1.Introduction

In this paper we use a fixed point theorem due to Krasnoselskii to study the existence of positive periodic solutions of the system of neutral differential equations

The scalar version of (1.1) arises in food-limited population models (^{3},^{4},^{7},^{8},^{9}) and blood cell models (^{2}) Recently, Raffoul, (^{21}) obtained sufficient conditions for the existence of positive periodic solutions for the scalar neutral nonlinear differential equation

In the current paper we extend the results in (^{21}) to systems of equations. It must be noted that if *τ*(*t*) = 0 in the first term on the right hand side of (1.1) and *n* = 1, then (1.1) reduces to (1.2). Thus, even for *n* = 1 our results obtained in this paper are more general than that obtained in (^{21}) Let **R**
_{+}=
For each, *x* = (_{
x1,x2,x3,…,xn
} )^{
T
} ∈ **R**
^{
n
} , the norm of *x* is defined as |x| = Σ_{
nj
}
_{=1}|x_{
j
} |. **R**
^{
n
}
_{+} = {(_{
x1,x2,x3,…,xn
} )^{
T
} ∈ **R**
^{
n
}: x_{j} ≥ 0, *j* = 1,2,3,…,*n*}. We say that *x* is "positive" whenever x ∈ **R**
_{
n+
} .

In this paper we make the following assumptions.

2.Preliminaries

Let **S**
_{ω} = {*ϕ* ∈ *C*(**R**,**R**
^{
n
} ): *ϕ (t + ω = ϕ(t)* for *t* ∈ **R**}, be endowed with the usual linear structure as well as the norm

Then **S**
_{ω} is a Banach space.

We assume that all functions in (1.1) are continuous with respect to their arguments.

We also assume that for all *t* ∈ **R**,

Also, let

_{
mj ≤ Gj (t,s) ≤ Mj.
}

It is clear that _{
Gj(t
} + *ω, s* +_{
ω) = Gj(t,s)
} and so *G(t* + *ω, s* + *ω) = G (t,s)* for all *(t,s)* ∈ **R**
^{
2
}
**.**

**Lemma 2.1.** Suppose *(2.1)-(2.4)* hold. Suppose also that *τ*’(*t*) ≠ 1 for all *t* ∈ **R**. If *x*(t). If x(t) ∈ *S*
_{ω}, then *x*
_{j} (*t*) is a solution of *(2.7)* if and only if

**Proof.**

gives

This completes the proof.

We next state Krasnoselskii's Theorem which is the main mathematical tool in this paper in the following lemma.

**Lemma 2.3** (Krasnoselskii's ) Let **M** be a closed convex nonempty subset of a Banach space (_{
Sω
} ,||.||). Suppose that *J* and *D* map **M** into _{
Sω
} such that

(i) *x,y* ∈ **M**, implies *Jx* + *Dy* ∈ **M**,

(ii) *D* is continuous and *D*
**M** is contained in a compact set,

(iii) *J* is a contraction mapping.

Then there exists *z* ∈ **M** with *z* = *Jz* + *Dz*.

3.Main Results

For some non-negative constant *L* and a positive constant *K* define the set

which is a closed convex and bounded subset of the Banach space _{
Sω
} . We also assume that for all *s* ∈**
R, ρ
** ∈

**M**

Where *j* = 1,2,…*n*.

Define the map *D*: **M** → **S**
_{ω} by

**Lemma 3.1.** Suppose that *(2.1)*-*(2.4)*, *(3.1)*, *(H1), (H2), (H3)* and *(H5)*
*hold*. Then the operator *D* is completely continuous on **M**.

Where *a*
^{*}= min_{1≤j≤n} (*a*
_{j}ϒ). It therefore follows that

||(*D*ϕ|| ≤ *K*.

This shows that *D*(**M**) is uniformly bounded.

We will next show that *D*(**M**) is equi-continuous. Let ϕ ∈ **M**. Then differentiating (3.2) with respect to *t* gives

Where *M = max*{_{
M1, M2,…,Mn
} }. Thus showing that *D*(**M**) is equicontinuous. Then using Ascoli-Arzela theorem we obtain that *D* is a compact map. Due to the continuity of all the terms in (3.2), we have that *D* is continuous.

**Lemma 3.2** Suppose that (H2) and (H5) hold. Then the operator *J* is a contraction.

**Proof.** For ϕ,ψ ∈ **M**

Where a = *max*{_{
a1ϒ,…,anϒ
} }. This completes the proof of lemma 3.2.

**Theorem 3.3** Suppose (H1), (H2), (H3), (H4), (H5), and (3.1) hold. Also suppose that the hypotheses of Lemma 3.2 and Lemma 3.3 hold. Then (1.1) has a positive periodic solution *x* satisfying *L* ≤ ||*x*||≤ *K*.

**Proof.** Let ϕ,ψ ∈ **M.** Then

This completes the proof of theorem 3.3.