Mathematical Problems in EngineeringVolume 2011 (2011), Article ID 567319, 10 pagesdoi:10.1155/2011/567319

Research Article

Positive Almost Periodic Solution on a Nonlinear Differential Equation

Ni Hua, Tian Li-xin, and Liu Xun

Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China

Received 17 March 2011; Accepted 10 July 2011

Academic Editor: Jitao Sun

Copyright © 2011 Ni Hua et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the following nonlinear equation , by using fixed point theorem, the sufficient conditions of the existence of a unique positive almost periodic solution for above system are obtained, by using the theories of stability, the sufficient conditions which guarantee the stability of the unique positive almost periodic solution are derived.

1. Introduction

Let us consider the following Logistic-type equation When the external perturbation and are positive constants, (1.1) is the typical Logistic equation. It was firstly introduced as a mathematical model for studying population dynamics and has become a classic topic in the textbooks on ordinary differential equations and its quality theory (see [1, 2]). By using the method of separation of variables and integration by partial fractions, we can get explicitly all the solutions of the typical Logistic equation and completely analyze the behavior of all the solutions. But when are no longer constants and there exists some perturbation, the problem is not so simple as no explicit solutions can be found in general. In [2–6], the time-periodic case was considered. In [7], the existence of positive almost periodic solutions was considered when are almost periodic and there is no perturbation. When satisfy the assumption ,  , and there is no external perturbation , but there is a nonlinear perturbation , Nkashama [8] obtained the existence of bounded and positive almost periodic solutions. In [9], Zhu et al. considered the case that satisfy the assumption ,   and there exists an external perturbation , and they got the existence and uniqueness of positive periodic solutions and almost periodic solutions of (1.1).

In this paper, we consider the following more complex system: where ,  , are all continuous almost periodic functions, and is almost periodic in and uniformly with respect to .

In this paper, we use the fixed point theorem and get the existence and uniqueness of positive almost periodic solution for (1.2), the stability of the unique positive almost periodic solution of (1.2) is also discussed, and some new results are obtained.

2. The Existence and Uniqueness of Positive Almost Periodic Solution

Before we start with our main results, for the sake of convenience, suppose that is a continuous bounded function, and we denote and ; first, we introduce some lemmas.

Lemma 2.1 (see [10]). Consider the following equation: where are continuous almost periodic functions; if , then (2.1) exists a unique almost periodic solution , and can be written as follows: where is the real part of .

Lemma 2.2. Consider (1.2); if ; then the domain is a positive invariant with respect to (1.2).

Proof. Since , it follows that thus we have the assertion is valid for all . The proof is completed.

Theorem 2.3. Consider (1.2); is a constant and , are continuous almost periodic functions, is a continuous almost periodic function in and uniformly with respect to ; if the following conditions hold: (1),  ,  ,(2), where is a positive number,(3), where ,  , then (1.2) exists a unique positive almost periodic solution , and .

Proof. Let , since we only consider positive solutions of (1.2), by Lemma 2.2, it follows that , then (1.2) can be written as follows: Define where denotes the set of almost periodic functions in , the norm is defined as , thus is a Banach space, given any , consider the following equation: since , it follows that , thus , from (2.7), by virtue of Lemma 2.1, we can know that (2.7) has a unique almost periodic solution , it can be written as follows: Now, define an operator as follows: note that Since the condition holds, it follows that ; thus, we have Also Since , it follows that ; thus, we have Hence ; therefore, . For any , it follows that According to mean value theorem, we can get where ,   or ,  . Notice that ,  ,  ,  ; it follows that Note that , and is a positive number; it follows that therefore, is a contraction mapping on , that is to say, has a unique fixed point on , the unique fixed point is the unique positive almost periodic solution of (2.5), and . Notice that , and, by Lemma 2.2, (1.2) has a unique positive almost periodic solution , and . This completes the proof of Theorem 2.3.

3. The Uniqueness of the Solution with Initial Value Problem

Consider (2.5); if given initial value , and , then we have the following theorem.

Theorem 3.1. Consider (2.5); is a constant and are continuous functions, is a continuous function in and uniformly with respect to ; if the following conditions hold: (1),  ,  ,(2), where is a positive number,(3), where ,  , then (2.5) exists a unique continuous solution with initial value and .

Proof. The initial value problem of (2.5) is equivalent to the solution of the following integral equation: Define an operator as follows: The following proof is similar as the proof of Theorem 2.3, so we omit it here.

4. The Stability of the Positive Almost Periodic Solution

Theorem 4.1. Consider (1.2); is a constant and , are continuous almost periodic functions, is a continuous almost periodic function in and uniformly with respect to ; if the following conditions hold: (1),  ,  ,(2), where is a positive number,(3), where ,  , then there exists a region , and, in this region, the unique positive almost periodic solution of (1.2) is uniformly asymptotically stable.

Proof. From Theorem 3.1, we know when , (2.5) exists a unique solution with initial value , and , since the transformation , it follows that , so (1.2) exists a unique solution with initial value , and . It is easy to know that, the conditions of Theorem 4.1 are met with Theorem 2.3, so (2.5) exists a unique positive almost periodic solution ; from (2.9), the unique positive almost periodic solution of (2.5) can be expressed by integral equation as follows: The solution of (2.5) with initial value is given as follows: from (4.1) and (4.2), when , we have According to mean value theorem, we can get where ,   or  ,  . Notice that ,  ,  ,  ; it follows that Multiplying both sides of the above inequality by , we have According to Bellman’s inequality, we can obtain Multiplied both sides of the above inequality by , we get Notice that , hence , and is the unique positive almost periodic solution of (1.2), so we have where or , note that the condition (3) holds, it follows that therefore, there must be a small positive constant such that the following equality holds: From (4.11), we know that the unique positive almost periodic solution of (1.2) is uniformly asymptotically stable.

References

M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York, NY, USA, 1974.

J. K. Hale and H. Koçak, Dynamics and Bifurcations, vol. 3 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1991.

Y. Chen, “Periodic solutions of a delayed periodic logistic equation,” Applied Mathematics Letters, vol. 16, no. 7, pp. 1047–1051, 2003.

Y. Chen, “Existence of positive periodic solutions for a periodic Logistic equation,” Applied Mathematics Letters, vol. 16, pp. 1047–1051, 2003.

X. Liu and L. Chen, “Global dynamics of the periodic logistic system with periodic impulsive perturbations,” Journal of Mathematical Analysis and Applications, vol. 289, no. 1, pp. 279–291, 2004.

P. De Mottoni and A. Schiaffino, “Competition systems with periodic coefficients: a geometric approach,” Journal of Mathematical Biology, vol. 11, no. 3, pp. 319–335, 1981.

Z. Teng, “On the positive almost periodic solutions of a class of Lotka-Volterra type systems with delays,” Journal of Mathematical Analysis and Applications, vol. 249, no. 2, pp. 433–444, 2000.

M. N. Nkashama, “Bounded and almost-periodic solutions of nonlocal perturbations of nonautonomous logistic-type equations,” Acta Applicandae Mathematicae, vol. 65, no. 1–3, pp. 283–293, 2001.

W. Z. Zhu, S. G. Ji, and B. F. Liu, “The existence of periodic and almost periodic solutions of logistic type equation,” Northeastern Mathematical Journal, vol. 23, no. 4, pp. 298–310, 2007.

C. Y He, Almost Periodic Differential Equations, Higher Education Press, Beijing, China, 1992.