Czechoslovak Mathematical Journal, Vol. 69, No. 1, pp. 99-116, 2019


On Kneser solutions of the $n$-th order nonlinear differential inclusions

Martina Pavlačková

Received April 24, 2017.   Published online July 13, 2018.

Abstract:  The paper deals with the existence of a Kneser solution of the $n$-th order nonlinear differential inclusion $\align{x}^{(n)}(t)\in-A_1(t,x(t),\ldots,x^{(n-1)}(t))x^{(n-1)}(t)-\ldots-A_n(t,x(t),\ldots,&x^{(n-1)}(t))x(t) \\ &\text{for a.a.} t\in[a,\infty),$ where $a\in(0,\infty)$, and $A_i [a,\infty) \times\mathbb{R}^n\to\mathbb{R}$, $i=1,\ldots,n,$ are upper-Carathéodory mappings. The derived result is finally illustrated by the third order Kneser problem.
Keywords:  asymptotic $n$-th order vector problems; $R_{\delta}$-set; inverse limit technique; Kneser problem
Classification MSC:  34A60, 34B15, 34B40
DOI:  10.21136/CMJ.2018.0191-17

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] R. P. Agarwal, D. O'Regan: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht (2001). DOI 10.1007/978-94-010-0718-4 | MR 1845855 | Zbl 0988.34002
[2] J. Andres, G. Gabor, L. Górniewicz: Topological structure of solution sets to multi-valued asymptotic problems. Z. Anal. Anwend. 19 (2000), 35-60. DOI 10.4171/ZAA/937 | MR 1748055 | Zbl 0974.34045
[3] J. Andres, G. Gabor, L. Górniewicz: Acyclicity of solution sets to functional inclusions. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 49 (2002), 671-688. DOI 10.1016/S0362-546X(01)00131-6 | MR 1894303 | Zbl 1012.34011
[4] J. Andres, L. Górniewicz: Topological Fixed Point Principles for Boundary Value Problems. Topological Fixed Point Theory and Its Applications 1, Kluwer Academic Publishers, Dordrecht (2003). DOI 10.1007/978-94-017-0407-6 | MR 1998968 | Zbl 1029.55002
[5] J. Andres, M. Pavlačková: Asymptotic boundary value problems for second-order differential systems. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71 (2009), 1462-1473. DOI 10.1016/j.na.2008.12.013 | MR 2524361 | Zbl 1182.34038
[6] J. Andres, M. Pavlačková: Boundary value problems on noncompact intervals for the $n$-th order vector differential inclusions. Electron. J. Qual. Theory Differ. Equ. 2016 (2016), 19 pages. DOI 10.14232/ejqtde.2016.1.60 | MR 3547436 | Zbl 06806157
[7] J. Appell, E. De Pascale, N. H. Thái, P. P. Zabreǐko: Multi-valued superpositions. Diss. Math. 345 (1995), 97 pages. MR 1354934 | Zbl 0855.47037
[8] J.-P. Aubin, A. Cellina: Differential Inclusions. Set-Valued Maps and Viability Theory. Grundlehren der Mathematischen Wissenschaften 264, Springer, Berlin (1984). DOI 10.1007/978-3-642-69512-4 | MR 0755330 | Zbl 0538.34007
[9] M. Bartušek, M. Cecchi, M. Marini: On Kneser solutions of nonlinear third order differential equations. J. Math. Anal. Appl. 261 (2001), 72-84. DOI 10.1006/jmaa.2000.7473 | MR 1850957 | Zbl 0995.34025
[10] M. Bartušek, Z. Došlá: Oscillation of third order differential equation with damping term. Czech. Math. J. 65 (2015), 301-316. DOI 10.1007/s10587-015-0176-3 | MR 3360427 | Zbl 1363.34095
[11] K. Borsuk: Theory of Retracts. Monografie Matematyczne 44, PWN, Warszawa (1967). MR 0216473 | Zbl 0153.52905
[12] M. Cecchi, M. Furi, M. Marini: About the solvability of ordinary differential equations with asymptotic boundary conditions. Boll. Unione Mat. Ital., VI. Ser., C, Anal. Funz. Appl. 4 (1985), 329-345. MR 0805224 | Zbl 0587.34013
[13] E. Fermi: Un metodo statistico per la determinazione di alcune prioriet á dell'atomo. Rend. R. Accad. Nat. Lincei 6 (1927), 602-607. (In Italian.)
