On the two-point boundary value problem for quadratic second-order differential equations and inclusions on manifolds

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

ON THE TWO-POINT BOUNDARY VALUE PROBLEM FOR QUADRATIC SECOND-ORDER DIFFERENTIAL EQUATIONS AND INCLUSIONS ON MANIFOLDS

The two-point boundary value problem for second-order differential inclusions of the form (D/dt)ṁ(t) ∈ F(t,m(t),ṁ(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 setvalued 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