Positive periodic solution of first-order neutral differential equation with infinite distributed delay and applications

1.   Introduction

This paper is devoted to investigate the existence of positive $\omega$-periodic solutions of the following first-order neutral differential equation with infinite distributed delay

where $c$ is a constant with $|c|\neq1$, $P(t)\in C((-\infty, 0], [0, +\infty))$ with $\int^{0}_{-\infty}P(\sigma)d\sigma = 1$, $a(t)\in C(\mathbb{R}, \mathbb{R})$, $b(t)\in C(\mathbb{R}, (0, +\infty))$ and the nonlinear term $f \in C(\mathbb{R}\times\mathbb{R}, \mathbb{R})$ are $\omega$-periodic functions with respect to $t$ where $\int^{\omega}_{0}a(t)dt > 0$.

Equation (1.1) includes many mathematical ecological models and population models. For example, the Hematopoiesis model ^[1,2,3]

where $\beta(t)\in C(\mathbb{R}, \mathbb{R})$ is a continuous $\omega$-periodic function, the Nicholson's blowflies model ^[4,5,6]

and the model of blood cell production ^[7,8,9]

It is well known that neutral equations play a significant role in the applied science. Many scholars have studied the above equations from different perspectives ^{[5,7,10,11,12,13,14,15,16,17]}. For Eq (1.1), using Krasnoselskii's fixed-point theorem in cones, Li and Wang ^[7] proved the existence of positive $\omega$-periodic solutions, where $c\in[0, 1)$ and $a(t)\in C(\mathbb{R}, (0, +\infty))$. Luo et al. ^[5] obtained sufficient conditions for the existence of positive $\omega$-periodic solutions for $|c| < 1$ and $a(t)\in C(\mathbb{R}, (0, +\infty))$. However, the above results are only related to the case of sublinearity, and can not applicable to models (1.2)-(1.4). A naturally question is that whether or not there is any positive $\omega$-periodic solution for (1.1) with semi-linearity and super-linearity? Especially for models (1.2)-(1.4)?

In this paper, we provide some sufficient conditions for the existence of positive $\omega$-periodic solutions of Eq (1.1) where the nonlinear term $f$ may satisfy sub-linearity, semi-linearity and super-linearity conditions. The main tool is the fixed point theorem of Leray-Schauder type. As applications, we prove that models (1.2)-(1.4) exist positive $\omega$-periodic solutions.

Compared with ^[5,7], we have following five differences. Firstly, we give the property of neutral operator $(Au)(t): = u(t)-c\int^{0}_{-\infty} P(\sigma)u(t+\sigma)d\sigma$ for the first time. Secondly, we enlarge the range of the parameter $c$, i.e., $|c| < 1$ and $|c| > 1$. Thirdly, we weaken conditions of the nonlinear term $f$, i.e., $f$ satisfies sub-linearity, semi-linearity and super-linearity conditions. Fourthly, we prove that the models (1.2)-(1.4) have at least one positive $\omega$-periodic solution for the first time. Fifthly, in addition to $a(t)$ is a positive function, we study that $a(t)$ may change sign.

2.   Preliminaries

Define

with $\|u\|: = \max\limits_{t\in\mathbb{R}}|u(t)|.$ Obviously, $(C_\omega, \|\cdot\|)$ is a Banach space.

Firstly, we recall a fixed point theorem of Leray-Schauder type, which will be used in our proof.

Lemma 2.1. [18,Theorem 5] Let $B(0, r_1)$ (respectively, $B[0, r_1]$) be the open ball (respectively, the closed ball) in a Banach space $X = (X, \|\cdot\|)$ with center 0 and radius $r_1$. Suppose $\mathcal{A}, \mathcal{B}:X\rightarrow X$ are two operators satisfying the following conditions:

(a) $\mathcal{A}$ is a contraction;

(b) $\mathcal{B}$ is continuous and completely continuous.

