About DML-CZ | FAQ | Conditions of Use | Math Archives | Contact Us
  Previous |  Up |  Next

Article

Pavlačková, Martina
On Kneser solutions of the $n$-th order nonlinear differential inclusions. (English). Czechoslovak Mathematical Journal, vol. 69 (2019), issue 1, pp. 99-116
MSC: 34A60, 34B15, 34B40 | MR 3923578 | Zbl 07088773 | DOI: 10.21136/CMJ.2018.0191-17
Keywords:
asymptotic $n$-th order vector problems; $R_{\delta }$-set; inverse limit technique; Kneser problem
Summary:
The paper deals with the existence of a Kneser solution of the $n$-th order nonlinear differential inclusion \begin {eqnarray} {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)\nonumber \\ &\text {for a.a.} \ t\in [a,\infty ),\nonumber \end {eqnarray} where $a\in (0,\infty )$, and $A_i\colon [a,\infty ) \times \mathbb {R}^{n}\to \mathbb {R}$, $i=1,\ldots ,n,$ are upper-Carathéodory mappings. The derived result is finally illustrated by the third order Kneser problem.
References:
[1] Agarwal, R. P., O'Regan, D.: 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] Andres, J., Gabor, G., Górniewicz, L.: 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] Andres, J., Gabor, G., Górniewicz, L.: 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] Andres, J., Górniewicz, L.: 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] Andres, J., Pavlačková, M.: 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] Andres, J., Pavlačková, M.: 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] Appell, J., Pascale, E. De, Thái, N. H., ıko, P. P. Zabre\v: Multi-valued superpositions. Diss. Math. 345 (1995), 97 pages. MR 1354934 | Zbl 0855.47037
[8] Aubin, J.-P., Cellina, A.: 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] Bartušek, M., Cecchi, M., Marini, M.: 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] Bartušek, M., Došlá, Z.: 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] Borsuk, K.: Theory of Retracts. Monografie Matematyczne 44, PWN, Warszawa (1967). MR 0216473 | Zbl 0153.52905
[12] Cecchi, M., Furi, M., Marini, M.: 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] Fermi, E.: Un metodo statistico per la determinazione di alcune prioriet á dell'atomo. Rend. R. Accad. Nat. Lincei 6 (1927), 602-607 Italian.
[14] Filippov, A. F.: 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] Gabor, G.: 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] Górniewicz, L.: 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] Graef, J. R., Henderson, J., Ouahab, A.: 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] Hartman, P., Wintner, A.: 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] Kantorovich, L. V., Akilov, G. P.: 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] Kiguradze, I. T., Chanturia, T. A.: 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] Kiguradze, I. T., Shekhter, B. L.: Singular boundary value problems for second-order ordinary differential equations. J. Soviet Math. 43 (1988), 2340-2417 English. Russian original translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 30 1987 105-201. DOI 10.1007/BF01100361 | MR 0925830 | Zbl 0631.34021
[22] Kneser, A.: 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 German \99999JFM99999 27.0253.03. MR 1580424
[23] Kozlov, V. A.: 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] Kurzweil, J.: 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] O'Regan, D., Petruşel, A.: 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] Padhi, S., Pati, S.: Theory of Third-Order Differential Equations. Springer, New Delhi (2014). DOI 10.1007/978-81-322-1614-8 | MR 3136420 | Zbl 1308.34002
[27] Partsvania, N., Sokhadze, Z.: 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] Thomas, L. H.: The calculation of atomic fields. Proceedings Cambridge 23 (1927), 542-548 \99999JFM99999 53.0868.04. DOI 10.1017/S0305004100011683
[29] Vrabie, I. I.: 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
Partner of
EuDML logo