[14] A. F. Filippov: Differential Equations with Discontinuous Right-Hand Sides. Mathematics and Its Applications (Soviet Series) 18 Kluwer Academic Publishers, Dordrecht (1988). DOI 10.1007/978-94-015-7793-9 | MR 1028776 | Zbl 0664.34001
[15] G. Gabor: On the acyclicity of fixed point sets of multivalued maps. Topol. Methods Nonlinear Anal. 14 (1999), 327-343. DOI 10.12775/TMNA.1999.036 | MR 1766183 | Zbl 0954.54022
[16] L. Górniewicz: Topological Fixed Point Theory of Multivalued Mappings. Mathematics and Its Applications 495, Kluwer Academic Publishers, Dordrecht (1999). DOI 10.1007/978-94-015-9195-9 | MR 1748378 | Zbl 0937.55001
[17] J. R. Graef, J. Henderson, A. Ouahab: Impulsive Differential Inclusions. A Fixed Point Approach. De Gruyter Series in Nonlinear Analysis and Applications 20, De Gruyter, Berlin (2013). DOI 10.1515/9783110295313 | MR 3114179 | Zbl 1285.34002
[18] P. Hartman, A. Wintner: On the non-increasing solutions of $y"=f(x,y,y')$. Am. J. Math. 73 (1951), 390-404. DOI 10.2307/2372184 | MR 0042004 | Zbl 0042.32601
[19] L. V. Kantorovich, G. P. Akilov: Functional Analysis in Normed Spaces. International Series of Monographs in Pure and Applied Mathematics 46, Pergamon Press, Oxford (1964). MR 0213845 | Zbl 0127.06104
[20] I. T. Kiguradze, T. A. Chanturia: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Mathematics and Its Applications (Soviet Series) 89, Kluwer Academic Publishers, Dordrecht (1993). DOI 10.1007/978-94-011-1808-8 | MR 1220223 | Zbl 0782.34002
[21] I. T. Kiguradze, B. L. Shekhter: Singular boundary value problems for second-order ordinary differential equations. J. Soviet Math. 43 (1988), 2340-2417 English. Russian original \kern3sp translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 30 (1987), 105-201. MR 0925830 | Zbl 0631.34021
[22] A. Kneser: Untersuchung und asymptotische Darstellung der Integrale gewisser Differentialgleichungen bei grossen reellen Werten des Arguments I, II. J. für Math. 116 (1896), 178-212 117 (1896), 72-103. (In German.) JFM 27.0253.03
[23] V. A. Kozlov: On Kneser solutions of higher order nonlinear ordinary differential equations. Ark. Mat. 37 (1999), 305-322. DOI 10.1007/BF02412217 | MR 1714766 | Zbl 1118.34317
[24] J. Kurzweil: Ordinary Differential Equations. Introduction to the Theory of Ordinary Differential Equations in the Real Domain. Studies in Applied Mechanics 13, Elsevier Scientific Publishing, Amsterdam; SNTL Publishers of Technical Literature, Praha (1986). MR 0929466 | Zbl 0667.34002
[25] D. O'Regan, A. Petruşel: Leray-Schauder, Lefschetz and Krasnoselskii fixed point theory in Fréchet spaces for general classes of Volterra operators. Fixed Point Theory 9 (2008), 497-513. MR 2464132 | Zbl 1179.47049
[26] S. Padhi, S. Pati: Theory of Third-Order Differential Equations. Springer, New Delhi (2014). DOI 10.1007/978-81-322-1614-8 | MR 3136420 | Zbl 1308.34002
[27] N. Partsvania, Z. Sokhadze: Oscillatory and monotone solutions of first-order nonlinear delay differential equations. Georgian Math. J. 23 (2016), 269-277. DOI 10.1515/gmj-2016-0015 | MR 3507955 | Zbl 1342.34091
[28] L. H. Thomas: The calculation of atomic fields. Proceedings Cambridge 23 (1927), 542-548. DOI 10.1017/S0305004100011683 | JFM 53.0868.04
[29] I. I. Vrabie: Compactness Methods for Nonlinear Evolutions. Pitman Monographs and Surveys in Pure and Applied Mathematics 75, Longman Scientific & Technical, Harlow; John Wiley & Sons, New York (1995). MR 1375237 | Zbl 0842.47040

Affiliations:   Martina Pavlačková, Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic, e-mail: martina.pavlackova@upol.cz


 
PDF available at: