Let $t,t_{0}\in\mathbb{R},\;t<t_{0},\;\tau_{1},\tau_{2}>0,\;\tau_{1}\leq\tau_{2},\;f_{i}\in C([t_{0},b]\times\mathbb{R}^{4})$, $i=1,2$, $\varphi\in C[t_{0}-\tau_{1},t_{0}]$, $\psi\in C[t_{0}-\tau_{2},t_{0}]$ be given.

There have been many studies on this subject (see [2], [5], [8]). The fact that time delays are harmless for the uniform persistence of solutions, is established by Wang and Ma for a predator-prey system, by Lu and Takeuchi and Takeuchi for competitive systems.

Recently, Saito, Hara and Ma [8] have derived necessary and sufficient conditions for the permanence (uniform persistence) and global stability of a symmetrical Lotka-Volterra-type predator-prey system with two delays.

For a nonautonomous competitive Lotka-Volterra system with no delays, recently Ahmad and Lazer have established the average conditions for the persistence, which are weaker than those of Gopalsamy and Tineo and Alvarez for periodic or almost-periodic cases.

Here we study the existence and uniqueness of the solution using the contraction principle and the data dependence using Lemma 2 for the problem (1.1)+(1.2).

The purpose of this section is to find the conditions for the existence and uniqueness of the solution of problem (1.1)+(1.2).

Let $(x,y)$ be a solution of (1.1)+(1.2). The problem (1.1)+(1.2) is equivalent with

where $x\in C[t_{0}-\tau_{1},b]$ and $y\in C[t_{0}-\tau_{2},b]$.

We consider the operator

and we remark that it follows

where

Consider the Banach space $C[t_{0},b]$ with Bielecki norm $\left\|\cdot\right\|_{B}$ defined by

For $t\in[t_{0}-\tau_{1},t_{0}]$ we have $\left|A_{f}(x,y)(t)-A_{f}(\overline{x},\overline{y})(t)\right|=0.$

For $t\in[t_{0}-\tau_{2},t_{0}]$ we have $\left|A_{f}(x,y)(t)-A_{f}(\overline{x},\overline{y})(t)\right|=0.$

For $t\in[t_{0},b],$ let $(X,d)$ be a metric space with $X=(C[t_{0},b],\left\|\cdot\right\|_{B})$ and $\;(x,y),(\overline{x},\overline{y})\in X\times X,$ then:

Consequently,

where $\pi_{1}$ is the first projection for $A_{f}(x,y)$ from (2.4).

By similar calculations we obtain

where $\pi_{2}$ is the second projection for $A_{f}(x,y)$ from (2.4). We deduce

Then $A_{f}$ is Lipschitz with a Lipschitz constant $L_{A_{f}}=\dfrac{4L}{\rho}$. For $\rho=4L+1,$ $A_{f}$ is a contraction. By the contraction principle we have:

In this section we shall discus a theorem of data dependence for the solution of problem (1.1)+(1.2). To prove data dependence relation we need the following lemma:

We have

•   Proof.

  Consider that we are under the hypothesis of Theorem 1. If $(x_{1}^{*},y_{1}^{*})$ solution of problem (1.1)+(1.2) with data $f_{i}^{1},\varphi^{1},\psi^{1},A_{f}^{1},\;i=1,2,$ and if $(x_{2}^{*},y_{2}^{*})$ solution of problem (1.1)+(1.2) with data $f_{i}^{2},\varphi^{2},\psi^{2},A_{f}^{2},\;i=1,2,$ then it follows that

