Existence Of Mild Solutions For The Initial Value Problem Of Fractional Integro-diferential Evolution Equation

The subject of fractional evolution equations is a feld that have abundant con-tents. Fractional evolution equations provide an excellent instrument for descriptionof memory and hereditary properties of various physical processes. In some cases,fractional evolution equations are more widely than traditional integer-order evo-lution equations. However, compare with integer-order evolution equations, thetheories of fractional evolution equations are still unperfect. Many problems havenot been considered.In this paper, by using the semigroup of linear operator theory, the method ofupper and lower solutions and the fxed point index theory, we study the existenceof mild solutions of the initial value of two class of fractional evolution equations.The main results of this paper are as follows:1. With the aid of the existence and uniqueness of mild solution of the ini-tial value problem of corresponding linear fractional evolution equations, we obtainthe existence of mild solution of fractional integro-diferential evolution equationnonlocal problems by using the fxed point index theory.2. Under the case of C₀-semigroup, we obtain the existence of mild solutionsfor the initial value problem of impulsive fractional integro-diferential evolutionequation by using the method of discussing interval by interval.