Electronic Journal of Qualitative Theory of Differential Equations

We are concerned here with the existence of monotonic and uniformly asymptotically stable solution of an initial-value problem of non-autonomous delay differential equations of arbitrary (fractional) orders.


Introduction
In a number of papers [1,2,3,7] stability and existence of solution of some equations of fractional order has been investigated.
In [2] the authors proved the stability (and some other properties concerning the existence and uniqueness) of solution of the problem of non-autonomous system D α t 0 X(t) = A(t) X(t) + f (t), α ∈ (0, 1], X(t 0 ) = X 0 , EJQTDE, 2008 No. 16, p. 1 where the coefficients of the matrix functions A(t) and f (t) are absolutely continuous functions.And the problem where the coefficients of the matrix A(t) are bounded and measurable and the function f (t) is integrable.
Let α i ∈ (0, 1], β j ∈ [0, 1), i = 1, 2, . . ., n, j = 1, 2, . . ., m, and r i , σ j are positive constants.Consider the initial-value problem We prove here the existence of uniformly asymptotically stable and monotonic increasing As an application the special (linear) case and the initial value problem of the multidimensional neutral delay differential equation will be studied where Γ(.) is the gamma function.

Existence of solution
Let C[0, T ] be the class of continuous functions.For x ∈ C[0, T ] we use the norm Proof.Let y(t) = d dt x(t) then x(t) = x 0 + I y(t), which implies from which we obtain Similarly we obtain Now equation (1.1) can be written as Define the operator   2) can be written as x(t) = x 0 + I y(t), this implies that dx dt = y > 0 which prove that dx dt ∈ C[0, T ] and x is monotonic increasing.

Stability
In this section we study the stability of the solution of the initial value problem (1.1) - where therefore lim t→∞ x − x * 1 = 0, then the solution of the system (1.1) is uniformly asymptotically stable.EJQTDE, 2008 No. 16, p. 7