The Existence Of The Solutions To The Sturm-Liouville Boundary Value Problem In Banach Space

Based on the partial order theory, Kuratowski measure of noncompactness, fixed point theorem of condensing mapping and the fixed point index theory in cones, the paper discussed the existence of solutions to the Sturm-Liouville problemsin Banach spaces and the main results are as follows:1. The existence of the solutions to the Sturm-Liouville problem in Banach space is discussed by establishing a new maximum principle.where f: I x E -E, which does not require to be continuous, only satisfies the condition of weak Caratheodory condition. The author obtains the existence results of upper and lower solutions by using the monotone iterative method with upper and lower solutions.2. By doing precise computation of spectral radius of linear operator of linear equation and by using measure of noncompactness and Leray-schauder type fixed point theorem of condensing mapping , the existence and the uniqueness of the solutions are obtained, which extend the results recently achieved in this field.3. As for the existence of positive solutions, the cone theory and the fixed point index of condensing mapping are employed, and the results of the existence of positive solutions are obtained in the case of superlinear and sublinear. The conclusions extend and improve the existence theorem which Lou Ben-dong extablished in 1996 about the question of Sturm-Liouville of Banach spaces.