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On the Neumann problem for an ordinary differential equation with discontinuous right-hand side. (English. Russian original) Zbl 1091.34012
Differ. Equ. 41, No. 6, 791-796 (2005); translation from Differ. Uravn. 41, No. 6, 755-760 (2005).
The authors consider the following Neumann problem with discontinuous right-hand side \[ x''\in g(t,x,x'),\tag{1} \]
\[ x'(0)=r,\;x'(T)=s, \tag{2} \] where \(g:U\rightarrow \mathbb{R}\) is a multifunction, \(U=(a,b)\times \mathbb{R}\times \mathbb{R}\), \(a<0<T<b\) and \(r,\;s\in\mathbb{R}\).
By considering \(y=x'\) and \(z=(x,y)\in\mathbb{R}\), the original problem becomes
\[ z'\in h(t,z),\tag{3} \]
\[ y(0)=r,\;\;y(T)=s,\tag{4} \] with \(h(t,z)=(y,g(t,x,y))\).
The authors prove the existence of solutions of problem (3), (4), assuming that the solutions of the corresponding Cauchy problem exist, are continuous and the solution set of each specific Cauchy problem is acyclic. To do so they use, among other tools, a version of the principle of continuation with respect to a parameter.
34B15 Nonlinear boundary value problems for ordinary differential equations
34A60 Ordinary differential inclusions
34A36 Discontinuous ordinary differential equations
Full Text: DOI
[1] Filippov, V.V., Mat. Sb., 1997, vol. 188, no.6, pp. 139–160.
[2] Filippov, V.V., Mat. Sb., 2002, vol. 193, no.11, pp. 139–160.
[3] Filippov, V.V., Differents. Uravn., 1997, vol. 33, no.8, pp. 1057–1068.
[4] Filippov, V.V., Differents. Uravn., 2001, vol. 37, no.8, pp. 1049–1061.
[5] Filippov, V.V., Mat. Zametki, 2003, vol. 73, no.6, pp. 910–922.
[6] Filippov, V.V., Prostranstva reshenii obyknovennykh differentsial’nykh uravnenii (Solution Spaces of Ordinary Differential Equations), Moscow, 1993.
[7] Filippov, V.V., Uspekhi Mat. Nauk, 1993, vol. 48, no.1, pp. 103–154.
[8] Filippov, V.V., Basic Topological Structures of Ordinary Differential Equations, Kluwer, 1998. · Zbl 0905.34001
[9] Filippov, V.V., Fund. Prikl. Matematika, 1994, vol. 3, no.2, pp. 587–623.
[10] Filippov, V.V., Differents. Uravn., 1997, vol. 33, no.4, pp. 75–79.
[11] Seifert, H. and Threlfall, W., Lehrbuch der Topologie, Leipzig: Teubner, 1934. Translated under the title Topologiya, Leningrad, 1938.
[12] Granas, A., Guenter, R.B., and Lee, J.W., Pacific J. of Math., 1978, vol. 74, no.2, pp. 67–82.
[13] Granas, A., Guenter, R.B., and Lee, J.W., Pacific J. of Math., 1983, vol. 104, no.1, pp. 95–109.
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