The analysis of some special results of a Lasota–Wazewska model with mixed variable delays

This study is about getting some conditions that guarantee the existence and uniqueness of the weighted pseudo almost periodic (WPAP) solutions of a Lasota–Wazewska model with time-varying delays. Some adequate conditions have been obtained for the existence and uniqueness of the WPAP solutions of the Lasota–Wazewska model, which we dealt with using some differential inequalities, the WPAP theory, and the Banach fixed point theorem. Besides, an application is given to demonstrate the accuracy of the conditions of our main results.


Introduction
In 1976, Wazewska and Lasota [1] presented the delayed logistic differential model z (t) = -(t)z(t) + p k=1 κ k (t)e -η k (t)z(t-ρ k (t)) (1.1) The author obtained some conditions on the almost periodic solution of this model using the fixed point theorem in cones. In [7], the researchers established some qualitative behaviors of PAP solutions of the following equation with constant delays: 3) The study of almost periodic (AP) and pseudo almost periodic (PAP) differential equations is one of the most interesting issues for the study of almost periodic of many mathematicians: indeed, they are of great importance even in probability for investigating stochastic processes in stability problems tied to oscillatory phenomena [1,3,6,[8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], and [25]. In [26], Diagana familiarized the concept of (WPAP) functions, which is a natural generalization of the concept of (PAP) functions. Since then, some interesting and remarkable results concerning composition theorem, translation invariance, and the ergodicity of (WPAP) have been obtained [26][27][28][29]. It is clear that under some limitations of weight function, many of the properties of almost periodic (AP) and pseudo almost periodic (PAP) are valid in this type of class. Thanks to the invariant property under translation, it is quite simple to investigate such solutions in delayed differential equations. For some works on the pseudo almost periodic solutions, oscillation of solutions, and so fourth of various differential equations, see [4,5,24,25,[30][31][32][33][34].
Our main purpose is to obtain some sufficient conditions for the existence, uniqueness, and global exponential stability of (WPAP) solutions of the following Lasota-Wazewska model with mixed variable delays: As far as we know, there are no studies related to the (WPAP) solutions of (1.4) with variable delays. Therefore, the results attained here are new and complementary to previous studies.

Preliminary results
Definition 2.1 ([8]) A function f ∈ C(R, R) is called almost periodic if for any ε > 0 there exists a trigonometric polynomial T ε such that

Definition 2.2 ([27]) A function η ∈ C(R, R) is called (PAP) if it can be written as
with η 1 ∈ AP(R, R) and η 2 ∈ PAP 0 (R, R), where space PAP 0 is defined by Let be the set of functions (weight) υ : R → (0, ∞) which are integrable on (-∞, ∞). If υ ∈ and Q := [-q, q] for q > 0, we then set The space of weights ∞ is defined by A continuous function is called WPAP if it can be written as with η 1 ∈ AP(R, R) and η 2 ∈ PAP 0 (R, R), where space PAP 0 is defined by

Lemma 3.1 Suppose that
Define a nonlinear operator G for each z ∈ PAP(R, R, υ) Then Gz ∈ PAP(R, R, υ).
Proof Because of M[a] > 0 in [8] and Lemma 3.1 in [7], we have that According to Lemma 2.1, Lemma 2.2, we obtain that there are z 11 (t) ∈ AP(R, R) and z 12 (t) ∈ PAP 0 (R, R, υ) such that is a solution of the following almost periodic differential equation: By using a similar manner in the proof of Theorem 3.5 in [7], it can be displayed that z 12 Now, we shall prove that L 1 = L 2 = 0, From Lemma 2.1 the function z 12 (tξ ) ∈ PAP 0 (R, R, υ), we obtain that Therefore, Notice that |z 12 | ∞ = sup t∈R |z 12 (t)| = M and by (3.1), then Proof First, let us prove that G ∈ PAP(R, R + , υ) into itself. It is clear that Obviously, for x 1 , x 1 ∈ [0, +∞], Therefore, By AA we can see that (1 -(δ - 1), and hence G is a contraction mapping of C * . Subsequently, G has a unique fixed point z * ∈ C * that is G(z * ) = z * . Thus, z * is the unique WPAP solution of (1.4) in C * .