodic solutions; given sets; Caratheodory systems; degree argu´ ments; Wazewski-type result. ˙
1. INTRODUCTION The study of periodic solutions to differential systems which remain in a given set is mostly related to the Žpositive or negative. flow invariance of this set. M. Nagumo’s classical result, already formulated in 1942, is a typical example of a so-called weak flow invariance, when there is not necessarily uniqueness for Cauchy problems. This theorem has been 1
Supported by the Council of the Czech Government ŽJ 14r98:153100011. and by a grant ˇ ŽF 0158r2000.. from the FRVS 2 Supported by the Council of the Czech Government ŽCEZ MSM272400019.. 495 0022-247Xr01 $35.00 䊚 2001 Elsevier Science All rights reserved.
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redeveloped and extended several times, with the use of various types of tangent cones, which allows us to obtain the existence results for periodic solutions by means of degree arguments. For the related references, see, e.g., the handbook wHPx. On the other hand, the case where no invariance is assumed is still far from obvious. Usually, the transversality behavior on the boundary of a given domain is required in terms of so-called canonical domains, Wazew˙ ski sets, bound sets, etc., verified by means of guiding or bounding functions, etc. Žsee, e.g., wA1, A2, AGJ, AMT, B, BK, BS, C1, C2, FZ, GGL, GM, GS, Kr1, Kr2, LW, M, R, Sa, Sr1᎐Sr3, T, Zx.. In wAK, Kcx, we considered entirely bounded solutions in given sets, when using simultaneously compression- and expansion-type conditions, similar to typical Wazewaki pictures Žalthough not applying the Wazewski ˙ ˙ topological principle, as developed and improved in wW, BS, H, JKx, etc... These results can also be used, for planar systems with uniqueness for Cauchy problems, to guarantee periodic oscillations, on the basis of the well-known Massera transformation theorem, which, however, is no longer true in ⺢ n, where n ) 2. Here we investigate again the problem given in the title by means of the Wazewski-type approach combined with degree arguments. Although the ˙ main result seems to be practically covered by geometric methods of R. Srzednicki in wSr1, Sr3x Žsee Remark 3 here., the new ideas in the proof make it original and still rather general. We also add an illustrating example in the concluding part.
2. SOME PRELIMINARIES Consider the differential system xX s f Ž t , x . ,
Ž 1.
where f : w0, 1x = ⺢ n ª ⺢ n satisfies Ži. f Ž t, ⭈ . is continuous, for a.a. t g w0, 1x; Žii. f Ž⭈, x . is Lebesgue measurable, for every x g ⺢ n ; Žiii. for every r g ⺢q, there exists r g L Žw0, 1x, ⺢q. such that, for a.a. t g w0, 1x, the implication <
«
f Ž t , . F r Ž t . .
. solutions x g A C Žw0, 1x, ⺢ n . of holds. We are interested in ŽCaratheodory ´ Ž1., satisfying x Ž 0. s x Ž 1. , Ž 2. the values of which are located in a given set ⌰.
PERIODIC SOLUTIONS FOR DIFFERENTIAL SYSTEMS
497
DEFINITION 1. Given a bounded open set ⭋ / ⌰ ; ⺢ n with the boundary frŽ ⌰ ., Ži. We call a point g frŽ ⌰ . an ingress point Žinto ⌰ . if, for every tU g Ž0, 1x and every solution ␥ of Ž1. in w0, 1x, the implication
␥ Ž tU . s
« ᭚⑀ g ⺢q
᭙ t g Ž tU y ⑀ , tU . : ␥ Ž t . g ⺢ n _cl Ž ⌰ .
takes place. Žii. We call a point g frŽ ⌰ . an egress point Žfrom ⌰ . if, for every U t g w0, 1. and every solution ␥ of Ž1. in w0, 1x, the implication
␥ Ž tU . s
« ᭚⑀ g ⺢q
᭙ t g Ž tU , tU q ⑀ . : ␥ Ž t . g ⺢ n _cl Ž ⌰ .