Then either

(i) $\exists\; u \in B[0, r_1]$ with $u = \mathcal{A}u+\mathcal{B}u$; or

(ii) $\exists\; u \in \partial B[0, r_1]$ and $\lambda \in (0, 1)$ with $u = \lambda \mathcal{A}(\frac{u}{\lambda})+ \lambda \mathcal{B}u.$

Lemma 2.2. [5,Remark 2.2] The following first-order linear differential equation

has an $\omega$-periodic solution

where

and $h\in C^+_\omega: = \{h\in C(\mathbb{R}, (0, \infty)):\; h(t+\omega) = h(t), \; \mathit{\mbox{for}}\; t\in\mathbb{R}\}.$ Moreover, since $\int_{0}^{\omega}a(t)dt > 0$, it is clear that $G(t, s)$ is positive for all $(t, s)\in[0, \omega]\times[0, \omega]$.

Next, we give the property of operator $A$.

Lemma 2.3. If $|c| < 1$, then the operator $A$ has a continuous inverse $A^{-1}$ on $C_\omega$ satisfying

Proof. First, define an operator $S:C_\omega\rightarrow C_\omega$ by

Then, we arrive that

Therefore, we get

Since $A = I-S$, where $I$ is an identity operator, and

we obtain that $A$ has a continuous inverse $A^{-1}:C_\omega\rightarrow C_\omega$ by

Thus we have

Therefore, we obtain

3.   Equation (1.1) with small constant $c$

In this section, we consider the existence of a positive $\omega$-periodic solution of Eq (1.1). Moreover, we suppose the absolute value of constant $c$ is smaller than $1$, to be precise, $c\in(-\frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}, \frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}),$ where $\bar{a}: = \frac{1}{\omega}\int^{\omega}_{0}a(t)dt$. We divide the discussion into the following two cases $c\in\left(0, \frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}\right)$ and $c\in\left(-\frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}, 0\right]$.

3.1.   Equation (1.1) in the case that $c\in\left(0, \frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}\right)$

Theorem 3.1. Suppose $c\in\left(0, \frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}\right)$ holds. Furthermore, assume that there exists a constant $r > 0$ such that

($H_{1}$) There exist continuous, non-negative functions $q(u)$ and $k(t)$ such that

where $q(u)$ is non-decreasing in $[0, r]$.

($H_{2}$) The following inequality holds

where $K(t): = \int^{\omega}_{0}{G}(t, s)k(s)ds,$ and $K^*: = \max \limits_{t\in[0, \omega]}K(t).$

Then Eq (1.1) has at least one positive $\omega$-periodic solution with $u\in[0, r]$.

Proof. Consider Eq (1.1)

and a family of the equations

Let $v(t) = (Au)(t).$ From Lemma 2.3, we have $u(t) = (A^{-1}v)(t).$ Then Eq (1.1) and (3.1) can be written in the following forms

and

where

Let

then $h(t)\in C_{\omega}^+$ and Eq (3.2) can be written as the following linear differential equation

Define operators $\mathcal{T}, \mathcal{N}:C_\omega\rightarrow C_\omega$ by

where $G(t, s)$ is defined in Lemma 2.2. Therefore, $v(t)$ satisfied by

is the positive $\omega$-periodic solution of Eq (3.1). Moreover, Eq (3.5) is equivalent to

Since $c\in\left(0, \frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}\right)$, using Lemma 2.2, we obtain

Case 1: If $0\leq s\leq t\leq\omega$,

Case 2: If $0\leq t\leq s\leq\omega$, similarly, we obtain the same result that $\|\mathcal{T}\mathcal{N}\| < 1$.

Hence, $I-\mathcal{T}\mathcal{N}$ is an invertible linear operator and

Now we define $\mathcal{P}:C_\omega\rightarrow C_\omega$ by

Since $\|\mathcal{T}\mathcal{N}\| < 1$, applying Neumann expansion of $\mathcal{P}$, we have

From inequality (3.6), we obtain that

for all $h(t)\in C_\omega^+.$ Define operators $\mathcal{A}, \; \mathcal{B}:C_\omega\rightarrow C_\omega$ by

According to the above analysis, the existence of a positive $\omega$-periodic solution of Eq (3.1) is just a fixed point of the following operator equation

in $C_\omega$. Similarly, the existence of a positive $\omega$-periodic solution of Eq (1.1) is just a fixed point of the following operator equation

in $C_\omega$.

