Multifunction fixed point theory and its application to an initial-value program

In this dissertation the primary concern is with showing the existence of a solution to the initial-value problem x(t) [epsilon] F(t,x(t)) x(0) = x[lowered o]. The function x is once-differentiable on a closed interval of real numbers having left endpoint zero into a Banach space. The multifunction F maps the cross product of the interval with the Banach space into the Banach space and x[lowered o] is in the Banach space. The initial-value problem is transposed, using the Bochner integral, into a multifunction fixed point problem in the space of continuous functions on the interval into the Banach space. Several multifunction fixed point theorems are obtained in solving the transposed problem. Each of these results is dependent, either directly or indirectly, on the multifunction being condensing with respect to a measure of non-compactness. As a result, both the concept of a measure of non-compactness and the concept of a condensing multifunction are treated. In addition, the idea of a monotone multifunction is developed and a role found for it in the fixed point theory. Finally, the topological structure of the solution set to the initial-value problem is investigated.