takes place. The set of ingress or egress points of ⌰ will be denoted by ⌫q or ⌫y, respectively. DEFINITION 2. Given a bounded open set ⭋ / ⌰ ; ⺢ n with the boundary frŽ ⌰ ., we say that the quadruple Ž ⌰, h, ⍀, g . is admissible w.r.t. Ž1. if the following conditions are satisfied: Ži. every boundary point Ži.e., of frŽ ⌰ .. is either an ingress or an egress point of ⌰ Žw.r.t. solutions of Ž1..; Žii. h is a homeomorphism of ⺢ n into itself and hŽ ⍀ . s ⌰; Žiii. g: ⺢ n ª ⺢ n is a continuous mapping such that Ža. if hŽ . g ⌰ j ⌫q and g w0, 1., then hŽ y Ž1 y . g Ž .. f q ⌫ ; Žb. if hŽ . g ⌫y and g w0, 1., then hŽ y Ž1 y . g Ž .. g n ⺢ _clŽ ⌰ .; Žc. degŽ g, ⍀, 0. / 0, where deg stands for the Brouwer degree. Remark 1. Denote wy1, 1x 0 = wy1, 1x s wy1, 1x = wy1, 1x 0 s wy1, 1x and assume that n
hy1 Ž ⌰ . s ⍀ s Ž y1, 1 . , hy1 Ž ⌫q . s
s g 0, . . . , n4 ,
nys
D Ž wy1, 1x sqiy1 = y1, 14 = wy1, 1x nysyi . , is1
hy1 Ž ⌫y . s
sy1
D Ž wy1, 1x j = y1, 14 = wy1, 1x nyjy1 . . js0
Defining g: ⺢ ª ⺢ , n
n
g Ž 1 , . . . , n . s 12 Ž y 1 , . . . , y s , sq1 , . . . , n . ,
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one can easily check that g satisfies conditions Žiii.Ža., Žb. and, since g is odd, Žiii.Žc.. Another possibility for satisfying Žiii.Žc. is that ⍀ is a so-called AR-space Žin particular, a retract of a convex set. and g is a self-mapping Ži.e., into. Žcf., e.g., wGx..
3. MAIN THEOREM THEOREM. Let Ž ⌰, h, ⍀, g . be an admissible quadruple w.r.t. Ž1.. Then problem Ž1. ᎐ Ž2. admits a solution with ¨ alues located in ⌰. Proof. We proceed in eight steps. ⺢
n.
Ži. At first, define ᒏ: C Žw0, 1x, clŽ ⍀ .. = clŽ ⍀ . = w0, 1x ª C Žw0, 1x, by ᒏ Ž q, , . Ž t . s
½
hy1 Ž ␥ Ž t . . ,
for t g w 0, x ,
hy1 Ž ␥ Ž . . ,
for t g Ž , 1 x ,
where ␥ g A C Žw0, 1x, ⺢ n .,
␥ Ž t . s hŽ . q
t
H0 f Ž s, h Ž q Ž s . . . d s.
Žii. Furthermore, define ᒐ: C Žw0, 1x, clŽ ⍀ .. = clŽ ⍀ . = w0, 1x ª ⺢ n by ᒐ Ž q, , . s g Ž ᒏ Ž q, , . Ž . . . Žiii. At last, define ᑪ s Ž ᑪ 1 , ᑪ 2 .: C Žw0, 1x, clŽ ⍀ .. = clŽ ⍀ . = w0, 1x ª C Žw0, 1x, ⺢ n . = ⺢ n, by ᑪ 1 Ž q, , . s q y ᒏ Ž q, , . , ᑪ 2 Ž q, , . s y ᒏ Ž q, , . Ž . y Ž 1 y . ᒐ Ž q, , . . Živ. Let us fix g w0, 1x, q g C Žw0, 1x, clŽ ⍀ .., and g clŽ ⍀ . and denote ␥ s h( q. One can then observe that ᑪ Ž q, , . s 0
Ž 3.
X
␥ Ž t. s f Ž t, ␥ Ž t. . ,
«
for a.a. t g w 0, x , for every t g Ž , 1 x ,
Ž 4.
␥ Ž t . s ␥ Ž . ,
Ž 5.
q Ž 0. s s q Ž . y Ž 1 y . g Ž q Ž . . .
