SMOOTH DEPENDENCE ON PARAMETERS FOR A DIFFERENTIAL SYSTEM WITH DELAYS FROM POPULATION DYNAMICS

Diana Otrocol

Abstract.

The purpose of this paper is to present a differential systems with delays from population dynamics. For this system we study the smooth dependence on parameters for the solution of the systems using the fibre contraction principle.

1. Introduction

Time delays occur in differential equations arising in many fields of applied mathematics: chemistry, ecology, biology, population dynamics, electric engineering. In population dynamics they may model the gestation or maturation time of a species, or time taken for food resources to regenerate.

The purpose of this paper is to study the dependence on parameter $\lambda$ for the solution of the problem:

(1)

\begin{split}x_{i}^{\prime}(t)=f_{i}(t,x_{1}(t;\lambda),x_{2}(t;\lambda),x_{1}(t-\tau_{1};&\lambda),x_{2}(t-\tau_{2};\lambda);\lambda),\\ &\;t\in[t_{0},b],\;i=1,2;\end{split}

(2)

\left\{\begin{array}[c]{c}x_{1}(t)=\varphi(t),\;t\in[t_{0}-\tau_{1},t_{0}],\\ x_{2}(t)=\psi(t),\;t\in[t_{0}-\tau_{2},t_{0}],\end{array}\right.

Some problems concerning this techniques was study in [4], [5], [6], [1], [8].

2. Weakly Picard operators

I.A. Rus introduced the Picard operators class (PO) and the weakly Picard operators class (WPO) for the operators defined on a metric space and he gave basic notations, definitions and results in this field in many papers [4]–[6].

Let $(X,d)$ be a metric space and $A:X\rightarrow X$ an operator. We shall use the following notations:

$F_{A}:=\{x\in X\mid A(x)=x\}$ - the fixed point set of $A$ ;

$I(A):=\{Y\subset X\mid A(Y)\subset Y,Y\neq\emptyset\}$ - the family of the nonempty invariant subset of $A$ ;

$A^{n+1}:=A\circ A^{n},\;A^{0}=1_{X},\;A^{1}=A,\;n\in\mathbb{N}$ ;

$P(X):=\{Y\subset X\mid Y\neq\emptyset\}$ - the set of the parts of $X;$

Definition 1.

Let $(X,d)$ be a metric space. An operator $A:X\rightarrow X$ is a Picard operator (PO) if there exists $x^{\ast}\in X$ such that:

(i)

$F_{A}=\{x^{\ast}\},$
(ii)

the sequence $(A^{n}(x_{0}))_{n\in\mathbb{N}}$ converges to $x^{\ast}$ for all $x_{0}\in X$ .

Remark 1.

Accordingly to the definition, the contraction principle insures that, if $A:X\rightarrow X$ is an $\alpha$ -contraction on the complete metric space $X$ , then it is a Picard operator.

Definition 2.

Let $(X,d)$ be a metric space. An operator $A:X\rightarrow X$ is a weakly Picard operator (WPO) if the sequence $(A^{n}(x))_{n\in\mathbb{N}}$ converges for all $x\in X$ , and its limit ( which may depend on $x$ ) is a fixed point of $A$ .

Theorem 1.

Let $(X,d)$ be a metric space and $A:X\rightarrow X$ an operator. The operator $A$ is weakly Picard operator if and only if there exists a partition of $X$ ,

X=\underset{\lambda\in\Lambda}{\cup}X_{\lambda}

where $\Lambda$ is the indices set of partition, such that:

(a)

$X_{\lambda}\in I(A),\ \lambda\in\Lambda$ ;
(b)

$A|_{X_{\lambda}}:X_{\lambda}\rightarrow X_{\lambda}$ is a Picard operator for all $\lambda\in\Lambda$ .

Definition 3.

If $A$ is weakly Picard operator then we consider the operator $A^{\infty}$ defined by

A^{\infty}:X\rightarrow X,\ A^{\infty}(x):=\underset{n\rightarrow\infty}{\lim}A^{n}(x).

It is clear that $A^{\infty}(X)=F_{A}.$

Definition 4.

Let $A$ be a weakly Picard operator and $c>0.$ The operator $A\$ is $c$ -weakly Picard operator if

d(x,A^{\infty}(x))\leq cd(x,A(x)),\ \forall x\in X.

Example 1.

Let $(X,d)$ be a complete metric space and $A:X\rightarrow X$ a continuous operator. We suppose that there exists $\alpha\in[0,1)$ such that

d(A^{2}(x),A(x))\leq\alpha(x,A(x)),\ \forall x\in X.

Then $A$ is $c$ -weakly Picard operator with $c=\dfrac{1}{1-\alpha}.$

Theorem 2.

