Mathematical Sciences: Non-Autonomous Ordinary and Partial Differential Equations

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Nonlinear nonautonomous differential equations have attracted the attention of many mathematicians and mathematical scientists. The description and modeling of physical phenomena frequently involve such equations. Progress has not been easy but topological and variational methods have been relatively successful in yielding existence and multiplicity information regarding boundary value problems and periodic solutions. However, progress has been much slower in our understanding of the dynamics of such systems. Many classes of these problems have a rather sparse literature, perhaps due to the fact that the topological degree and variational methods usually yield little or no information on dynamics. The proposed research will develop a unified approach to these difficulties through the application of homotopy index theory and analytical methods to parameter dependent skew product flows associated with the nonautonomous differential equations,as well as topological degree, semigroup methods, and monotone iterative techniques. Continuation methods for asymptotic stability, existence of bounded solutions and other dynamical questions will be developed.