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Repelling conditions for boundary sets using Liapunov-like functions. I: Flow-invariance, terminal value problem and weak persistence. (English) Zbl 0672.34048

Noncontinuable solutions x(\(\cdot)\) of the equation (1) \(x'=f(t,x)\) satisfying the initial condition (2) \(x(t_ 0)=x_ 0\in \Omega\) \((t_ 0\in J)\) are considered. In (1), (2) f: \(J\times \Omega \to {\mathbb{R}}^ d\) is a continuous function, J is a nondegenerate real interval, \(\Omega \subset {\mathbb{R}}^ d\) is a nonempty open set and \({\mathbb{R}}^ d\) is the d- dimensional real euclidean space. Let additionally \(G,S\subset {\mathbb{R}}^ d\) be nonempty sets, with \(G\subset \Omega\) and \(S\cap G=\emptyset\). Then a purpose of the article is to find various criteria relating behaviour of the solution of (1), (2) for \(x(t_ 0)\in G\), with the set S. The following situations 1) solutions of (1) never reach S from the set G; 2) there are no solutions x(\(\cdot)\) of (1), with x(t)\(\in G\) for all \(t\in I_ x\) and such that \(\lim_{t\to t_ x^-}x(t)=u\in S;\) 3) solutions of (1), with x(t)\(\in G\) for all \(t\in I_ x\), are asymptotically far from S; 4) S is a repeller for the solutions of (1) which remain in \(G(t_ x:=Sup I_ x,I_ x\) is the right maximal interval of existence of x(\(\cdot))\), are discussed.
Reviewer: S.G.Zhuravlev

MSC:

34D20 Stability of solutions to ordinary differential equations
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References:

[1] H. Amann , Gewöhnliche Differentialgleichungen , Walter de Gruyter , Berlin , 1983 . MR 713040 | Zbl 0823.34001 · Zbl 0823.34001
[2] J.-P. Aubin - A. CELLINA, Differential inclusions , Springer-Verlag , Berlin and New York , 1984 . MR 755330 | Zbl 0538.34007 · Zbl 0538.34007
[3] S.R. Bernfeld - R.D. Driver - V. Lakshmikantham , Uniqueness for ordinary differential equations , Math. Systems Theory , 9 ( 1976 ), pp. 359 - 367 . MR 447673 | Zbl 0329.34003 · Zbl 0329.34003 · doi:10.1007/BF01715361
[4] J.M. Bownds , A uniqueness theorem for y’ = f (x, y) using a certain factorization of f , J. Differential Equations , 7 ( 1970 ), pp. 227 - 231 . MR 254305 | Zbl 0194.11701 · Zbl 0194.11701 · doi:10.1016/0022-0396(70)90107-5
[5] G. Butler - H.I. Freedman - P. Waltman , Uniformly persistent systems , Proc. Amer. Math. Soc. , 96 ( 1986 ), pp. 425 - 430 . MR 822433 | Zbl 0603.34043 · Zbl 0603.34043 · doi:10.2307/2046588
[6] F. Cafiero , Sui teoremi d’unicità relativi ad un’equazione differenziale ordinata del primo ordine , Giorn. Mat. Battaglini , 78 ( 1948 ), pp. 193 - 215 . MR 32083 | Zbl 0032.41103 · Zbl 0032.41103
[7] K.W. Chang - F.A. Howes , Nonlinear singular perturbation phenomena: theory and applications , Springer-Verlag , Berlin and New York , 1984 . MR 764395 | Zbl 0559.34013 · Zbl 0559.34013
[8] M.G. Crandall , A generalization of Peano’s existence theorem and flow-invariance , Proc. Amer. Math. Soc. , 36 ( 1972 ), pp. 151 - 155 . MR 306586 | Zbl 0271.34084 · Zbl 0271.34084 · doi:10.2307/2039051
[9] M.L.C. Fernandes , Invariant sets and periodic solutions for differential systems , Magister Ph. Thesis, I.S.A.S ., Trieste , 1986 .
