Fractional Cauchy Problems With Almost Sectorial Operators

Fractional derivatives describe the property of memory and heredity of material-s, and it is the major advantage of fractional derivatives compared with integer orderderivatives. The properties of fractional derivativesare sufcient to describe practical sit-uation. More and more mathematical modeling and simulation of systems and processesbase on the description of fractional evolution equations. Therefore, we make scientificresearches on the abstract Cauchy problem of fractional evolution equations which areprovided with important theoretical significance and practical value. Although, in recentyears, there has been a significant development in the theory of fractional evolution equa-tions, efective general methods for the study of theory of fractional evolution equationscannot be found even in the most recent works.This paper concerns the abstract Cauchy problem of fractional evolution equation-s with almost sectorial operators. The suitable mild solutions of evolution equationswith Riemann-Liouville derivative and Caputo derivative are introduced respectively byconsidering probability density. The existence theorems of mild solutions of Cauchyproblems are established. The results obtained here improve and generalize some knownresults. Finally, an example is given for demonstration.