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  (3.1)

  $$\begin{split}&\left|\pi_{1}A_{f}^{1}(x,y)(t)-\pi_{1}A_{f}^{2}(x,y)(t)\right|\\ &=\left|\varphi^{1}(t_{0})+{\displaystyle\int_{t_{0}}^{t}}f_{1}^{1}(s,x(s),y(s),x(s-\tau_{1}),y(s-\tau_{2}))\mathrm{d}s\right.\\ &\left.-\varphi^{2}(t_{0})-{\displaystyle\int_{t_{0}}^{t}}f_{1}^{2}(s,x(s),y(s),x(s-\tau_{1}),y(s-\tau_{2}))\mathrm{d}s\right|\\ &\leq\left|\varphi^{1}(t_{0})-\varphi^{2}(t_{0})\right|+{\displaystyle\int_{t_{0}}^{t}}\left|f_{1}^{1}(s,x(s),y(s),x(s-\tau_{1}),y(s-\tau_{2}))\right.\\ &-\left.f_{1}^{2}(s,x(s),y(s),x(s-\tau_{1}),y(s-\tau_{2}))\right|\mathrm{d}s\leq\eta_{1}+\eta_{3}(t-t_{0}).\end{split}$$

  We have

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  (3.2)		$\displaystyle\left|\pi_{1}A_{f}^{1}(x,y)(t)-\pi_{1}A_{f}^{2}(x,y)(t)\right|\leq\eta_{1}+\eta_{3}(t-t_{0})$
  (3.3)		$\displaystyle\left|\pi_{1}A_{f}^{1}(x,y)(t)-\pi_{1}A_{f}^{2}(x,y)(t)\right|e^{-\rho(t-t_{0})}\leq\eta_{1}+\eta_{3}(t-t_{0})$
  (3.4)		$\displaystyle\left\|\pi_{1}A_{f}^{1}(x,y)(t)-\pi_{1}A_{f}^{2}(x,y)(t)\right\|_{B}\leq\eta_{1}+\eta_{3}(t-t_{0}).$

  Analogously

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  $$\left\|\pi_{2}A_{f}^{1}(x,y)(t)-\pi_{2}A_{f}^{2}(x,y)(t)\right\|\leq\eta_{2}+\eta_{3}(t-t_{0}).$$

  Then

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  $$\left\|A_{f}^{1}(x,y)-A_{f}^{2}(x,y)\right\|_{B}\leq\eta_{1}+\eta_{2}+2\eta_{3}(t-t_{0}).$$

  From Lemma 2 we have:

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  $$\left\|(x_{1}^{*},y_{1}^{*})-(x_{2}^{*},y_{2}^{*})\right\|_{B}\leq\frac{\eta_{1}+\eta_{2}+2\eta_{3}(t-t_{0})}{1-\dfrac{4L}{4L+1}}$$

  So the proof is complete. ∎

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Let $\mu\in\mathbb{R},\;\varphi\in C[-1,0],\;\psi\in C[-2,0]$ be given. We consider the problem

Then

We note that if we take $A:C[-1,2]\times[-2,2]\rightarrow C[-1,2]\times[-2,2]$ defined by

then the problem (4.1) is equivalent with

From Theorem 1 the problem (4.1) has a unique solution.

In what follows we discuss the data dependence of the solution.
Let $\varphi^{1},\;\varphi^{2},\;\psi^{1},\;\psi^{2}.$ We suppose that there are $\delta_{i}>0,\;i=1,2,3$ such that

Let us consider the problems:

If $(x_{1}^{*},y_{1}^{*})$ is a solution for the problem (4.4) and $(x_{2}^{*},y_{2}^{*})$ is the solution for the problem (4.5), we look for a estimation of $\left\|(x_{1}^{*},y_{1}^{*})-(x_{2}^{*},y_{2}^{*})\right\|$. We have the operators $A_{f}^{1}(x,y)(t)$ and $A_{f}^{2}(x,y)(t)$ It follows that

From Theorem 3, we have

Let $x_{1}(t),x_{2}(t),\dots,x_{n}(t)$ be lows of growing, continuous and derivable. Then we have the system

where $t\in[t_{0},b],\;i=1,\dots,n,\;f_{i}\in C([t_{0},b]\times\mathbb{R}^{n}\times\mathbb{R}^{n},\mathbb{R}),\;i=1,\dots,n$, and the initial conditions

The problem is to determine $x_{i}\in C([t_{0}-\tau_{i},b])\cap C^{1}[t_{0},b],\;i=1,\dots,n$ that suits the problem (5.1)+(5.2).

By the contraction principle we have

Applying Lemma 2 we have