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

Conditions for global existence of solutions of ordinary differential, stochastic differential, and parabolic equations

First, we prove a necessary and sufficient condition for global in time existence of all solutions of an ordinary differential equation (ODE). It is a condition of one-sided estimate type that is formulated in terms of so-called proper functions on extended phase space. A generalization of this idea to stochastic differential equations (SDE) and parabolic equations (PE) allows us to prove similar necessary and sufficient conditions for global in time existence of solutions of special sorts: L1-complete solutions of SDE (this means that they belong to a certain functional space of L1 type) and the so-called complete Feller evolution families giving solutions of PE. The general case of equations on noncompact smooth manifolds is under consideration

Global and Stochastic Analysis with Applications to Mathematical Physics

Methods of global analysis and stochastic analysis are most often applied in mathematical physics as separate entities, thus forming important directions in the field. However, while combination of the two subject areas is rare, it is fundamental for the consideration of a broader class of problems This book develops methods of Global Analysis and Stochastic Analysis such that their combination allows one to have a more or less common treatment for areas of mathematical physics that traditionally are considered as divergent and requiring different methods of investigation Global and Stochastic Analysis with Applications to Mathematical Physics covers branches of mathematics that are currently absent in monograph form. Through the demonstration of new topics of investigation and results, both in traditional and more recent problems, this book offers a fresh perspective on ordinary and stochastic differential equations and inclusions (in particular, given in terms of Nelson's mean derivatives) on linear spaces and manifolds. Topics covered include classical mechanics on non-linear configuration spaces, problems of statistical and quantum physics, and hydrodynamics A self-contained book that provides a large amount of preliminary material and recent results which will serve to be a useful introduction to the subject and a valuable resource for further research. It will appeal to researchers, graduate and PhD students working in global analysis, stochastic analysis and mathematical physic

Total and local topological indices for maps of Hilbert and Banach manifolds

Total and local topological indices are constructed for various types of continuous maps of infinite-dimensional manifolds and ANR's from a broad class. In particular the construction covers locally compact maps with compact sets of fixed points (e.g. maps having a certain finite iteration compact or having compact attractor or being asymptotically compact etc.); condensing maps ($k$-set contraction) with respect to Kuratowski's or Hausdorff's measure of non-compactness on Finsler manifolds; maps, continuous with respect to the topology of weak convergence, etc. The characteristic point is that all conditions are formulated in internal terms and the index is in fact internal while the construction is produced through transition to the enveloping space. Examples of applications are given

On a two-point boundary value problem for second-order differential inclusions on Riemannian manifolds

We consider second-order differential inclusions on a Riemannian manifold with lower semicontinuous right-hand sides. Several existence theorems for solutions of two-point boundary value problem are proved to be interpreted as controllability of special mechanical systems with control on nonlinear configuration spaces. As an application, a statement of controllability under extreme values of controlling force is obtained

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

Topological index for condensing maps on Finsler manifolds with applications to functional-differential equations of neutral type

The topological index for maps of infinite-dimensional Finsler manifolds, condensing with respect to internal Kuratowski's measure of non-compactness, is constructed under the hypothesis that the manifold can be embedded into a certain Banach linear space as a neighbourhood retract so that the Finsler norm in tangent spaces and the restriction of the norm from enveloping space on the tangent spaces are equivalent. It is shown that the index is an internal topological characteristic, i.e. it does not depend on the choice of enveloping space, embedding, etc. The total index (Lefschetz number) and the Nielsen number are also introduced. The developed machinery is applied to investigation of functional-differential equations of neutral type on Riemannian manifolds. A certain existence and uniqueness theorem is proved. It is shown that the shift operator, acting in the manifold of $C^1$-curves, is condensing, its total index is calculated to be equal to the Euler characteristic of (compact) finite-dimensional Riemannian manifold where the equation is given. Some examples of calculating the Nielsen number are also considered