Research Article | Open Access
Mean-Square Almost Periodic Solutions for Impulsive Stochastic Host-Macroparasite Equation on Time Scales
We consider an impulsive stochastic host-macroparasite equation on time scales. By use of the Banach fixed point theorem and Gronwall-Bellman’s inequality technique on time scales, we obtain the existence and exponential stability of mean-square almost periodic solutions for the host-macroparasite equation on time scales. Finally, we give an example to illustrate the feasibility of our results.
Many important human diseases, particularly in tropical and subtropical regions, arise from infection by macroparasites or metazoan organisms. These organisms tend to have much larger generation times and more complex life cycles than microparasites. In life cycles, there are two or more obligatory host species together with the final host (humans). Macroparasitic infections are generally chronic in form and they are more a cause of morbidity than mortality and tend to be persistent in character in areas where they are endemic. The final hosts of parasites are usually humans (the hosts in which the parasite attains reproductive maturity) and they gain entry to the definitive host as a consequence of developmental changes which normally occur before the organism arrives at its preferred site and attains reproductive maturity. Because of this fact, the host-macroparasite system has been attracting the attention of many researchers (see [1–4]). A time delay, therefore, exists between entry to the definitive host and the point when the parasite begins the production of eggs or larvae for transmission to other hosts (see [5–7]). In , by means of a continuation theorem in coincidence degree theory, the authors considered the oscillation and global attractivity of the nonlinear delay host-macroparasite model with periodic coefficients.
On the one hand, the theory of impulsive differential equations is now being recognized to not only be richer than the corresponding theory of differential equations without impulses, but also represent a more natural framework for mathematical modelling of many real-world phenomena, such as population dynamical models and neural networks. Since many dynamical processes are characterized by the fact that, at certain moments of time, they undergo abrupt changes of state, with the development of the theory of impulsive differential equations (see [8, 9]), various models of impulsive differential equations have been proposed and studied extensively (see [10–14]). For example, authors of  considered the nonlinear impulsive delay host-macroparasite model with periodic coefficients: where is a positive integer, , , and are -periodic functions. By use of the continuation theorem of coincidence degree, some sufficient conditions are obtained for the global attractivity and oscillation of positive periodic solutions.
In fact, both continuous and discrete systems are very important in implementation and applications. But it is troublesome to study the dynamical properties for continuous and discrete systems, respectively. Therefore, it is significant to study that on time scales which can unify the continuous and discrete cases. In , the author considered the following host-macroparasite equation on time scales: By using the contraction principle and Gronwall-Bellman’s inequality on time scale, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solutions.
On the other hand, as a matter of fact, population systems are often subject to environmental noise; that is, due to environmental fluctuations, parameters involved in population models are not absolute constants, and they may fluctuate around some average values. Based on these factors, more and more people began to be concerned about stochastic population systems (see [16–20]). Meanwhile, almost periodicity is universal than periodicity, and the mean-square almost periodicity is important in probability for investigating stochastic processes. To the best of our knowledge, there exist few results for mean-square almost periodic solutions for impulsive stochastic process models with delays; one can see some results in [21–23]. However, there exists no result on the existence and uniqueness of mean-square almost periodic solutions for impulsive stochastic host-macroparasite equation on time scales. Motivated by the above, we consider the following impulsive stochastic host-macroparasite equation on time scales: where denotes a -stochastic differential of , , is Borel measurable, , , , , and . Let be a complete probability space furnished with a complete family of right continuous increasing sub--algebras satisfying . is a standard Brownian motion over .
Throughout this paper, we assume the following.
() , , , are all mean-square almost periodic functions, , , , for , is mean-square almost periodic in uniformly for , , and there exists constant such that
() The sequences , are almost periodic in and the sequences are almost periodic in uniformly for and there exists constant such that
() The set of sequences , , , are equipotentially almost periodic and , where will be introduced in Section 2.
For convenience, we denote where is a mean-square almost periodic function and is a mean-square uniformly almost periodic function on time scales.
Let be a mean-square almost periodic function. Denote by the solution of (3) satisfying the initial conditions: where .
Our main purpose of this paper is to study the existence and exponential stability of mean-square almost periodic solutions to (3) by means of the Banach fixed point theorem and Gronwall-Bellman’s inequality technique.
In this section, we shall recall some basic definitions and lemmas which are used in what follows.
A time scale is an arbitrary nonempty closed subset of the real numbers; the forward and backward jump operators , and the forward graininess are defined, respectively, by
A point is called left-dense if and , left-scattered if , right-dense if and , and right-scattered if . If has a left-scattered maximum , then ; otherwise, . If has a right-scattered minimum , then ; otherwise, .
A function is right-dense continuous provided it is continuous at right-dense point in and its left-side limits exist at left-dense points in . If is continuous at each right-dense point and each left-dense point, then is said to be continuous function on .
For and , we define the delta derivative of , , to be the number (if it exists) with the property that, for a given , there exists a neighborhood of such thatfor all .
If is continuous, then is right-dense continuous, and if is delta differentiable at , then is continuous at .
Let be right-dense continuous; if , then we define the delta integral by
Lemma 1 (see ). Assume are delta differentiable at ; then,(i);(ii);(iii)if , then ;(iv)if and are continuous, then .
A function is called regressive provided , for all . The set of all regressive and rd-continuous functions will be denoted by . We define the set .
If , then the generalized exponential function is defined by for all , with the cylinder transformation
Let be two regressive functions; we defineThen, the generalized exponential function has the following properties.
Lemma 2 (see ). Assume that are two regressive functions; then, (i) and ;(ii);(iii);(iv);(v);(vi);(vii).