Next, we use a fixed point theorem of Leray-Schauder type, see Lemma 2.1, to prove the existence of fixed point of Eq (3.9). Define

where $r$ is defined in Theorem 3.1. Obviously, $B[0, r]$ is a bounded closed convex set in $C_\omega$. Then, we obtain at

for any $u\in B[0, r]$, and $t\in\mathbb{R}$. Obviously, $(\mathcal{A}u)(t)$ and $(\mathcal{B}u)(t)$ are $\omega$-periodic. Moreover, we obtain

for any $u_1$, $u_2 \in B[0, r]$. Since $c\in\left(0, \frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}\right)$, we know that $\mathcal{A}$ is contractive. Moreover, it is easy to obtain that $\mathcal{B}$ is completely continuous (for details, please see [10,Theorem 3.1]).

On the other hand, we claim that any fixed point $u$ of Eq (3.8) for any $\lambda\in(0, 1)$ must satisfy $\|u\|\neq r$. Through the reverse proving, we assume that the above claim does not holds. Then, there exists a fixed point $u$ of Eq (3.8) for some $\lambda\in(0, 1)$ such that $\|u\| = r$. From Eq (3.7), conditions $(H_1)$ and $(H_2)$, we obtain

Thus, $r = \|u\| < r$, this is a contradiction. Using Lemma 2.1, we obtain that $u = \mathcal{A}u+\mathcal{B}u$ has a fixed point $u$ in $B[0, r]$. Therefore, Equation (1.1) has at least one positive $\omega$-periodic solution $u$ with $u\in[0, r]$.

In the following, applying Theorem 3.1, we consider the existence of positive $\omega$-periodic solutions of the Hematopoiesis model (1.2), the Nicholson's blowflies model (1.3) and the model of blood cell production (1.4).

Corollary 3.1. Assume $c\in\left(0, \frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}\right)$ holds, then model (1.2) has at least one positive $\omega$-periodic solution.

Proof. We apply Theorem 3.1 in which we set. Let us set

Then condition $(H_1)$ is satisfied and the existence condition $(H_2)$ becomes

We can choose $r$ appropriately large such that (3.11) holds.

Corollary 3.2. Assume $c\in\left(0, \frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}\right)$ holds. Furthermore, the following inequality holds:

Then, model (1.3) has at least one positive $\omega$-periodic solution.

Proof. We apply Theorem 3.1 in which we set. Let us set

Then condition $(H_1)$ is satisfied and the existence condition $(H_2)$ becomes

From (3.12), we know $\ln^{\frac{1-e^{-\bar{a}\omega}+c(e^{-\bar{a}\omega}-1-\|a\|\omega)}{\|b\|\omega}} > 0,$ we can take $r$ appropriately small such that (3.13) holds.

Corollary 3.3. Assume $c\in\left(0, \frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}\right)$ and (3.12) hold. Then, model (1.4) has at least one positive $\omega$-periodic solution.

Proof. We apply Theorem 3.1 in which we set. Let us set

Then condition $(H_1)$ is satisfied and the existence condition $(H_2)$ becomes

3.2.   Equation (1.1) in the case that $c\in\left(-\frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}, 0\right]$

Theorem 3.2. Suppose $c\in\left(-\frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}, 0\right]$ and ($H_1$) hold. Furthermore, assume that the following condition is satisfied:

($H_{3}$) There exists a constant $r > 0$ such that

Then Eq (1.1) has at least one positive $\omega$-periodic solution with $u\in[0, r]$.

Proof. We follow the same notations and use a similar method as in the proof of Theorem 3.1. For Eq (3.8)

we claim that any fixed point $u$ of Eq (3.8) for any $\lambda\in(0, 1)$ must satisfy $\|u\|\neq r$. Through the reverse proving, we assume that the above claim does not holds. Then, there exists a fixed point $u$ of Eq (3.8) for some $\lambda\in(0, 1)$ such that $\|u\| = r$. From Eq (3.7), conditions ($H_1)$ and ($H_3)$, we get

where

Thus, $r = \|u\| < r$, this is a contradiction. Therefore, using Lemma 2.1, we obtain that $u = \mathcal{A}u+\mathcal{B}u$ has a fixed point $u$ in $B[0, r]$. Hence, Equation (1.1) has at least one positive $\omega$-periodic solution $u$ with $u\in[0, r]$.

