Positive Solutions for the Initial Value Problems of Fractional Evolution Equation

This paper discusses the existence of positive solutions for the initial value problem of fractional evolution equation with noncompact semigroup D(q)u(t) + Au(t) = f(t, u(t)), t >= 0; u(0) = u(0) in a Banach space X, where D-q denotes the Caputo fractional derivative of order q epsilon (0,1), A : D(A) subset of X -> X is a closed linear operator, -A generates an equicontinuous. C-0 semigroup, and f : [0,infinity) x X -> X is continuous. In the case where.. satisfies a weaker measure of noncompactness condition and a weaker boundedness condition, the existence results of positive and saturated mild solutions are obtained. Particularly, an existence result without using measure of noncompactness condition is presented in ordered and weakly sequentially complete Banach spaces. These results are very convenient for application. As an example, we study the partial differential equation of parabolic type of fractional order.