## Indicators

• Similars in SciELO

## Print version ISSN 0716-0917

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

#### http://dx.doi.org/10.4067/S0716-09172017000300423

Articles

Positive periodic solutions for neutral functional differential systems

1 University of Cape Coast, Department of Mathematics and Statistics, Ghana, email: ernestoyank@gmail.com

2 University of Cape Coast, Department of Mathematics and Statistics, Ghana, email: sammyassabil@gmail.com

Abstract:

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.

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 : 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.

2.Preliminaries

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

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 ( ,||.||). 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.

References

[1] T. A. Burton, Stability by Fixed Point Theory for functional Differential Equations, Dover, New York, (2006). [ Links ]

[2] E. Beretta, F. Solimano, Y. Takeuchi, A mathematical model for drug administration by using the phagocytosis of red blood cells, J. Math Biol. 10 Nov;35 (1), pp. 1-19, (1996). [ Links ]

[3] Y. Chen, New results on positive periodic solutions of a periodic integro-differential competition system, Appl. Math. Comput., 153 (2), pp. 557-565, (2004). [ Links ]

[4] F. D. Chen, Positive periodic solutions of neutral Lotka-Volterra system with feedback control, Appl. Math. Comput., 162 (3), pp. 1279-1302, (2005). [ Links ]

[5] F. D. Chen, Periodicity in a nonlinear predator-prey system with state dependent delays, Acta Math. Appl. Sinica English Series, 21 (1) (2005), pp. 49-60, (2005). [ Links ]

[6] F. D. Chen, S. J. Lin, Periodicity in a logistic type system with several delays, Comput. Math. Appl., 48 (1-), pp. 35-44, (2004). [ Links ]

[7] F. D. Chen, F. X. Lin, X. X. Chen, Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control, Appl. Math. Comput., 158 (1), pp. 45-68, (2004). [ Links ]

[8] M. Fan, K. Wang, Global periodic solutions of a generalized n-species Gilpin-Ayala competition model, Comput. Math. Appl., 40, pp. 1141-1151, (2000). [ Links ]

[9] M. Fan, P. J. Y. Wong , Periodicity and stability in a periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments, Acta Math. Sinica, 19 (4), pp. 801-822, (2003). [ Links ]

[10] M. E. Gilpin, F. J. Ayala, Global Models of Growth and Competition, Proc. Natl. Acad. Sci., USA 70, pp. 3590-3593, (1973). [ Links ]

[11] A. Datta and J. Henderson, Differences and smoothness of solutions for functional difference equations, Proceedings Difference Equations 1, pp. 133-142, (1995). [ Links ]

[12] J. Henderson and A. Peterson, Properties of delay variation in solutions of delay difference equations, Journal of Differential Equations 1, pp. 29-38, (1995). [ Links ]

[13] D. Jiang, J. Wei, B. Zhang, Positive periodic solutions of functional differential equations and population models, Electron. J. Diff. Eqns., Vol. No. 71, pp. 1-13, (2002). [ Links ]

[14] M. A. Krasnosel'skii, Positive solutions of operator Equations Noordhoff, Groningen, (1964). [ Links ]

[15] L. Y. Kun, Periodic solution of a periodic neutral delay equation, J. Math. Anal. Appl., 319, pp. 315-325, (2006). [ Links ]

[16] Y. Li, Y. Kuang, Periodic solutions of periodic delay Lotka-Volterra equations and systems, J. Math. Anal. Appl., 255, pp. 260-280, (2001). [ Links ]

[17] M. Maroun and Y. Raffoul , Periodic solutions in nonlinear neutral difference equations with functional delay, Journal of Korean Mathematical Society 42, pp. 255-268, (2005). [ Links ]

[18] Y. Raffoul, Periodic solutions for scaler and vector nonlinear difference equations, Pan-American Journal of Mathematics 9, pp. 97-111, (1999). [ Links ]

[19] Y. N. Raffoul, Periodic solutions for neutral nonlinear differential equations with delay, Electron. J. Diff. Eqns., Vol. No. 102, pp. 1-7, (2003). [ Links ]

[20] Y. Raffoul, Positive periodic solutions in neutral nonlinear differential equations, Electronic Journal of Qualitative Theory of Differential Equations 16, pp. 1-10, (2007). [ Links ]

[21] Y. Raffoul, Existence of positive periodic solutions in neutral nonlinear equations with functional delay, Rocky Mount. Journal of Mathematics 42(6), pp. 1983-1993, (2012). [ Links ]

[22] N. Zhang, B. Dai, Y. Chen, Positive periodic solutions of nonautonomous functional differential systems, J. Math. Anal., 333, pp. 667-678, (2007). [ Links ]

Received: November 2015; Accepted: June 2017