Initial Value Problems For Fractional Functional Differential Equations

In this paper, we study the existence and uniqueness of positive solution about high order fractional differential equations and fractional functional differential equations, also show the continuous dependence of the solution on the order and the initial condition for some fractional delay differential equations.First, using a nonlinear alternative of Leray-Schauder type and the generalization of Gronwall's lemma for singular kernels, we show the existence of positive solutions for the fractional delay differential equationwhere 0 <α< 1, we remove the condition that the function / be nondecreasing, which improves the previous results in [24].Second, we consider more general fractional functional differential equation of low orderwhere L(D) = D^s_n -α_n-1D^s_n-1 -…-α₁D^s₁,0 < s₁< s₂ <…< s_n < 1,α_j > 0, j = 1,2,…, n - 1, and show its existence of a positive solution by using a nonlinear alternative of Leray-Schauder type. By using a suitable extension of the Lipschitz condition and the Banach fixed point theorem, we obtain the uniqueness of a positive solution to this equation under the weaker condition comparing with the Lipschitz condition. Further, when the conditionα_j > 0, j = 1,2,…,n - 1, is relaxed, we show that it has a unique solution which may not necessarily be positive.Third, we show the existence of positive solutions about high order fractional differential equation with initial condition u(0) = 0, (?), (?)= (?), j=2,3…,n-1,n-1＜αj＞0,j=1,2,…,n-1,D^α-j denotes the standard Riemann-Liouville fractional derivative, f : [0,1]×[0, +∞)→[0, +∞] is a given continuous function. In [43], the authors study the existence of positive solutions to this equation by using the sub- and super-solution method, requiring an essential condition that f(t,·) be nondecreasing and bounded. Therefore, it is desirable to remove this condition, and this constitutes the purpose of this paper. Equations of this form are encountered in dynamical systems of fractional order and hence are of interest. Therefore, this paper provide insights in the equations encountered in dynamical systems and controllers of fractibnal order which further maybe explored.Lastly, we study the continuous dependence of the solution on the order and the initial condition for some fractional differential equations with delay and give some estimates about the solution dependence on the order and the initial condition by using the generalized Grbnwall inequality[19]. We use the estimate to obtain some appropriate solution for such fractional delay differential equation. Further, an Mittag-Leffler estimate of solution is obtained by application of the generalized Gronwall inequality.