On Some Fractional-Order Equations of Evolution

In this thesis we study, under certain conditions, the existence of a unique solution of the nonhomogeneous fractional order evolution equation D^¿ u(t)=Au(t)+f(t),u(0)=u_o,t¿J=[0,T],¿¿(0,1), the nonhomogeneous fractional order evolutionary integral equation D^¿ u(t)=f(t)+¿_0^t¿¿ h(t-s)Au(s)ds,u(0)=u_o,¿¿(0,1),t¿J=[0,T] and the nonhomogeneous fractional order evolutionary integro-differential equation D^¿ u(t)=¿Au(t)+¿_0^t¿¿ k(t-s)Au(s)ds+f(t), u(0)=x,u'(0)=y,¿¿(1,2),¿¿0, where A is a closed linear operator with dense domain D(A)=X_A in the Banach space X. Also we prove the continuation properties of the solution u_¿ (t) and its fractional derivative D^¿ u_¿ (t) in the first two problems as ¿¿1^- and in the third problem we prove the continuation properties of the solution u_¿ (t) and its fractional drerivative D^¿ u_¿ (t) as ¿¿1^+ and as ¿¿2^-. Finally we prove the maximal regularity property of the solution of each problem and give some examples of the three problems.