| As it is well known, many real world processes and phenomena exhibit periodic movements. The mathematical models describing these phenomena can be regarded as differential equations with periodic boundary conditions. Numerous results have been obtained from the study of such model problems during the past decades and the results have proven to be both fundamental and practical. But some of these processes may possess properties other than just periodicity. One such property is anti-periodicity. To study it, we use models such as differential equations with anti-periodic boundary conditions.; It is clear that an anti-periodic function is also a periodic one, which implies that anti-periodic problems are closely related to periodic ones. But the field study is by no means either trivial or extensive. To date, only a few abstract results are available (cf. (1) for references). In this dissertation, we offer a unified study for first order, second order and parabolic differential equations with anti-periodic boundary conditions. A new approach, utilizing the method of upper and lower solutions and monotone iterative techniques, is developed. The method of (generalized) quasilinearization is employed to obtain quadratic convergence of the monotone sequences. The construction of anti-periodic Green's functions is also discussed for second order, higher order and parabolic equations. Some existence and uniqueness results are then obtained by applying Banach's fixed point theorem. |