[10] M.L.C. Fernandes - F. Zanolin , Remarks on strongly flow-invariant sets , J. Math. Anal. Appl. , 128 ( 1987 ), pp. 176 - 188 . MR 915976 | Zbl 0657.34045 · Zbl 0657.34045 · doi:10.1016/0022-247X(87)90223-X
[11] M.L.C. Fernandes - F. Zanolin , On periodic solutions, in a given set, for differential systems (preprint). Zbl 0725.34039 · Zbl 0725.34039
[12] M.L.C. Fernandes - F. Zanolin , Repelling conditions for boundary sets using Liapunov-like functions. - II: Persistence and periodic solutions (preprint). MR 1061888 | Zbl 0719.34092 · Zbl 0719.34092 · doi:10.1016/0022-0396(90)90039-R
[13] A. Fonda , Uniformly persistent semi-dynamical systems , Proc. Amer. Math. Soc. (to appear). MR 958053 | Zbl 0667.34065 · Zbl 0667.34065 · doi:10.2307/2047471
[14] H.I. Freedman - P. Waltman , Mathematical analysis of some three species food-chain models , Math. Biosci. , 33 ( 1977 ), pp. 257 - 276 . MR 682262 | Zbl 0363.92022 · Zbl 0363.92022 · doi:10.1016/0025-5564(77)90142-0
[15] R.R. Gaines - J. Mawhin , Coincidence degree and nonlinear differential equations , Lecture Notes in Mathematics , vol. 568 , Springer-Verlag , Berlin , 1977 . MR 637067 | Zbl 0339.47031 · Zbl 0339.47031
[16] T. Gard , A generalization of the Naguno uniqueness criterion , Proc. Amer. Math. Soc. , 70 ( 1978 ), pp. 166 - 172 . MR 470288 | Zbl 0389.34003 · Zbl 0389.34003 · doi:10.2307/2042082
[17] T.C. Gard , Strongly flow-invariant sets , Appl. Analysis , 10 ( 1980 ), pp. 285 - 293 . MR 580813 | Zbl 0438.34039 · Zbl 0438.34039 · doi:10.1080/00036818008839310
[18] T.C. Gard - T.G. Hallam , Persistence in food webs-1. Lotka Volterra food chains , Bull. Math. Biol. , 41 ( 1979 ), pp. 877 - 891 . MR 640001 | Zbl 0422.92017 · Zbl 0422.92017
[19] P.M. Gruber , Aspects of convexity and its applications , Expo. Math. , 2 ( 1984 ), pp. 47 - 83 . MR 783125 | Zbl 0525.52001 · Zbl 0525.52001
[20] T.G. Hallam , A comparison principle for terminal value problems in ordinary differential equations , Trans. Amer. Math. Soc. , 169 ( 1972 ), pp. 49 - 57 . MR 306611 | Zbl 0257.34012 · Zbl 0257.34012 · doi:10.2307/1996229
[21] P. Hartman , Ordinary differential equations , Wiley , New York , 1964 . MR 171038 | Zbl 0125.32102 · Zbl 0125.32102
[22] J. Hofbauer , A general cooperation theorem for hypercycles , Monatsh. Math ., 91 ( 1981 ), pp. 233 - 240 . MR 619966 | Zbl 0449.34039 · Zbl 0449.34039 · doi:10.1007/BF01301790
[23] V. Hutson , A theorem on average Liapunov functions , Monatsh. Math ., 98 ( 1984 ), pp. 267 - 275 . Article | MR 776353 | Zbl 0542.34043 · Zbl 0542.34043 · doi:10.1007/BF01540776
[24] M.A. Krasnosel’skii , The operator of translation along trajectories of of differential equations , Amer. Math. Soc. , Providence, R.I. , 1968 . MR 223640
[25] V. Lakshmikantham - S. Leela , Differential and integral inequalities , vol. I , Academic Press , New York , 1969 . Zbl 0177.12403 · Zbl 0177.12403
[26] J.P. La Salle , The stability of dynamical systems , Reg. Conf. Ser. in Math., SIAM , Philadelp ia , 1976 .
