BOUNDARY VALUE PROBLEM FOR FUNCTIONAL DIFFERENTIAL INCLUSIONS ON MANIFOLDS AND FIXED POINTS OF INTEGRAL-TYPE OPERATORS

We investigate the boundary value problem for second order functional differential inclusions of the form D/dt<(m)over dot>(t) is an element of F(t, m(t)(theta), <(m)over dot>t (theta)) on a complete Riemannian manifold for a C-1-smooth curve phi : [-h, 0] -> M as initial value, and a point m(1) that is non-conjugate with phi(0) along at least one geodesic of Levi-Civita connection. Here D/dt is the covariant derivative of Levi-Civita connection and F(t, m(theta), X(theta)) is a set-valued vector field with closed convex values that satisfies upper Caratheodory condition and is given on couples: a continuous curve m(theta) in M, theta is an element of [-h, 0], and a vector field X(theta) along m(theta) that is continuous from the left and has limits from the right, under the assumption that F has uniformly quadratic or less than quadratic growth in velocity. Some conditions on certain geometric characteristics and on the distance between phi(0) and m(1), under which the problem is solvable, are found. The solution is constructed from a fixed point of an integral-type operator.