Abstract and Applied Analysis
Volume 2006 (2006), Article ID 30395, 9 pages
doi:10.1155/AAA/2006/30395
On the two-point boundary value problem for quadratic second-order differential equations and
inclusions on manifolds
1Mathematics Faculty, Voronezh State University, Universitetskaya pl. 1, Voronezh 394006, Russia
2Physics and Mathematics Faculty, Kursk State University, ul. Radishcheva 33, Kursk 305416, Russia
Received 14 January 2005; Accepted 1 March 2005
Copyright © 2006 Yuri E. Gliklikh and Peter S. Zykov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The two-point boundary value problem for second-order
differential inclusions of the form (D/dt)m˙(t)∈F(t,m(t),m˙(t)) on complete Riemannian manifolds is
investigated for a couple of points, nonconjugate along at least
one geodesic of Levi-Civitá connection, where D/dt is the covariant derivative of Levi-Civitá connection and F(t,m,X) is a set-valued vector with quadratic or less than quadratic growth in
the third argument. Some interrelations between certain geometric
characteristics, the distance between points, and the norm of
right-hand side are found that guarantee solvability of the above
problem for F with quadratic growth in X. It is shown that
this interrelation holds for all inclusions with F having less than quadratic growth in X, and so for them the problem is solvable.