This paper discusses the existence of positive solutions for the initial value problem of fractional evolution equation with noncompact semigroupDqu(t)+Au(t)=f(t,u(t)),t≥0;u(0)=u0in a Banach spaceX, whereDqdenotes the Caputo fractional derivative of orderq∈(0,1),A:D(A)⊂X→Xis a closed linear operator,-Agenerates an equicontinuousC0semigroup, andf:[0,∞)×X→Xis continuous. In the case wherefsatisfies 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.
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