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499
In particular, for s 1, we have
␥X Ž t. s f Ž t, ␥ Ž t. . ,
for a.a. t g w 0, 1 x , ␥ Ž 0 . s ␥ Ž 1 . ,
i.e., the solution of Ž1. ᎐ Ž2.. To verify the solvability of ᑪ Ž q, , 1. s 0, we show that deg Ž ᑪ Ž ⭈, ⭈ , 1 . , C Ž w 0, 1 x , ⍀ . = ⍀ , 0 . / 0, where deg stands for the Leray᎐Schauder degree. Žv. Now we show that, for every g w0, 1x and every pair Ž q, . g frŽ C Žw0, 1x, ⍀ . = ⍀ ., we have ᑪ Ž q, , . / 0. Ža. At first, we prove by contradiction the implication
g w 0, 1 x ,
q g fr Ž C Ž w 0, 1 x , ⍀ . . ,
g⍀
«
ᑪ Ž q, , . / 0.
Let there exist g w0, 1x, q g frŽ C Žw0, 1x,⍀ .., and g ⍀ such that ᑪ Ž q, , . s 0. Then q Ž . g frŽ ⍀ ., for some g w0, 1x, and, according to assumption Ži. in Definition 2, hŽ q Ž .. is an ingress or egress point of ⌰. For g Ž0, ., Ž3. implies ᭚⑀ g ⺢q such that either hŽ q Ž t .. g n ⺢ _clŽ ⌰ ., for all t g Ž y ⑀ , ., or hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž , q ⑀ ., takes place, respectively. This is, however, a contradiction of the choice of q g frŽ C Žw0, 1x, ⍀ ... Let s 0, g w0, 1.. If s 0, then Ž5. implies g Ž q Ž0.. s 0, which is a contradiction of condition Žiii.Žc. in Definition 2 Žactually, it is rather a contradiction of conditions Žiii.Ža. and Žiii.Žb., which guarantee that the degree in Žiii.Žc. is defined.. For g Ž0, 1. in Ž5. we know that q Ž0. s q Ž . y Ž1 y . g Ž q Ž .., where Ž q . g clŽ ⍀ .. If hŽ q Ž0.. g ⌫y, then ᭚⑀ g ⺢q : hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for every t g Ž0, ⑀ ., which is a contradiction of the choice of q g frŽ C Žw0, 1x, ⍀ ... Therefore, we can assume that hŽ q Ž0.. g ⌫q, which, however, contradicts conditions Žiii.Ža., Žb. in Definition 2. Let s 0, s 1. Since Ž5. implies q Ž0. s q Ž1., we can proceed quite analogously to the foregoing case. So, ᭚⑀ g ⺢q such that either hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž1 y ⑀ , 1., or hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž0, ⑀ ., take place, respectively. However, this is again a contradiction of the choice of q g frŽ C Žw0, 1x, ⍀ ... Let Ž0, 1x 2 G g w0, 1.. If s 0, then can be taken, according to Ž4., to be equal to 0 as well. Put g Ž0, 1.. Then Ž4. implies that q Ž . s q Ž . and Ž3. implies that hŽ q Ž .. is not an ingress point into ⌰. Otherwise, ᭚⑀ g ⺢q such that hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž y ⑀ , .. 䢇
䢇
䢇
䢇
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If hŽ q Ž .. s hŽ q Ž .. is an egress point from ⌰, then in view of Ž5. and condition Žiii.Žb. in Definition 2, the initial value hŽ q Ž0.. does not belong to clŽ ⌰ ., which is a contradiction of the choice of q g frŽ C Žw0, 1x, ⍀ ... Let s s 1. Then Ž5. implies q Ž0. s q Ž1. and Ž3. implies ᭚⑀ g ⺢q such that either hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž1 y ⑀ , 1., or hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž0, ⑀ ., takes place, respectively. This is, however, a contradiction of the choice of q g frŽ C Žw0, 1x, ⍀ ... Žb. Now we prove again by contradiction the implication 䢇
g w 0, 1 x ,
q g C Ž w 0, 1 x , cl Ž ⍀ . . ,
g fr Ž ⍀ .
