Existence-Uniqueness And Stability Of Solutions For Fractional Differential Equations

In this paper, we mainly study the existence and uniqueness of solutions for frac-tional integro-differential equations with Caputo derivative, and consider about the stability of solutions for a fractional non-linear delay differential equations with Ca-puto derivative.First, the existence and uniqueness of solutions for a kind of fractional integro-differential equations with Caputo fractional derivative is studied: where f:[a, b]×R→R is a continuously differentiable function, and K:[a, b]×[a, b]×R→R is a continuous function. The inhomogeneous term of the equations includes the fractional derivative of lower orders. Under several types of sufficient conditions, the existence and uniqueness of solutions for this kind of fractional differential equations are proved by the Leray-Schauder nonlinear alternative theorem and the contraction principle.Furthermore, by setting up the comparison principle, the stability of solutions for a fractional non-linear delay differential equations with Caputo derivative is investigated: where f:[0,+∞) xC→R is completely continuous,f(t,0)≡0, t∈[0,+∞).φ∈C, C=C([-γ,0],R), and xt0(θ)=x(t0+9),-γ≤θ≤0.