In the following, we will present some basic concepts and results concerning stochastic differential equations on time scales which will be essential to prove our main results. For more details, the reader may want to consult .
Definition 3 (see ). For a function , one can define the extension by , for all .
Let be a probability space, and stands for a space that consists of all -valued random variables with the normLet be a standard Wiener process and suppose is independent of , where is a filtration on , and with , we mean the -algebra generated by . We denote -stochastic integral on , by .
Definition 4 (see ). One can say that the random process belongs to the class if the following conditions hold: (i) is adapted to ; that is, is -measurable, for all .(ii).
Definition 5 (see ). One can say that the random process has a -stochastic integral on , provided the corresponding process has a stochastic integral in the common sense on and then one can have .
Lemma 6 (see ). The -stochastic integral has the following properties: (i)If and , then(ii)If , then and the Itô-isometry holds; that is,
Definition 7 (see ). If, for all and , such that where is Lebesgue integrable on and , then we say that the process has a -stochastic differential indicated by the notation
We consider the impulsive stochastic dynamic system on :with initial value , where is a random variable, independent of .
Definition 8 (see ). One can say that a random process with is a solution of the impulsive stochastic dynamic system (19) on if the following conditions hold:(i) is adapted to the filtration .(ii)For all , we have almost surely where is the Cauchy matrix of the following system:
Definition 9 (see ). One can say is rd-piecewise continuous with respect to a sequence which satisfies , , if is continuous on and rd-continuous on . Furthermore, , , are called intervals of continuity of the function .
For convenience, denotes the set of all piecewise continuous functions with respect to a sequence , . For any integers and , denote .
Definition 10 (see ). A time scale is called an almost periodic time scale if
Definition 11 (see ). For any , let be a set of real numbers and . One can say , , is equipotentially almost periodic on an almost periodic time scale if, for , there exists at least one integer such that
Definition 12 (see ). Let be an almost periodic time scale and assume that satisfying the derived sequence , , is equipotentially almost periodic. One can call a stochastic process mean-square almost periodic if (i)for any , there is a positive number such that if the points and belong to the same interval of continuity and , then ;(ii)for any , there is relative dense set such that if , then , for all , which satisfy the condition , .
Definition 13 (see ). Let be an almost periodic time scale and assume that satisfying the derived sequence , , is equipotentially almost periodic. One can call a stochastic process mean-square almost periodic in uniformly for if for any and for each compact subset of there existed the following:(i)for any , there is a positive number such that if the points and belong to the same interval of continuity and , then ;(ii)for any , there is relative dense set such that if , then , for all , which satisfy the condition , ,where or .
Definition 14. The equation (3) is said to be exponentially stable if, for all , there exist and such that if , then, for all ,
In order to study (3), we consider the linear system: Now let us consider the equations and their solutions
Similar to the proofs of Lemma 3.1 and Lemma 3.2 in , one can easily show the following two lemmas, respectively.
Lemma 15. For system (3), let hold. Then, for each , there exist , , a relative dense set of real numbers and , such that the following relations are fulfilled:(a), , , , , , , , , , ;(b), , , , , , ;(c), , , .
Lemma 16. For (3), let hold. Then, the Cauchy matrix of (3) satisfies the inequality and for any , , , , , there exists a relatively dense set of -almost periods of the function and a positive constant such that, for ,
Definition 17. The equation (3) is said to be exponentially stable if, for all , there exist and such that if , then, for all ,
3. Existence of Almost Periodic Solution
In this section, we will study the existence and exponential stability of mean-square almost periodic solutions of (3) by using the Banach fixed point theorem and Gronwall-Bellman’s inequality technique on time scales.
Theorem 18. Assume that hold and there exist two positive constants , such that the following conditions hold.
, where , whereThen,(1)There exists a unique square-mean almost periodic solution of (3) in the region .(2)The solution in the region is exponentially stable.
Proof. We define an operator in aswhereFor any , by means of the proof of Theorem 1 in , we haveFirstly, we will show that is a self-mapping from to . In fact, for any , by use of (37), we haveOn the other hand, let and , where the set is determined in Lemma 15. Then,whereSince , it follows that By conditions and Lemmas 15 and 16, we can obtain thatwhereConsequently, by (39) and (42), we obtain that .
Next, we prove that the mapping is a contraction mapping of . In fact, in view of , for any , we can easily obtain Notice thatThis implies that the mapping is a contraction mapping. Hence, there exists a unique mean-square almost periodic solution of (3) by the Banach fixed point theorem. Suppose that is an arbitrary solution of (3) with initial conditions: Since every solution of system (3) can be represented as we obtain that Since , it follows that Let ; then, From Gronwall-Bellman’s inequality on time scales, one has that is,whereThus, by condition of the theorem, the mean-quare almost periodic solution of (3) is exponentially stable. This completes the proof.
Remark 19. According to the conditions of Theorem 18, we can find that the uniqueness and exponential stability of the mean-square almost periodic solution for the impulsive stochastic host-macroparasite equation on time scales are independent of the magnitude of delays but are dependent on the magnitude of noise and impulse.
Remark 20. It is the first time that the sufficient conditions for the existence and exponential stability of piecewise mean-square almost periodic solutions for the impulsive stochastic host-macroparasite equation on time scales are investigated. The obtained results are essentially new. Without considering impulsive and stochastic effects on (3), then (3) reduce to (2).
4. An Example
Consider the following impulsive stochastic host-macroparasite equation on time scales: where , , , and
By calculating, we have
It is easy to see that