«
ᑪ Ž q, , . / 0.
Hence, let there exist g w0, 1x, q g C Žw0, 1x, clŽ ⍀ .., and g frŽ ⍀ . such that ᑪ Ž q, , . s 0. Let us employ that s q Ž0.. Let g w0, 1.. If s 0, then Ž5. implies that g Ž q Ž0.. s 0, which contradicts Žiii.Žc. in Definition 2. For g Ž0, 1., we know according to Ž5. that s q Ž0. s q Ž . y Ž1 y . g Ž q Ž .., where q Ž . g clŽ ⍀ .. If hŽ q Ž0.. g ⌫y, then ᭚⑀ g ⺢q : hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž0, ⑀ ., which is a contradiction of the choice of q g C Žw0, 1x, clŽ ⍀ ... Therefore, we can assume that hŽ q Ž0.. g ⌫q, which is, however, a contradiction of conditions Žiii.Ža., Žb. in Definition 2. Let s 1. Then Ž5. implies q Ž0. s q Ž1.. Thus, ᭚⑀ g ⺢q such that either hŽ q Ž t .. g ⺢ n _clŽ ⌰ ., for all t g Ž1 y ⑀ , 1., or hŽ q Ž t .. g ⺢ n _ clŽ ⌰ ., for all t g Ž0, ⑀ ., takes place, respectively, which is a contradiction of the choice of q g C Žw0, 1x, clŽ ⍀ ... Žvi. Furthermore, we show that the image of the operator 䢇
䢇
ᑯ : C Ž w 0, 1 x , cl Ž ⍀ . . = cl Ž ⍀ . = w 0, 1 x ª C Ž w 0, 1 x , ⺢ n . = ⺢ n ,
Ž 6.
ᑯ Ž q, , . s Ž q, . y ᑪ Ž q, , .
is relatively compact. q⬁ Let us consider the sequence Ž z n , n .4ns1 of points from the image of q⬁ ᑯ. Let Ž qn , n , n .4ns1 be a sequence such that Ž qn , n . y ᑪ Ž qn , n , n . s Ž z n , n ., for all positive integers n. q⬁ Ža. At first, let us consider the sequence ␥n4ns1 of absolutely continuous solutions of related problems, i.e., the function sequence defined by
␥n Ž t . s h Ž n . q
t
H0 f Ž s, h Ž q Ž s . . . d s, n
We prove that ␥n : n g ⺞4 is relatively compact.
t g w 0, 1 x .
501
PERIODIC SOLUTIONS FOR DIFFERENTIAL SYSTEMS
According to the well-known Ascoli lemma, it is sufficient to show that all functions from our set are uniformly bounded and equicontinuous. For a suitable r g ⺢q, it obviously holds that ᭙ t g w 0, 1 x : ␥n Ž t . s h Ž n . q
F h Ž n . s F K⌰ q
1
t
H0 f Ž s, h Ž q Ž s . . . d s n
1
H0
f Ž s, h Ž qn Ž s . . . d s
r Ž s . d s,
H0
where K ⌰ s max< < : g clŽ ⌰ .4 - q⬁ and, due to the Caratheodory ´ conditions, the Lebesgue integral on the right-hand side is finite. The equicontinuity follows easily as well; because of a finiteness of the integral of r , for every g ⺢q there is ␦ g ⺢q such that, for all n g ⺞ and t, p from w0, 1x with < t y p < - ␦ , we arrive at
␥n Ž t . y ␥n Ž p . F
t
Hp
f Ž s, h Ž qn Ž s . . . d s F
t
HP Ž s . d s r
- .
q⬁ Žb. Let n k 4ks1 be a sequence of positive integers such that
lim ␥n k s ␥ ,
lim n k s .
kªq⬁
kªq⬁
Then
¡h Ž ␥ Ž t . s~ y1
znk
¢h
y1
nk
for t g 0, n k ,
Ž t..,
Ž ␥n Ž n . . , k
for t g Ž n k , 1 .
k
Putting zŽ t. s
½
hy1 Ž ␥ Ž t . . ,
for t g w 0, x ,
hy1 Ž ␥ Ž . . ,
for t g Ž , 1 x ,
we show that lim z n k s z.