[27] J. Massera , Contributions to stability theory , Ann. Math. , 64 ( 1956 ), pp. 182 - 206 . MR 79179 | Zbl 0070.31003 · Zbl 0070.31003 · doi:10.2307/1969955
[28] J. Mawhin , Functional analysis and boundary value problems, in Studies in ordinary differential equations , vol. 14 (J. K. Hale, ed.), The Math. Assoc. of America , U.S.A., 1977 . MR 473303 | Zbl 0371.34017 · Zbl 0371.34017
[29] J. Mawhin , Topological degree methods in nonlinear boundary value problems, Reg. Conf. Ser . in Math., CBMS no. 40 , Amer. Math. Soc. , Providence, R.I. , 1979 . MR 525202 | Zbl 0414.34025 · Zbl 0414.34025
[30] M. Nagumo , Eine hinreichende Bedingung für die Unität der Lösung von Differentialgleichungen erster Ordnung , Japan J. Math. , 3 ( 1926 ), pp. 107 - 112 . JFM 52.0438.01 · JFM 52.0438.01
[31] M. Nagumo , Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen, Proc. Phys .- Math. Soc. Japan , 24 ( 1942 ), pp. 551 - 559 . MR 15180 | Zbl 0061.17204 · Zbl 0061.17204
[32] L.C. Piccinini - G. Stampacchia - G. Vidossich , Ordinary differential equations in Rn, problems and methods , Springer-Verlag , Berlin and New York , 1984 . MR 740539 | Zbl 0535.34001 · Zbl 0535.34001
[33] R.M. Redheffer - W. Walter , Flow-invariant sets and differential inequalities in normed spaces , Appl. Analysis , 5 ( 1975 ), pp. 149 - 161 . MR 470401 | Zbl 0353.34067 · Zbl 0353.34067 · doi:10.1080/00036817508839117
[34] R. Reissig - G. Sansone - R. Conti , Qualitative Theorie nichtlinearer Differentialgleichungen , Cremonese , Roma , 1963 . MR 158121 | Zbl 0114.04302 · Zbl 0114.04302
[35] N. Rouche - P. Habets - M. Laloy , Stability theory by Liapunov’s Direct Method , Springer-Verlag , Berlin and New York , 1977 . MR 450715 | Zbl 0364.34022 · Zbl 0364.34022
[36] L. Salvadori , Famiglie ad un parametro di funzioni di Liapunov, nello studio della stabilità , Symposia Math. , 6 ( 1971 ), pp. 309 - 330 . MR 279396 | Zbl 0243.34099 · Zbl 0243.34099
[37] P. Schuster - K. Sigmund - R. Wolff , Dynamical systems under constant organization. - III: Cooperative and competitive behavior of hypercycles , J. Differential Equations , 32 ( 1979 ), pp. 357 - 368 . MR 535168 | Zbl 0384.34029 · Zbl 0384.34029 · doi:10.1016/0022-0396(79)90039-1
[38] M. Turinici , A singular perturbation result for a system of ordinary differential equations , Bull. Math. Soc. Sci. Math. R. S. Roumanie , 27 ( 1983 ), pp. 273 - 282 . MR 724153 | Zbl 0532.34040 · Zbl 0532.34040
[39] G. Vidossich , Solutions of Hallam’s problem on the terminal comparison principle for ordinary differential inequalities , Trans. Amer. Math. Soc. , 220 ( 1976 ), pp. 115 - 132 . MR 412524 | Zbl 0346.34007 · Zbl 0346.34007 · doi:10.2307/1997636
[40] P. Volkmann , Über die positive Invaranz einer abgeschlossenen Teilmenge eines Banachschen Raumes bezüglich der Differentialgleichung u’ = f(t, u) , J. reine angew. Math. , 285 ( 1976 ), pp. 59 - 65 . MR 415033 | Zbl 0326.34081 · Zbl 0326.34081 · doi:10.1515/crll.1976.285.59
[41] D.V.V. Wend , Existence and uniqueness of solutions of ordinary differential equations , Proc. Amer. Math. Soc. , 23 ( 1969 ), pp. 27 - 33 . MR 245879 | Zbl 0183.35604 · Zbl 0183.35604 · doi:10.2307/2037480
[42] J.A. Yorke , Invariance for ordinary differential equations , Math. Systems Theory , 1 ( 1967 ), pp. 353 - 372 . MR 226105 | Zbl 0155.14201 · Zbl 0155.14201 · doi:10.1007/BF01695169
[43] T. Yoshizawa , Stability theory by Liapunov’s second method , The Math. Soc. of Japan , Tokyo , 1966 . MR 208086 | Zbl 0144.10802 · Zbl 0144.10802
[44] F. Zanolin , Bound sets, periodic solutions and flow-invariance for ordinary differential equations in Rn some remarks, in Colloquium on Topological Methods in BPVs for ODEs , ISAS, Rend. Ist. Mat. Univ . Trieste , 19 ( 1987 ), pp. 76 - 92 . MR 941094 | Zbl 0651.34049 · Zbl 0651.34049
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