Existence Of Solutions To Several Classes Of Fractional Differential Equations

In this dissertation,we study the existence of solutions to several classes of fractional dif-ferential equations by using semigroup theory,measures of noncompactness,fixed point theory,upper and lower solutions and monotone iterative methodIn Chapter ?,we introduce the background and status of the research,the structure arrange-ment and some notations of this dissertationChapter ? introduces preliminary facts of this dissertation includes the definitions of frac-tional calculus and some properties,Mittag-Leffier functions,Kuratowskii measures of noncom-pactness and fixed point theorem,almost sectorial operator,conformable derivativesChapter ? discusses two classes of abstract fractional evolution equation with infinite delay on infinite interval in Banach space.In Section 3.2,the evolution operator A generates a uniformly bounded C0-semigroup.We consider the case of compact semigroups and not compact semigroup-s.The existence of global mild solutions is established by Schauder's and Darbo-Sadovskii fixed point theorem.The operator A is an almost sectorial operator in section 3.3 and generates an analytic semigroup.The existence results obtain by Schauder's fixed point theoremIn Chapter ?,the general conformable fractional derivatives is used in the classical diffu-sion equations.First,we prove the theorem of term-by-term integration and differentiation of the general conformable factional calculus.Then,we obtain the existence of classical solution to the diffusion equations with general conformable fractional derivatives.Finally,we get the asymptotic behavior of the classical solution.In Chapter ?,the existence and uniqueness of a class of nonlinear diffusion equations with the conformable variable order derivative is studied.Based on the conformable derivative,we introduce the concept of conformable variable order derivative.The conformable derivative is a local operator.It has many properties like usual integer derivative's ones and we study some properties of the conformable variable order derivative.We investigate the fundamental solutions to initial value problem for linear homogeneous and inhomogeneous linear diffusion differential equations with the conformable variable order derivative.In addition,using upper and lower solu-tions and monotone iterative method,we consider the existence and uniqueness of solutions to an initial value problem for nonlinear diffusion differential equations involving with the conformable variable order derivative.