(Fibre contraction principle). Let $(X,d)$ and $(Y,\rho)$ be two metric spaces and $A:X\times Y\rightarrow X\times Y,\ A=(B,C),\ (\ B:X\rightarrow X,\ C:X\times Y\rightarrow Y\ )$ a triangular operator. We suppose that

(i)

$(Y,\rho)$ is a complete metric space;
(ii)

the operator $B$ is Picard operator;
(iii)

there exists $l\in[0,1)$ such that $C(x,\cdot):Y\rightarrow Y$ is a $l$ -contraction, for all $x\in X$ ;
(iv)

if $(x^{\ast},y^{\ast})\in F_{A}$ , then $C(\cdot,y^{\ast})$ is continuous in $x^{\ast}$ .

Then the operator $A$ is Picard operator.

3. Main results

Consider the following problem with parameter

(3)

\begin{split}x_{i}^{\prime}(t)=f_{i}(t,x_{1}(t),x_{2}(t),x_{1}(t-&\tau_{1}),x_{2}(t-\tau_{2});\lambda),\\ &\;t\in[t_{0},b],\;i=1,2,\end{split}

(4)

\left\{\begin{array}[c]{c}x_{1}(t)=\varphi(t),\;t\in[t_{0}-\tau_{1},t_{0}],\\ x_{2}(t)=\psi(t),\;t\in[t_{0}-\tau_{2},t_{0}]\end{array}\right..

We suppose that

$(H_{1})\;\tau_{1}\leq\tau_{2};\;t_{0}<b;\;J\subset\mathbb{R}$ , a compact interval;

$(H_{2})\;f_{i}\in C^{1}([t_{0},b]\times\mathbb{R}^{4}\times J),\;i=1,2$ ;

$(H_{3})\;\exists L_{f}>0$ such that $\left\|\dfrac{\partial f_{i}}{\partial u_{j}}(t,u_{1},u_{2},u_{3},u_{4};\lambda)\right\|_{\mathbb{R}^{n}}\leq L_{f},\;\newline$ for all $\ t\in[t_{0},b],\;u_{j}\in R^{n},\;j=\overline{1,4},\;\lambda\in J,i=1,2$ ;

$(H_{4})\;\varphi\in C([t_{0}-\tau_{1},t_{0}]),\;\psi\in C([t_{0}-\tau_{2},t_{0}])$ ;

We have

Theorem 3.

([2]) We suppose that the conditions ( $H_{1}$ )-( $H_{4}$ ) are satisfied.

Then the problem (3)–(4) has a unique solution. Moreover, if $(x_{1}^{\ast},x_{2}^{\ast})$ the unique solution of (3)–(4), then

(x_{1}^{\ast},x_{2}^{\ast})=\underset{n\rightarrow\infty}{\lim}A_{f}^{n}(x_{1},x_{2}),\;\text{for all }x_{1}\in C[t_{0}-\tau_{1},b]\text{, }x_{2}\in C[t_{0}-\tau_{2},b].

From the Theorem 3, in [2], we have that the problem (3)–(4) has a unique solution, $(x_{1}(t;\lambda),x_{2}(t;\lambda))$ .

Now we prove that

	$\displaystyle x_{1}(t;\cdot)$	$\displaystyle\in C^{1}(J)\text{, for all }t\in[t_{0}-\tau_{1},b],$
	$\displaystyle x_{2}(t;\cdot)$	$\displaystyle\in C^{1}(J)\text{, for all }t\in[t_{0}-\tau_{2},b].$

For this we consider the system

(5)

x_{i}^{\prime}(t)=f_{i}(t,x_{1}(t;\lambda),x_{2}(t;\lambda),x_{1}(t-\tau_{1};\lambda),x_{2}(t-\tau_{2};\lambda);\lambda),

where $\;t\in[t_{0},b],\;\lambda\in J,x_{1}\in C([t_{0}-\tau_{1},b]\times J)\cap C^{1}[t_{0},b],\;x_{2}\in C([t_{0}-\tau_{2},b]\times J)\cap C^{1}[t_{0},b]$ .

Theorem 4.

Consider the problem (5)–(4), in the conditions $(H_{1})-(H_{4})$ . Then the problem (5)–(4) has a unique solution $(x_{1}^{\ast},x_{2}^{\ast}),\;x_{1}^{\ast}\in C([t_{0}-\tau_{1},b]\times J)\cap C^{1}[t_{0},b],\;x_{2}^{\ast}\in C([t_{0}-\tau_{2},b]\times J)\cap C^{1}[t_{0},b]$ and the solution is derivable on $\lambda$ .

Proof.

