an:05563005
Zbl 1177.34015
Andres, Jan; Ko??u??n??kov??, Martina; Malaguti, Luisa
Bound sets approach to boundary value problems for vector second-order differential inclusions
EN
Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 1-2, 28-44 (2009).
00249962
2009
j
34A60 34B15 47H04 47N20
upper-Carath??odory differential incusions; Floquet problem; viability result
The paper deals with the second-order boundary value problem
\[
\ddot x(t)\in F(t,x(t),\dot x(t))\quad \text{for a.a.}\;t\in J, x\in S, \leqno(1)
\]
where \(J=[t_0,t_1]\) is a compact interval, \(F: J\times \mathbb{R}^n\times \mathbb{R}^n \multimap \mathbb{R}^n\) is an upper-Carath??odory mapping and \(S\) is a subset of \(AC^1(J,\mathbb{R}^n)\). The authors develop a continuation principle for the solvability of (1) using fixed point index arguments. The main assumption which yields the possibility to apply the continuation principle is the transversality condition which is guaranted here by means of Liapunov-like bounding functions. In addition, for the Floquet semi-linear problem
\[
\ddot x(t)+A(t)\dot x(t)+B(t)x(t)\in F(t,x(t),\dot x(t))\quad \text{for a.a.}\;t\in J,
\]
\[
x(t_1)=Mx(t_0),\quad \dot x(t_1)=N\dot x(t_0),
\]
where \(A,B:J\to \mathbb{R}^n\times \mathbb{R}^n\) are integrable matrix functions and \(M\) and \(N\) are real \(n\times n\) matrices with \(M\) non-singular, a viability result will be obtained by means of a bound sets technique.
Irena Rach??nkov?? (Olomouc)