New insights on weighted pseudo almost periodic solutions in a Lasota–Wazewska system

We study the weighted pseudo almost periodic solutions of a Lasota–Wazewska system. With the aid of fixed point theory and differential inequality strategies, we give a set of new sufficient criteria that guarantee the existence and global exponential stability of weighted pseudo almost periodic solutions to a Lasota–Wazewska system. The obtained results of this manuscript are completely innovative and complement the work of Shao (Appl. Math. Lett. 43:90–95, 2015) to some degree. So far, no scholars have investigated this aspect.


Introduction
In 1988, Wazewska-Czyzewska and Lasota [2] proposed the delayed differential model w(t) = -(t)w(t) + p k=1 θ k (t)e -η k (t)w(t-ρ k (t)) (1.1) to describe the survival of red cells in an animal [1]. In this model, p is a positive integer, w(t) stands for the number of red blood cells at time t, (t) stands for the death rate of the red blood cell, θ k (t) and η k (t) are related to the production of red blood cells per unit time, and ρ k (t) represents the time to produce a red blood cell. For details, see [2]. We know that the death rate or harvesting rate usually change under different seasonal fluctuations. In addition, the actual living environment of species have weighted pseudo almost periodic nature due to the effect of human activities and industrial production, for example, the exhaust emission and reconstruction of rivers. Based on this viewpoint, we think that it is reasonable to suppose that the coefficients in model (1.1) are weighted pseudo almost periodic functions, which can be expressed as an almost periodic component plus the weighted ergodic perturbation. Thus a key problem aries: seek the new suf-ficient conditions to ensure the existence and exponential stability of the weighted pseudo almost periodic solution for model (1.1).
Unfortunately, until now, no scholars have considered the weighted pseudo almost periodic solutions for model (1.1). To make up for this deficiency and inspired by the previous discussion, in this work, we concentrate on the weighted pseudo almost periodic solutions for model (1.1).
Let BC(R, R) denote the set of all bounded continuous functions from R to R, and letḡ = sup t∈R g(t) and g = inf t∈R g(t) for a bounded continuous function g(t). The initial condition of system (1.1) is given by where ψ ∈ BC([-ρ, 0], R + ), ρ = max k=1,2,...,pρk , and R + denotes the nonnegative real numbers.
We plan the paper as follows. In Sect. 2, we present some preliminary knowledge on weighted pseudo almost periodic solution. In Sect. 3, we investigate the existence and global exponential stability of weighted pseudo almost periodic solution to model (1.1). In Sect. 4, numerical simulations are put into effect. The conclusion is given to end this paper in Sect. 5.
Then there exists t 0 such that the solution w(t) of system We introduce the following assumptions: (A 2 ) υ ∈ U + ∞ , and there exist constants > 0 and ζ > 0 such that
Thus all the hypotheses of Theorem 3.1 are satisfied, and so system (4.1) has a unique weighted pseudo almost periodic solution, which is globally exponentially stable. Figure 1 reveals this fact.

Conclusions
In this paper, we have discussed the existence and globally exponential stability of weighted pseudo almost periodic solutions for a Lasota-Wazewska system. Using the fixed point theory and differential inequalities, we establish new sufficient criteria ensuring the existence and globally exponential stability of weighted pseudo almost periodic solutions for the Lasota-Wazewska model. The derived results complement some earlier publications to some extent. Up to now, to the best of our knowledge, it is the first time to deal with this aspect. In the near future, we will investigate the pseudo almost automorphic solutions for the Lasota-Wazewska model.