Global Solutions For Initial Value Problems Of Ordinary Differential Equations In Banach Spaces

In this thesis, by using the theory of noncompactness measure, Sadovskii fixed point theorem, S.Szuffla theorem, the theory of cones and the partial-ordered method combined with the upper and lower solutions and monotone iterative techniques. We have studied the existence and uniqueness of global solutions for the bellow first order initial value problem of ordinary differential equation in Banach space EHereThe main results are summarized as follows:1 .Under the compact type conditions, Existence of global solutions of (1) on [t0,+) is discussed. Theory of non-compactness measure, Sadovskii fixed point theorem of condensing map and S.Szuffla theorem are applied to these problems and four existence results are obtained. The results improve and extend the relevant conclusions in reference [1][2][7].2. Under the compact type condition a comparability theorem is founded on [0,-H) by using the theory of noncompactness measure and upper and lower solutions method with monotone iterative techniques, three existence results of global solutions for (1) are obtained. The results are new, improve the relevant conclusions in reference [8].3. The existence of global solutions on interval [0,a] (a > 0)for the first order initial value problem of discontinuous equations in Banach space (1) is discussed. The fixed-point theorem of T-monotone increasing operators and the partial-ordered method are applied to these problems.An existence result is obtained .The results extend the relevantconclusions in reference [7].4.By using the theory of cones and the partial-ordered method with upper solution or lower solution. Existence and uniqueness of global solutions on interval [0, a] (a > 0) for the first order initial valueproblem of discontinuous equations in Banach space (1) is discussed. Two results are obtained. The same method is applied to the first order periodic value boundary problem of discontinuous equations in Banach space. Two existence and uniqueness results are also obtained. The first two results improve the relevant conclusions in reference [10] [14]. Later two results are new and independent of other known results.