The problem (5)–(4) is equivalent with

(6)

\begin{array}[c]{l}x_{1}(t)=\left\{\begin{array}[c]{l}\varphi(t),\;t\in[t_{0}-\tau_{1},t_{0}],\\ \varphi(t_{0})+\\ +\int\nolimits_{t_{0}}^{t}f_{1}(s,x_{1}(s;\lambda),x_{2}(s;\lambda),x_{1}(s-\tau_{1};\lambda),x_{2}(s-\tau_{2};\lambda);\lambda)\mathrm{d}s,\\ \;\;\;\;\;\;\;t\in[t_{0},b],\lambda\in J,\end{array}\right.\\ x_{2}(t)=\left\{\begin{array}[c]{l}\psi(t),\;t\in[t_{0}-\tau_{2},t_{0}],\\ \psi(t_{0})+\\ +\int\nolimits_{t_{0}}^{t}f_{2}(s,x_{1}(s;\lambda),x_{2}(s;\lambda),x_{1}(s-\tau_{1};\lambda),x_{2}(s-\tau_{2};\lambda);\lambda)\mathrm{d}s,\\ \;\;\;\;\;\;\;t\in[t_{0},b],\lambda\in J.\end{array}\right.\end{array}

We consider the operator

\begin{array}[c]{l}B_{f}:C([t_{0}-\tau_{1},b]\times J)\times C([t_{0}-\tau_{2},b]\times J)\rightarrow\\ \rightarrow C([t_{0}-\tau_{1},b]\times J)\times C([t_{0}-\tau_{2},b]\times J)\end{array}

where $B_{f}(x_{1},x_{2})$ is defined by the second part of (6).

Let $X:=C([t_{0}-\tau_{1},b]\times J)\times C([t_{0}-\tau_{2},b]\times J)$ and let, $\left\|\cdot\right\|$ , be the Chebyshev norm on $X$ . It is clear that, in the conditions $(H_{1})-(H_{4})$ , the operator $B_{f}$ is Picard operator (Remark 6,[2]). Let $(x_{1}^{*},x_{2}^{*})$ be the unique fixed point of $B_{f}$ .

We suppose that there exists $\dfrac{\partial x_{1}}{\partial\lambda}$ , $\dfrac{\partial x_{2}}{\partial\lambda}$ . Then from (6) we have that:

\begin{split}&\frac{\partial x_{i}^{*}(t;\lambda)}{\partial\lambda}=\\ &=\int_{t_{0}}^{t}\frac{\partial f_{i}(s,x_{1}^{*}(s;\lambda),x_{2}^{*}(s;\lambda),x_{1}^{*}(s-\tau_{1};\lambda),x_{2}^{*}(s-\tau_{2};\lambda);\lambda)}{\partial x_{1}^{*}(s;\lambda)}\cdot\frac{\partial x_{1}^{*}(s,\lambda)}{\partial\lambda}ds\\ &+\int_{t_{0}}^{t}\frac{\partial f_{i}(s,x_{1}^{*}(s;\lambda),x_{2}^{*}(s;\lambda),x_{1}^{*}(s-\tau_{1};\lambda),x_{2}^{*}(s-\tau_{2};\lambda);\lambda)}{\partial x_{2}^{*}(s;\lambda)}\cdot\frac{\partial x_{2}^{*}(s,\lambda)}{\partial\lambda}ds\\ &+\int_{t_{0}}^{t}\frac{\partial f_{i}(s,x_{1}^{*}(s;\lambda),x_{2}^{*}(s;\lambda),x_{1}^{*}(s-\tau_{1};\lambda),x_{2}^{*}(s-\tau_{2};\lambda);\lambda)}{\partial x_{1}^{*}(s-\tau_{1};\lambda)}\cdot\frac{\partial x_{1}^{*}(s-\tau_{1},\lambda)}{\partial\lambda}ds\\ &+\int_{t_{0}}^{t}\frac{\partial f_{i}(s,x_{1}^{*}(s;\lambda),x_{2}^{*}(s;\lambda),x_{1}^{*}(s-\tau_{1};\lambda),x_{2}^{*}(s-\tau_{2};\lambda);\lambda)}{\partial x_{1}^{*}(s-\tau_{2};\lambda)}\cdot\frac{\partial x_{1}^{*}(s-\tau_{2},\lambda)}{\partial\lambda}ds,\end{split}

where $t\in[t_{0},b],\;\lambda\in J,\;i=1,2.$

This relations suggests us to consider the operator $C_{f}:X\times X\rightarrow X$ defined by

