Nonlinear evolution equations and singular perturbations of variational inequalities

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Since Newton's invention, differential equations, that are equations containing unknown functions and their derivatives, have been and remain the most useful tool to describe and study mathematically various phenomina arising in physics and engineerings. From the viewpoint of abstract nonlinear evolution equations, we considered variational inequalities, that are problems of differential equations with constrains given by inequalities and imposed upon the unknown functions, and proved the existence of the singular limit of a problem containing a parameter whose limit changes the type of equation. Also in the course of the study, we proposed the concept of quasi-variational analysis by synthesizing the principal tools of nonlinear analysis: variational principles and fixed point analysis, and introduced the concept of a quasi-subdifferential operator which is a new class of abstract nonlinear operators.