Research For Solutions Of Deflnite Problems For Two Fractional Differential Equations

After three centuries development, fractional calculus has been applied in more and more fields. Compared with the classical calculus, it can describe the processes with memory and hereditary in biology, physics, chemistry, hydrology, financial and so on, and these processes can be described by fractional differential equation. On the other hand, the properties of nonlocal and singular of the fractional differential operator make it difficult in theory research. Thus, it is important to study on the fractional differential equation. This thesis considers the existence of solutions for a class of abstract time fractional evolution equation Cauchy problem and a class of fractional differential equation boundary value problem.Firstly, we concerned with abstract time fractional evolution equation Cauchy problem of order a∈(1,2) with Caputo derivative. We introduce a family of bounded linear operator which is defined in terms of sectorial operator, Mittag-Leffler function and the curve integral, and present a deep analysis on the prop-erties for it. Based on these properties, we obtain the existence and uniqueness of mild solution and classical solution to the inhomogeneous linear problem and obtain the regularity of mild solution of it. We also consider the existence of classical solution of semilinear fractional abstract Cauchy problem.Secondly, we consider the Subordination principle of the fractional resolvent operator function and obtain that a sectorial operator can determine an analyti-cal fractional resolvent operator function. Using the properties of analytical frac-tional resolvent operator function, we consider a class of abstract time fractional evolution equation Cauchy problem of order α∈(1,2) with Riemann-Liouville derivative and obtain the regularity of its mild solution.Then we prove that β-times integrated α-resolvent operator function ((α, β)-R.OF) satisfies a functional equation which extend the corresponding property of integral semigroup. For a inhomogeneous a-Cauchy problem, if its coefficient op-erator generates an (a,β)-ROF Sα,β(t), we give the relation between the function v(t)=Sα,β(t)xo+(g1*Sα,β)(t)x1+(gα-1*Sα,β*f)(t) and mild solution and classical solution of it. And for another fractional Cauchy problem, we obtain that its coefficient operator generates an exponentially bounded (α,β)-ROF on X if and only if the problem has a unique exponentially bounded classical solution vx and Avx∈L1loc(R+,X).Finally, we concerned with the existence of solutions to a class of fractional differential equation boundary value problem with a parameter. We consider the eigenvalue problem associates with it and prove that there is a sequence of positive and increasing real eigenvalues, characterization of the first eigenvalue is also given. Then under different assumptions on the nonlinearity, the existence criteria of solutions of the problem when the parameter lies in different intervals are established. The main tools are variational method and critical point theory.