Repelling conditions for boundary sets using Liapunov-like functions. I. - Flow-invariance, terminal value problem and weak persistence
Fernandes, M. L. C. ; Zanolin, F.
Rendiconti del Seminario Matematico della Università di Padova, Tome 80 (1988), p. 95-116 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)
Publié le : 1988-01-01
@article{RSMUP_1988__80__95_0,
     author = {Fernandes, M. L. C. and Zanolin, Fabio},
     title = {Repelling conditions for boundary sets using Liapunov-like functions. I. - Flow-invariance, terminal value problem and weak persistence},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     volume = {80},
     year = {1988},
     pages = {95-116},
     mrnumber = {988116},
     zbl = {0672.34048},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RSMUP_1988__80__95_0}
}

Fernandes, M. L. C.; Zanolin, F. Repelling conditions for boundary sets using Liapunov-like functions. I. - Flow-invariance, terminal value problem and weak persistence. Rendiconti del Seminario Matematico della Università di Padova, Tome 80 (1988) pp. 95-116. http://gdmltest.u-ga.fr/item/RSMUP_1988__80__95_0/

[1] H. Amann, Gewöhnliche Differentialgleichungen, Walter de Gruyter, Berlin, 1983. | MR 713040 | Zbl 0823.34001

[2] J.-P. Aubin - A. CELLINA, Differential inclusions, Springer-Verlag, Berlin and New York, 1984. | MR 755330 | 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

[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

[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

[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

[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

[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

[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

[11] M.L.C. Fernandes - F. Zanolin, On periodic solutions, in a given set, for differential systems (preprint). | MR 1116980 | 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

[13] A. Fonda, Uniformly persistent semi-dynamical systems, Proc. Amer. Math. Soc. (to appear). | MR 958053 | Zbl 0667.34065

[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

[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

[16] T. Gard, A generalization of the Naguno uniqueness criterion, Proc. Amer. Math. Soc., 70 (1978), pp. 166-172. | MR 470288 | Zbl 0389.34003

[17] T.C. Gard, Strongly flow-invariant sets, Appl. Analysis, 10 (1980), pp. 285-293. | MR 580813 | Zbl 0438.34039

[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

[19] P.M. Gruber, Aspects of convexity and its applications, Expo. Math., 2 (1984), pp. 47-83. | MR 783125 | 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

[21] P. Hartman, Ordinary differential equations, Wiley, New York, 1964. | MR 171038 | Zbl 0125.32102

[22] J. Hofbauer, A general cooperation theorem for hypercycles, Monatsh. Math., 91 (1981), pp. 233-240. | MR 619966 | Zbl 0449.34039

[23] V. Hutson, A theorem on average Liapunov functions, Monatsh. Math., 98 (1984), pp. 267-275. | MR 776353 | Zbl 0542.34043

[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. | MR 379934 | Zbl 0177.12403

[26] J.P. La Salle, The stability of dynamical systems, Reg. Conf. Ser. in Math., SIAM, Philadelp ia, 1976. | MR 481301

[27] J. Massera, Contributions to stability theory, Ann. Math., 64 (1956), pp. 182-206. | MR 79179 | Zbl 0070.31003

[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

[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

[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

[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

[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

[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

[34] R. Reissig - G. Sansone - R. Conti, Qualitative Theorie nichtlinearer Differentialgleichungen, Cremonese, Roma, 1963. | MR 158121 | 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

[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

[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

[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

[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

[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

[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

[42] J.A. Yorke, Invariance for ordinary differential equations, Math. Systems Theory, 1 (1967), pp. 353-372. | MR 226105 | Zbl 0155.14201

[43] T. Yoshizawa, Stability theory by Liapunov's second method, The Math. Soc. of Japan, Tokyo, 1966. | MR 208086 | 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