FIXED POINT APPROACH TO SOME TWO-POINT BOUNDARY VALUE PROBLEMS FOR DIFFERENTIAL INCLUSIONS ON MANIFOLDS

We investigate the two-point boundary value problem for second order differential inclusions of the form D/dt (m) over dot(t) is an element of F(t, m(t), (m) over dot(t)) on a, complete Riemannian manifold for a couple of points, non-conjugate along at least one geodesic of Levi-Civita connection, where D/dt is the covariant derivative of Levi-Civita. connection and F(t, m, X) is convex valued and satisfies the upper Caratheodory condition or is almost lower semi-continuous set-valued vector field such that parallel to F(t, m, X)parallel to < a(t, m)parallel to X parallel to(2) with continuous a(t, m) > 0. Some conditions on certain geometric characteristics, On the distance between points and on a(t, m) are found, under which the problem is solvable on any time interval. The solution is constructed from a fixed point of a certain integral-type operator, acting in the space of continuous curves in the tangent space at initial point. The existence of fixed point is proved by application of Bohnenblust-Karlin and Schauder theorems.