kªq⬁
Choose g ⺢q. The function hy1 (␥ is uniformly continuous on w0, 1x, by which there exists ␦ g ⺢q such that < hy1 Ž␥ Ž t .. y hy1 Ž␥ Ž p ..< - 2 ,
502
ANDRES AND KRAJC
whenever < t y p < - ␦ , t g w0, 1x, p g w0, 1x. Let k 0 g ⺞ be such that < n y < - ␦ , k
sup tg w0, 1 x
hy1 Ž ␥n kŽ t . . y hy1 Ž ␥ Ž t . . -
2
,
for all ⺞ 2 k ) k 0 . Observe that the latter inequality is correct, because the set ␥n k : k g ⺞4 is bounded. If t g w0, min n k, 4x, then z n kŽ t . y z Ž t . s hy1 Ž ␥n kŽ t . . y hy1 Ž ␥ Ž t . . -
2
-,
for ⺞ 2 k ) k 0 . If t g Žmax n k, 4 , 1x, then z n kŽ t . y z Ž t . s hy1 Ž ␥n k Ž n k . . y hy1 Ž ␥ Ž . . F hy1 Ž ␥n k Ž n k . . y hy1 Ž ␥ Ž n k . . q hy1 Ž ␥ Ž n k . . y hy1 Ž ␥ Ž . . -
2
q
2
s,
for ⺞ 2 k ) k 0 .
If t g w n k, x, then < n k y t < - ␦ , and, subsequently, z n kŽ t . y z Ž t . s hy1 Ž ␥n k Ž n k . . y hy1 Ž ␥ Ž t . . F hy1 Ž ␥n k Ž n k . . y hy1 Ž ␥ Ž n k . . q hy1 Ž ␥ Ž n k . . y hy1 Ž ␥ Ž t . . -
2
q
2
s,
for ⺞ 2 k ) k 0 .
If t g w , n k x, then < y t < - ␦ , and, subsequently, z n kŽ t . y z Ž t . s hy1 Ž ␥n kŽ t . . y hy1 Ž ␥ Ž . . F hy1 Ž ␥n kŽ t . . y hy1 Ž ␥ Ž t . . q hy1 Ž ␥ Ž t . . y hy1 Ž ␥ Ž . . - q s, for ⺞ 2 k ) k 0 . 2 2
PERIODIC SOLUTIONS FOR DIFFERENTIAL SYSTEMS
503
Thus, we demonstrated that the set z n : n g ⺞4 is relatively compact. It just remains to realize that Ž< ␥nŽ n .< : n g ⺞4 is bounded. the set
n : n g ⺞ 4 s hy1 Ž ␥n Ž n . . y Ž 1 y n . g Ž hy1 Ž ␥n Ž n . . . : n g ⺞ 4 is relatively compact as well. Žvii. Here we show the continuity of the operator ᑯ defined by q⬁ means of Ž6.. Let Ž qn , n , n .4ns1 be a sequence, in the domain of ᑪ, such Ž . Ž that lim nªq⬁ qn , n , n s q, , .. We should prove that lim
nªq⬁
Ž Ž qn , n . y ᑪ Ž qn , n , n . . s Ž q, . y ᑪ Ž q, , . .
q⬁ Ža. At first, let us consider the sequence ␥n4ns1 of absolutely continuous solutions of the related problems, i.e., the function sequence defined by
␥n Ž t . s h Ž n . q
t
H0 f Ž s, h Ž q Ž s . . . d s, n
t g w 0, 1 x .
