About The Causal Operator Of Fractional Order Differential Equation

The term causal operators is adopted from the engineering literature and the theory of these operators has the powerful quality of unifying ordinary differential equations, integrodifferential equations, differential equations with finite or infinite delay, Valera integral equations, and neutral functional equations, to name but a few. The study of functional equations with causal operators has been rapid development in the past few years. Fractional calculus is the generalization of classical calculus, and fractional differential equations have unique advantage on describing those systems with memory or hereditary such as biological engineering, dynamic system, control system, signal processing and so on. For this reason, scientists have more interesting in fractional differential equations recent years. To our knowledge, fractional functional differential equations with causal operators in Banat spaces have not been studied extensively. Motivated by this, we study some fractional differential equations with causal operators. This thesis mainly studies a class of nonlinear causal fractional differential equations with finite delay in Banach spaces cDau(t)-(Qu)(t),a.e.t∈[0,T], u|[σ,0]=φ∈Cσ,where CDa is the Caputo fractional derivative of order0<a<1, Q:C([0,T];X)â†'Lq ([0,T];X) is a causal operator, T>0, Cσ=C([-σ,0];X), as well as a class of infinite delay nonlinear causal fractional differential equation D0ay=f(t,y1),t∈[0,T],0<α<1y(t)+φ(t),t∈(-∞,0]where Da is Riemman-Liouville fractional derivative, B is the phase space defined axiomatically. Using the theory of measure of noncompactness and related fixed point theorem, the existence of mild solutions of these two classes of differential equation is obtained, which improves and extends some previous results.