| Subject and Goal. The Linear pendulum equation is one of the first mathematical models that undergraduate students familiarize with in any introductory Differential Equations course. Boundary value/periodic problems for the nonlinear pendulum equation (or, more generally, second order nonlinear ODEs) have been the focus of nonlinear analysis study for a long time. An important step was done by P. Hartman who established the existence result for the boundary value problem y&d3;=ft,y, y&d2; y0=y1 =0, 1 where the function f : [0, 1] × Rn×R n→Rn satisfies the so-called Hartman-Nagumo conditions which, informally, means that f is a reasonable function having sub-quadratic growth on y˙. Although Hartman's existence result was extended to more general settings by many authors, the problem of estimating the minimal number of solutions to (1) has not been carefully studied. The goal of this dissertation is to show how the equivariant degree theory can be used for the systematic study of multiple solutions to several (symmetric) generalizations of (1) and for the classification of symmetric properties of these solutions.;Method. There are a lot of methods of Nonlinear Analysis that are used to study the impact of symmetries of equations on symmetric properties of solutions: equivariant singularity theory, equivariant Morse theory, Lusternik-Schnirelman theory, and Morse-Floer complex techniques, to mention a few. Although these methods are very effective in the standard settings, their application is either impossible or encounters serious difficulties if: (i) the group of symmetries is large, (ii) the dimension of the problem is high, and (iii) multiplicities of eigenvalues of linearizations are large, etc. During the last twenty years, the equivariant degree theory emerged in Nonlinear Analysis. The equivariant degree is a topological tool allowing "counting" orbits of solutions to (symmetric) equations in the same way as the usual Brouwer degree does, but according to their symmetric properties. In addition, the equivariant degree theory has all the attributes allowing its application in settings related to (i)–(iii) and, in many cases, allows a complete computerization.;Results. In the dissertation, we: (i) set up the abstract functional analysis framework for studying symmetric properties of multiple solutions to symmetric generalizations of the problem (1) via the equivariant degree approach; (ii) describe wide classes of second order boundary value problems admitting dihedral symmetries to which the abstract theory can be effectively applied; and (iii) apply the obtained results to several classes of implicit second order symmetric differential equations. The abstract results are supported by concrete examples and numerical computations. |
