**INTRODUCTION**

Fractional differential equations have been of great interest recently. In
cause, in part to both the intensive development of the theory of fractional
calculus itself and the applications of such constructions in various sciences
such as physics, chemistry, mechanics engineering, etc. For more details one
can see from Hadid *et al*. (1996a), Diethelm
and Ford (2002), Kilbas *et al*. (2006), Rabha
and Momani (2007), Lin (2007), Kosmatov
(2009) and Lakshmikantham and Vatsala (2008).

This study is concerned with the existence and uniqueness of solutions for initial value problems of fractional order of the form:

with conditions:

Here, zεR^{n}, tεI⊆R, t_{0} εI⊆R, z_{0}, z_{1}, ..., z_{m-1}εR^{n} and h is a function from IxR^{nm} to R^{n}

Upon substituting:

and

System (1)-(2) amounts to the system:

with the condition:

where,

and I_{n} is the identity matrix.

System Eq. 5 and 6 has the form:

with the condition:

The following existence and uniqueness theorems for system Eq.
8 and 9 were proved by Hadid (1995)
and Hadid *et al*. (1996b).

**Theorem 1 (Hadid, 1995):** If (t, x) is continuous
in a closed and bounded rectangular box:

Then there exists at least one solution x(t) of Eq. 8-9
on 0<t_{0}<t≤t_{0}+β for some β>0.

**Theorem 2 (Hadid ***et al*., 1996b): The
initial value problem Eq. 8-9 has a unique
solution defined on the interval 0<t_{0}<t≤t_{0}+a
if the function f(t, x) is continuous and bounded in the strip:

And satisfies in this strip the Lipschitz condition:

for some positive constant L.

The above theorems will be used to investigate the results.

**MAIN RESULTS**

In this section we give existence and uniqueness results for the IVP (Eq. 1 and 2).

**Theorem 3:** If h(t, Y) is continuous in a closed and bounded rectangular box:

where,

and

Then there exists at least one solution z(t) of (1)-(2) on 0<t_{0}<t≤t_{0} +βfor some β>0

**Proof:** By the substitution Eq. 3 and 4,
system Eq. 1 and 2 is converted into system
Eq. 5 and 6 which has the form of system
Eq. 8 and 9. Now using Theorem 1 with:

Due to the continuity of the function h(t, Y) in D, the function f is continuous in D. By Theorem 1 system (5)-(6) has at least one solution Y(t) on 0<t_{0}<t≤t_{0}+B for some β>0 So system (1)-(2) has at least one solution z(t) on 0<t_{0} <t≤t_{0} +β for some β>0.

**Theorem 4:** If the function h(t,Y) is continuous and bounded in the strip:

where,

and

and satisfies in this strip the Lipschitz condition:

for some positive constant L. Then the initial value problem (1)-(2) has a unique solution z(t) defined on the interval 0<t_{0}<t≤t_{0}+a

**Proof:** Upon substituting Eq. 3 and 4,
system Eq. 1 and 2 amounts to system Eq.
5 and 6. Common practice involves the function:

Since h is continuous and bounded in the strip 0<t_{0}<t≤t_{0}+a, ||Y||<∞ the function f satisfies in this strip:

for some positive number L. From Theorem 2, system Eq. 5
and 6 has a unique solution Y(t) on the interval 0<t_{0}<t≤t_{0}+a
and so system Eq. 1 and 2 has a unique solution
z(t) on this interval.

**Theorem 5:** If A_{0}(t), A_{1}(t), ..., A_{m-1} (t) are continuous and bounded nxn matrix functions on some interval 0<t_{0}<t≤t_{0}+a and g(t) is a continuous and bounded vector function on some interval, then the linear fractional differential equation:

has a unique solution z(t) on the interval 0<t_{0}<t≤t_{0}+a

**Proof:** By the substitutions Eq. 3 and 4, system Eq. 1 and 2 can be written in the form:

with the condition:

where,

Let

From the definition of the function h it seems clear that:

The desired result now follows from Theorem 2.