| In this thesis, the main research includes three aspects: the periodic boundary value problem (PBVP), the two-point boundary value problem (BVP) and reversed upper lower solutions for the differential equations in abstract spaces. At first, in ordinary Banach spaces, under the con- ditions of the normal cone or the regular cone and only existence one upper solution or one lower solution, the existence of the maximal and minimal solutions and the iterative sequence of corresponding solutions of the periodic boundary value problem for integro-differential equa- tions are investigated. In abstract spaces, when the equations is dis- continuous the exsistence of second order differential equations value problem is investigated. For this problem includes two aspects: without u′item and with u′item. When the second order differential equa- tions without u′item, by use of the theory and property of mixed monotone operator, the iterative sequence of solutions and the error estimations for the approximation solutions are given. When the differential equations with u′item applying the integral transform, the equations is became a integro-differential equation without u′item, the iterative sequence of generalized solutions and the error estimations for the convergent iterative sequence are aslo obtained, applying monotone iterative method. The last, dispart two conditions investigate the reversed upper solutions and lower solutions. The first order differential equations with the nonlinear boundary condition is investigated. In this conditions, by established a new comparation theorem solving the problem in the course of prove. By establishing a new comparation theorem, applying the monotone iterative method investigate the periodic boundary value problem for second order differential equations in Banach spaces the regular cone when the upper solutions is small than the lower solutions, critera on the existence of maximal solutions and minimal solutions are obtained. |
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