Abstract: We give the definition of a kind of new operator-convex-power condensing operator for the need of applied questions and generalize the definition of condensing operator. The fixed point theorem of this kind of new operator is proved. This generalizes the famous Schauder fixed point theorem and Sadovskii fixed point theorem. As applications, the existence of global mild solutions and positive mild solutions of initial value problem for a class of semilinear evolution equations with noncompact semigroup in Banach spaces is obtained.

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