On two-point boundary value problems for second-order differential inclusions on manifolds

Abstract

We investigate the two-point boundary value problem for second-order differential inclusions of the form on a complete Riemannian manifold for a couple of points, non-conjugate along at least one geodesic of Levi-Civita connection, where is the covariant derivative of Levi-Civita connection and F(t, m, X) is a set-valued vector field (it is either convexvalued and satisfies the upper Caratheodory condition or it is almost lower semi-continuous) such that where f : [0, ∞) → [0, ∞) is an arbitrary continuous function, increasing on [0, ∞). Some conditions on certain geometric characteristics, on the distance between points and on the length of time interval, under which the problem is solvable, are found. A generalization to inclusions of the same sort subjected to a non-holonomic constraint is also presented.

Keywords:

• Riemannian manifolds
• two-point boundary value problem
• second-order differential inclusions
• non-conjugate points
• constraint

AMS Subject Classifications::

• 58C06
• 34B15
• 70G45