By Theorem 3.2 and Corollary 3.1, Corollary 3.2, Corollary 3.3, we get the following conclusions.

Corollary 3.4. Assume $c\in\left(-\frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}, 0\right]$ holds, then model (1.2) has at least one positive $\omega$-periodic solution.

Corollary 3.5. Assume $c\in\left(-\frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}, 0\right]$ holds. Furthermore, The following inequality holds:

Then, model (1.3) has at least one positive $\omega$-periodic solution.

Corollary 3.6. Assume $c\in\left(-\frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}, 0\right]$ and (3.15) hold. Then, model (1.4) has at least one positive $\omega$-periodic solution.

Remark 3.1. If $|c| > 1$, from (3.10), we do not obtain that $\mathcal{A}$ is contractive. Therefore, the above method does not apply to the case that $|c| > 1$. Next, we use another way to get over this problem.

4.   Equation (1.1) with large constant $c$

In this section, we consider the existence of a positive $\omega$-periodic solution of Eq (1.1). Moreover, we suppose the absolute value of constant $c$ is larger than $1$. We divide the discussion into the following two cases $c\in\left(1, +\infty\right)$ and $c\in\left(-\infty, -1\right)$.

4.1.   Equation (1.1) in the case that $c\in(1, +\infty)$

Consider Eq (1.1), it can be transformed into

Define

then Eq (1.1) can be written as

Theorem 4.1. Suppose $c\in(1, +\infty)$ holds. Furthermore, assume that there exists a constant $r > 0$ such that

($H_4$) There exist continuous, non-negative functions $q(u)$ and $k(t)$ such that

where $h(u)$ is non-decreasing in $[0, r]$.

($H_5$) The following condition holds

where $K^*$ is defined by Theorem (3.1).

Then Eq (1.1) has at least one positive $\omega$-periodic solution $u$ with $u(t)\in[0, r]$.

Proof. Let us set

then Eq (4.2) can be written as the following form

Next we study the following equation

Then we obtain

Define operators $\tilde{\mathcal{A}}$, $\tilde{\mathcal{B}}: C_\omega\rightarrow C_\omega$ by

According to the above analysis, the existence of a positive $\omega$-periodic of Eq (4.3) is equivalent to the existence of solution for the operator equation

in $C_\omega$. Similarly, the existence of a positive $\omega$-periodic of Eq (1.1) is equivalent to the existence of solution for the operator equation

in $C_\omega$.

Next, we use a fixed point theorem of Leray-Schauder type, see Lemma 2.1, to prove the existence of fixed point of Eq (4.5). First, we have

for any $u\in B[0, r]$, and $t\in\mathbb{R}$. Obviously, $(\tilde{\mathcal{A}}u)(t)$ and $(\tilde{\mathcal{B}}u)(t)$ are $\omega$-periodic. Moreover, we get

for any $u\in B[0, r]$, and $t\in\mathbb{R}$. Thus, we know that $\tilde{\mathcal{A}}$ is contractive since $c\in(1, +\infty)$. By using the same notations and a similar method as in the proof of Theorem 3.1, we can get that $\tilde{B}$ is completely continuous.

Next, we claim that any fixed point $u$ of Eq (4.4) for any $\lambda\in(0, 1)$ must satisfy $\|u\|\neq r$. Through the reverse proving, we assume that the above claim does not holds. Then, there exists a fixed point $u$ of Eq (4.4) for some $\lambda\in(0, 1)$ such that $\|u\| = r$. From conditions ($H_4$) and ($H_5$), we have

Thus, $r = \|u\| < r$, this is a contradiction. Using Lemma 2.1, we see that $u = \tilde{\mathcal{A}}u+\tilde{\mathcal{B}}u$ has a fixed point $u$ in $B[0, r]$. Therefore, Equation (1.1) has at least one positive $\omega$-periodic solution $u(t)$ with $u(t)\in[0, r]$.

Corollary 4.1. Assume $c\in(1, +\infty)$ holds. Furthermore, The following inequality holds:

Then, models (1.2), (1.3) and (1.4) respectively have at least one positive $\omega$-periodic solution.

