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Abstract and Applied Analysis
Volume 2006 (2006), Article ID 30395, 9 pages

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.


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.