Non-linear Analysis and Dynamical System

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The aim of this research is to investigate how we can extend the linear operator theory to non-linear operator and to study chaos and fractal which are characteristic feature in non-linear theory, by using operator theory, probability theory, potential theory, differential equation theory and so on.In fact, we have classified the limit set of cellular automata, by using operator theory, into four types : stable state, cyclic state, chaotic state and others. As a result, we can extend some property of linear operators to non-linear operators. We also investigate the limit set of cellular automata by embedding them into finite-valued upper semi-continuous function spaces and defined the topology in which the cellular automata converge to the limit set, by considering how linear property can be extended to non-linear cases.Dynamical system of strongly continuous semi-groups on weighted functions has already been characterized with their spectral property in some sense. As another characterization, we give necessary and sufficient conditions for the semi-group to be super-cyclic, hyper-cyclic and chaotic, by using admissible weight functions. We can extend this result to the property of evolution equation, since its solution often corresponds to the strongly continuous semi-group. It is interesting that even if the semi-group consists of bounded linear operators, it sometimes occur chaotic. This may happen because the generator is bounded.As shown in the list of published papers, we get many other results in operator theory, probability theory, potential theory, differential equation theory, by considering how we can extend linear case to non-linear case or what is the essential property of non-linear analysis.