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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.36 no.3 Antofagasta Sept. 2017 


Positive periodic solutions for neutral functional differential systems

Ernest Yankson1 

Samuel E. Assabil2 

1 University of Cape Coast, Department of Mathematics and Statistics, Ghana, email:

2 University of Cape Coast, Department of Mathematics and Statistics, Ghana, email:


We study the existence of positive periodic solutions of a system of neutral differential equations. In the process we construct two mappings in which one is a contraction and the other compact. A Krasnoselskii's fixed point theorem is then used in the analysis.

Keywords : Krasnoselskii; Neutral Functional differential System; Positive periodic solutions.


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 : xj ≥ 0, j = 1,2,3,…,n}. We say that x is "positive" whenever x ∈ R n+ .

In this paper we make the following assumptions.


Let S ω = {ϕC(R,R n ): ϕ (t + ω = ϕ(t) for tR}, 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 tR,

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 tR. If x(t). If x(t) ∈ S ω, then x j (t) is a solution of (2.7) if and only if



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 ( ,||.||). Suppose that J and D map M into such that

(i) x,yM, implies Jx + DyM,

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

(iii) J is a contraction mapping.

Then there exists zM 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 . We also assume that for all s R, ρ M

Where j = 1,2,…n.

Define the map D: MS 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 *= min1≤j≤n (a jϒ). It therefore follows that

||(Dϕ|| ≤ 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.


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Received: November 2015; Accepted: June 2017

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