$ $
Observe that the functions f n, f nŽ t . s f Ž t, hŽ qnŽ t ..., t g w0, 1x, are measurable and have the common Lebesgue integrable majorant r . Moreover, $
lim f n Ž t . s f Ž t , h Ž q Ž t . . . ,
nªq⬁
for almost all t g w0, 1x. Thus, it follows, according to the Lebesgue dominant convergence theorem, that lim ␥n Ž t . s ␥ Ž t . ,
nªq⬁
for every t g w 0, 1 x ,
whenever ␥ Ž t . s hŽ . q H0t f Ž s, hŽ q Ž s ... d s, t g w0, 1x. Since ␥n : n g ⺞4 is Žsee part Žvi.Ža.. a bounded set of equicontinuous functions, our sequence tends to ␥ uniformly, on a compact interval w0, 1x. Žb. The remaining part of the proof can be performed quite analogously to the part Žvi.Žb.. Žviii. At last, we will show the non-triviality of the degree for the operator ᑪ Ž⭈, ⭈ , 0.. Ža. At first, observe that ᑯ Ž⭈, ⭈ , 0.: C Žw0, 1x, clŽ ⍀ .. = clŽ ⍀ . ª ⺢ n n = ⺢ and denote ⌬ s Ž C Ž w 0, 1 x , ⍀ . = ⍀ . l Ž ⺢ n = ⺢ n . , ᑫ s ᑪ Ž ⭈, ⭈ , 0 . < clŽ⌬ . .
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ANDRES AND KRAJC
By the reduction property of the degree Žsee, e.g., wD, p. 59x., we have deg Ž ᑪ Ž ⭈, ⭈ , 0 . , ⍀ , 0 . s deg Ž ᑫ , ⌬ , 0 . . Žb. Following step by step the proof of the uniqueness of the degree in a finite-dimensional space Žsee again, e.g., wD, pp. 5᎐12x., we see that we can assume, without any loss of generality, that ᑫ g C Ž1. Ž ⌬, ⺢ 2 n . l C ŽclŽ ⌬ ., ⺢ 2 n .. For the same reasons, we can assume that 0 is a regular point of ᑫ, i.e., detŽ ᑫX Ž x .. / 0, for every solution x of the equation ᑫ Ž x . s 0. Recall that ᑫ Ž , . s Ž y , g Ž . . ,
for Ž , . g cl Ž ⌬ . .
So, if Ž , . is a solution of our equation, then det Ž ᑫX Ž , . . s det
ž
I Ž 0.
M g Ž. X
/
s det Ž g X Ž . . ,
where I is a unit matrix of the type Ž n, n., and Ž0. is a zero matrix of the same type as M. This, jointly with the assumption Žiii.Žc. in Definition 2, already implies that deg Ž ᑫ , ⌬ , 0 . s deg Ž g , ⍀ , 0 . / 0, as claimed.
4. APPLICATION AND CONCLUDING REMARKS To describe an appropriate given set for differential system Ž1., the Liapunov-like functions method can be used. For the sake of simplicity, we restrict our demonstration to the systems with continuous right-hand sides only. Thus, the following approach can be based on the following simple lemma, which we state without the proof. LEMMA. Consider system Ž1., where f is, additionally, continuous. Let be a solution of Ž1. in an inter¨ al T ; w0, 1x. Assume that y: ⺢ n ª ⺢ is a continuously differentiable function. Let y Ž Ž tU .. s 0, for fixed tU g T. Then there exists ⑀ g ⺢q such that Ži. tU ) inf T , Ž grad y . Ž Ž tU . . ⭈ f Ž tU , Ž tU . . - 0 U
«
U
᭙ t g Ž t y ⑀ , t . : y Ž Ž t . . ) 0, Žii. tU - sup T , Ž grad y . Ž Ž tU . . ⭈ f Ž tU , Ž tU . . ) 0 U
U
᭙ t g Ž t , t q ⑀ . : y Ž Ž t . . ) 0.
«
PERIODIC SOLUTIONS FOR DIFFERENTIAL SYSTEMS
505
Now we are ready to give a non-trivial application of the theorem Žsee Remark 4.. EXAMPLE. Consider the system xX2 iy1 s x 2 i , xX2 i s f 2 i Ž t , x 1 , . . . , x 2 n . , where i s 1, . . . , n, and f s Ž x 2 , f 2 , x 4 , f 4 , . . . , x 2 n , f 2 n .: w0, 1x = ⺢ 2 n ª ⺢ 2 n is a continuous mapping. Denote ⌰ s Ž ␣ ,  . g ⺢ 2 : < ␣ < q <  < - 14 n. Assume that, for all t g w0, 1x, g clŽ ⌰ ., i s 1, . . . , n, the following inequalities take place:
Ž 7.