Proof. We apply Theorem 4.1 in which we set. Let us set

Then condition $(H_4)$ is satisfied and the existence condition $(H_5)$ becomes

since $1-e^{-\bar{a}\omega}-\omega\|a\| > 0$. Then, models (1.2), (1.3) and (1.4) respectively have at least one positive $\omega$-periodic solution.

Remark 4.1. If $c\in(-\frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}}, \frac{1-e^{-\bar{a}\omega}}{1+\|a\|\omega-e^{-\bar{a}\omega}})$, the method of proving the positive $\omega$-periodic solutions of models (1.2), (1.3) and (1.4) in Corollaries 1-6 is more general than the above method (Corollary 4.1).

4.2.   Equation (1.1) in the case that $c\in(-\infty, -1)$

Theorem 4.2. Suppose $c\in(-\infty, -1)$ and ($H_4$) hold. Furthermore, the following condition is satisfied:

($H_6$) There exists a constant $r > 0$ such that

Then Eq (1.1) has at least one positive $\omega$-periodic solution $u$ with $u(t)\in[0, r]$.

Proof. We follow the same notations and use the same method in the proof of Theorem 4.1. We claim that any fixed point $u$ of Eq (4.4) for any $\lambda\in(0, 1)$ must satisfy $\|u\|\neq r$. Through the reverse proving, we assume that the above claim does not holds. Then, there exists a fixed point $u$ of Eq (4.4) for some $\lambda\in(0, 1)$ such that $\|u\| = r$. From conditions $(H_4)$ and $(H_6)$, we get

Thus, $r = \|u\| < r$, this is a contradiction. Using Lemma 2.1, we see that $u = \tilde{\mathcal{A}}u+\tilde{\mathcal{B}}u$ has a fixed point $u$ in $B[0, r]$. Therefore, Equation (1.1) has at least one positive $\omega$-periodic solution $u(t)$ with $u(t)\in[0, r]$.

Remark 4.2. If $c\in(-\infty, -1)$, from the definition of $F(t, x)$ and models (1.2), (1.3) and (1.4), we can not find appropriate $k(t)$ and $h(u)$ such that conditions ($H_4$) and ($H_6$) are satisfied. Therefore, the above method does not apply to models (1.2), (1.3) and (1.4).

Finally, we present an example to illustrate our results.

Example 4.1. Consider the following neutral equation

Comparing Eq (4.6) to Eq (1.1), we have $\omega = \frac{\pi}{4}, \sigma = \frac{\pi}{5}$, $\bar{a} = 2$, $c = \frac{1}{20} < \frac{1-\frac{1}{e^{\frac{\pi}{2}}}}{1+\frac{3\pi}{4}-\frac{1}{e^{\frac{\pi}{2}}}}\approx0.9340$, $a(t) = \cos 8t+2$, $b(t) = \sin 8t+2$, $f(t, u) = (\cos 8t+2)8u^{2}(t+\frac{\pi}{5})$. Let $k(t) = \cos 8t+2$, $q(u) = 8u^{2}(t+\frac{\pi}{5})$, we get condition $(H_1)$ is satisfied. Let $r = \frac{1}{40}$, we can verify that condition $(H_2)$ is satisfied. Applying Theorem 3.1, Equation (4.6) has at least one $\frac{\pi}{4}$-periodic solution with $u\in[0, \frac{1}{40}]$.

5.   Conclusion

By virtue of a fixed point theorem of Leray-Schauder type, we prove the existence of positive periodic solutions of the following first-order neutral differential equation with infinite distributed delay

and we prove that Hematopoiesis model, Nicholson's blowflies model and the model of blood cell production have positive periodic solutions.

Acknowledgments

Research is supported by National Natural Science Foundation of China (11501170), China Postdoctoral Science Foundation funded project (2016M590886), Key Research Funds for the Universities of Henan Province (19A110018, 20B110006), Fundamental Research Funds for the Universities of Henan Province (NSFRF170302, NSFRF180320), Henan Polytechnic University Doctor Fund (B2016-58).

Conflict of interest

The authors declare that they have no competing interests in this paper.