Lagoda V. The nonlocal existence theorems of V-bounded solutions and invariant manifolds of nonlinear systems.

The thesis is devoted to studying the following problems for nonlinear systems: the existence of global solutions on the whole real axis with certain boundedness properties; the detection of invariant manifolds created by such solutions. We develop a V-W-pair technique to study strongly nonlinear nonautonomous systems by means of evaluating and guiding functions. For such systems, we generalize a number of previously known results concerning the existence and uniqueness of bounded and almost periodic solutions. Using the V-W-pair of auxiliary functions, new sufficient conditions for the existence of Lipschitzian invariant section for nonlinear indefinite monotone extension of dynamical system on m-dimensional torus are established. We prove a nonlocal theorem on the existence of invariant torus for indefinite monotone system on . New sufficient conditions for the existence of invariant section for an indefinite monotone nonlinear extension of dynamical system on noncompact manifold are established, and a nonlocal theorem is proved on the existence of invariant section for indefinite monotone system on trivial fiber bundle with the base .