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.

Dr. Mohamed Herzallah has obtained his Ph.D degree in functional analysis in 2005. He is interested in fractional evolution equation, fractional variational calculus and existence and uniqueness of fractional differential equations.