f 2 i Ž t , 1 , . . . , 2 iy1 , 1 y 2 iy1 , 2 iq1 , . . . , 2 n . ) 2 iy1 y 1, for 2 iy1 g w 0, 1 x ,
Ž 8.
f 2 i Ž t , 1 , . . . , 2 iy1 , 2 iy1 y 1, 2 iq1 , . . . , 2 n . ) 2 iy1 y 1, for 2 iy1 g w 0, 1 x ,
Ž 9.
f 2 i Ž t , 1 , . . . , 2 iy1 , y1 y 2 iy1 , 2 iq1 , . . . , 2 n . - 2 iy1 q 1, for 2 iy1 g w y1, 0 x ,
Ž 10 .
f 2 i Ž t , 1 , . . . , 2 iy1 , 1 q 2 iy1 , 2 iq1 , . . . , 2 n . - 2 iy1 q 1, for 2 iy1 g w y1, 0 x .
Then the above system admits a solution satisfying Ž2., the values of which are located in ⌰. At first, observe that ⌰ can be obtained as the intersection n
⌰s
F g ⺢ 2 n : yji Ž . - 0, for j s 1, . . . , 4 4 , is1
where, for i s 1, . . . , n, y 1 i : ⺢ 2 n ª ⺢,
y 1 i Ž . s 2 iy1 q 2 i y 1,
y 2 i : ⺢ 2 n ª ⺢,
y 2 i Ž . s 2 iy1 y 2 i y 1,
y 3 i : ⺢ 2 n ª ⺢,
y 3 i Ž . s y 2 iy1 y 2 i y 1,
y4 i : ⺢ 2 n ª ⺢,
y4 i Ž . s y 2 iy1 q 2 i y 1.
Hence, g frŽ ⌰ . if and only if there exist jU g 1, . . . , 44 and iU g 1, . . . , n4 such that y jU iU Ž . s 0 and y ji Ž . F 0, for all
Ž j, i . g Ž 1, 2, 3, 4 4 = 1, . . . n4 . _ Ž jU , iU . .
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ANDRES AND KRAJC
Now observe that if g frŽ ⌰ . and y 1 i Ž . s 0, for some i g 1, . . . , n4 , then 2 i s 1 y 2 iy1 , 2 iy1 g w0, 1x, and, under the inequality Ž7.,
Ž grad y 1 i . Ž . ⭈ f Ž t , . s 2 i q f 2 i Ž t , . s 1 y 2 iy1 q f 2 i Ž t , . ) 0, for all t g w 0, 1 x . Analogously, applying inequalities Ž8. ᎐ Ž10., we obtain, in the sequel, for all i g 1, . . . , n4 ,
g fr Ž ⌰ . , y 2 i Ž . s 0
«
2 iy1 g w 0, 1 x , Ž grad y 2 i . Ž . ⭈ f Ž t , . - 0, g fr Ž ⌰ . , y 3 i Ž . s 0
«
2 iy1 g w y1, 0 x , Ž grad y 3 i . Ž . ⭈ f Ž t , . ) 0, g fr Ž ⌰ . , y4 i Ž . s 0
«
2 iy1 g w y1, 0 x , Ž grad y4 i . Ž . ⭈ f Ž t , . - 0. Thus, under the foregoing lemma, the sets ⌫q, ⌫y can be described as
g ⌫q
m
g cl Ž ⌰ . ,
and there exists i g 1, . . . , n4 such that
y 2 i Ž . s 0, 2 iy1 g w 0, 1 x or y4 i Ž . s 0, 2 iy1 g w y1, 0 x ,
g ⌫y
m
g cl Ž ⌰ . ,
and there exists i g 1, . . . , n4 such that
y 1 i Ž . s 0, 2 iy1 g w 0, 1 x or y 3 i Ž . s 0, 2 iy1 g w y1, 0 x , Observe that frŽ ⌰ . s ⌫qj ⌫y. Furthermore, denote ⍀ s Žy1, 1. 2 n. It is easy to verify that, under homeomorphism h: ⺢ 2 n ª ⺢ 2 n , h Ž . s 12 Ž 1 q 2 , y 1 q 2 , . . . , 2 iy1 q 2 i , y 2 iy1 q 2 i , . . . , y 2 ny1 q 2 n . , hŽ ⍀ . s ⌰ holds for ⍀. Hence, we obtain the following equivalences for g clŽ ⍀ .:
g hy1 Ž ⌫q . g hy1 Ž ⌫y .
m there exists i g 1, . . . , n4 such that 2 iy1 g y1, 1 4 , m there exists i g 1, . . . , n4 such that 2 i g y1, 1 4 .
