H(o|¨)lder Regularity Aspects Of Fractional Evolution Equations With Nonhomogeneous In Banach Spaces

Fractional Calculus is about the theory of arbitrary order differential and integral. It isunified with the integer order calculus. Integer order calculus is an analytical mathematicaltool to describe the theory of classical physics and related disciplines. Mathematical modelsof many problems can ultimately come down to the definite solution problems of integer-order differential equations and it has perfect theories on theoretical analysis and numericalsolving. However, when people study complex systems and complex phenomenons, such asproblems from viscoelasticity, heat conduction in materials with memory, electrodynamicswith memory, they will encounter many problems to describe these systems using classicalinteger order calculus equation. In the past few decades, fractional calculus has been moreand more used in the mathematical modeling processes on many fields such as engineering,physics and economics. It is a very practical and effective tool.This paper is concerned with the theories of the solution of an fractional abstractCauchy problem with nonhomogeneous in Banach spaces. In particular, we mainly con-sider the Hoolder regularity aspects of the solution.In the process of studying the solution, fractional calculus and semigroup theory is thebasis knowledge we need. In order to study the expression of the solution, the paper willmainly introduce several key operators, including the resolvent family and characteristicsolution operator family, and at the same time their properties will be introduced to a certaindegree. When α∈(0,1), the fractional evolution equation with nonhomogeneous has twocommon forms of expression to the solution. One key point of this paper is to verify theequivalence of the two expressions and we will use the properties of the Gamma functions,subordination principle and convolution. The most important step is to discuss the Hoolderregularity for the solution. According to the inhomogeneous term f, we divide this step intotwo parts when f∈L~por f is Hoolder continuous. This is a promotion to the correspondingconclusions of the first-order evolution equation.