Finally, define g : ⺢2n ª ⺢2n,
g Ž . s 12 Ž 1 , y 2 , . . . , 2 iy1 , y 2 i , . . . , 2 ny1 , y 2 n .
PERIODIC SOLUTIONS FOR DIFFERENTIAL SYSTEMS
507
and observe that all assumptions in Definition 2 are satisfied for our quadruple Ž ⌰, h, ⍀, g .. Remark 2. One can easily formulate similar examples for higher-order systems. Remark 3. Obviously, the transversality requirements imposed in the theorem on the boundary frŽ ⌰ . of ⌰ cannot be avoided here, even if clŽ ⌰ . is positively or negatively strongly flow invariant. Thus, the comparison with analogous results obtained by different techniques has meaning only when the transversality occurs. In wB, FZ, GS, Kr1, Kr2, and Sax, given sets are intersections of cones or convex sets and sets lying between several level surfaces of Liapunov-like functions. Sometimes, further additional restrictions are imposed that require the associated operators to be, e.g., convexrconcave, as, for example, in wBx. On the other hand, in wBx multiplicity results are obtained for differential inclusions. In wR, Sr1᎐Sr3x, the criteria for the existence of periodic solutions seem to be more similar to ours than to those quoted above; in particular, those in wSr1x an wSr2x seem to be even more general. More precisely, in wSr1, Sr2x, it is still assumed that clŽ ⌰ . and clŽ ⌫ ". are compact absolute neighborhood retracts, but this is practically never an additional restriction. Moreover, since conditions Žiii.Ža. and Žiii.Žb. roughly mean that the vector yg Ž . is directed outward ⍀ in the set hy1 Ž ⌫y. and inward in the set hy1 Ž ⌫q. , we can apply a formula due to R. Srzednicki Žsee wSr2, p. 728x. for calculating the degree, n
deg Ž g , ⍀ , 0 . s Ž y1 . deg Ž yg , ⍀ , 0 . s Ž cl Ž ⌰ . . y Ž ⌫y . , where stands for the Euler characteristic. Thus, condition Žiii.Žc. means that ŽclŽ ⌰ .. y Ž ⌫y. / 0. This inequality appears in wSr1, Theorems 1 and 2x as well as in wSr2, Corollary 7.4x, instead of our Žiii.Ža. ᎐ Žc.. Remark 4. In this light, it can also easily be seen that the existence of a solution in the example immediately follows from the mentioned results in wSr1, Sr2x, because ŽclŽ ⌰ .. y Ž ⌫y. s Žy1. n.
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R. Srzednicki, Periodic and constant solutions via topological principle of Wazewski, ˙ Acta Math. Uni¨ . Iag. 26 Ž1987., 183᎐190. R. Srzednicki, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations, Nonlinear Anal. 22, No. 6 Ž1994., 707᎐737. R. Srzednicki, On solutions of two-point boundary value problems inside isolating segments, Topol. Methods Nonlinear Anal. 13 Ž1999., 73᎐89. V. Taddei, Bound sets for Floquet boundary value problems: the nonsmooth case, Discrete Cont. Dynam. Syst. 6 Ž2000., 459᎐473. T. Wazewski, Sur un principle topologique pour l’examen de allure asymptotique des ˙ integrales des ` equations differentiales ordinaires, Ann. Soc. Polon. Math. 20 Ž1947., ´ ´ 279᎐313. F. Zanolin, Bound sets, periodic solutions and flow-invariance for ordinary differential equations in ⺢ n : some remarks, Rend. Istit. Mat. Uni¨ . Trieste 19 Ž1987., 76᎐92.