\begin{split}C_{f}&(x_{1},x_{2},u,v)(t,\lambda):=\\ &=\int_{t_{0}}^{t}\frac{\partial f_{i}(s,x_{1}^{*}(s;\lambda),x_{2}^{*}(s;\lambda),x_{1}^{*}(s-\tau_{1};\lambda),x_{2}^{*}(s-\tau_{2};\lambda);\lambda)}{\partial u_{1}}\cdot u(s;\lambda)ds\\ &+\int_{t_{0}}^{t}\frac{\partial f_{i}(s,x_{1}^{*}(s;\lambda),x_{2}^{*}(s;\lambda),x_{1}^{*}(s-\tau_{1};\lambda),x_{2}^{*}(s-\tau_{2};\lambda);\lambda)}{\partial u_{2}}\cdot v(s;\lambda)ds\\ &+\int_{t_{0}}^{t}\dfrac{\partial f_{i}(s,x_{1}^{*}(s;\lambda),x_{2}^{*}(s;\lambda),x_{1}^{*}(s-\tau_{1};\lambda),x_{2}^{*}(s-\tau_{2};\lambda);\lambda)}{\partial u_{3}}\cdot u(s-\tau_{1};\lambda)ds\\ &+\int_{t_{0}}^{t}\frac{\partial f_{i}(s,x_{1}^{*}(s;\lambda),x_{2}^{*}(s;\lambda),x_{1}^{*}(s-\tau_{1};\lambda),x_{2}^{*}(s-\tau_{2};\lambda);\lambda)}{\partial u_{4}}\cdot u(s-\tau_{2};\lambda)ds.\end{split}

	$\displaystyle C_{f}(x_{1},x_{2},u,v)(t,\lambda)$	$\displaystyle:=0\text{ for all }t\in[t_{0}-\tau_{1},t_{0}],\;\lambda\in J,$
	$\displaystyle C_{f}(x_{1},x_{2},u,v)(t,\lambda)$	$\displaystyle:=0\;\text{for all }t\in[t_{0}-\tau_{2},t_{0}],\;\lambda\in J.$

In this way we have the triangular operator

A:X\times X\rightarrow X\times X

(x_{1},x_{2},u,v)\rightarrow(B_{f}(x_{1},x_{2}),C_{f}(x_{1},x_{2},u,v))

where $B_{f}$ is a Picard operator and $C_{f}(x_{1},x_{2},\cdot,\cdot):X\rightarrow X$ is an $\alpha$ -contraction, with $\alpha=\dfrac{4L_{f}}{\rho}$ .

From the theorem of fibre contraction we have that the operator $A$ is a Picard operator and $F_{A}={(x_{1}^{*},x_{2}^{*},u^{*},v^{*})}$ .

Let $(x_{1}^{*},x_{2}^{*},u^{*},v^{*})$ be the unique fixed point of the operator $A$ . So, the sequences

\begin{array}[c]{l}(x_{1,n+1},x_{2,n+1}):=B(x_{1,n},x_{2,n})\\ (u_{n+1},v_{n+1}):=C(x_{1,n},x_{2,n},u_{n},v_{n})\end{array},n\in\mathbb{N}

converges uniformly (with respect to $t\in X,\;\lambda\in J$ ) to $(x_{1}^{*},x_{2}^{*},u^{*},v^{*})\in F_{A}$ , for all $x_{1,0},x_{2,0},u_{0},v_{0}\in X$

If we take $x_{1,0}=0,\;x_{2,0}=0,\;u_{0}=\dfrac{\partial x_{1,0}}{\partial\lambda}=0,\;v_{0}=\dfrac{\partial x_{2,0}}{\partial\lambda}=0$ , then

u_{1}=\frac{\partial x_{1,1}}{\partial\lambda},\;v_{1}=\frac{\partial x_{2,1}}{\partial\lambda}.

By induction we prove that

u_{n}=\frac{\partial x_{1,n}}{\partial\lambda},\;v_{n}=\frac{\partial x_{2,n}}{\partial\lambda},\;\forall n\in\mathbb{N}.

Thus

	$\displaystyle x_{1,n}$	$\displaystyle\overset{unif}{\rightarrow}x_{1}^{*}\text{ as }n\rightarrow\infty,$
	$\displaystyle x_{2,n}$	$\displaystyle\overset{unif}{\rightarrow}x_{2}^{*}\text{ as }n\rightarrow\infty,$
	$\displaystyle\frac{\partial x_{1,n}}{\partial\lambda}$	$\displaystyle\overset{unif}{\rightarrow}u^{*}\text{ as }n\rightarrow\infty,$
	$\displaystyle\frac{\partial x_{2,n}}{\partial\lambda}$	$\displaystyle\overset{unif}{\rightarrow}v^{*}\text{ as }n\rightarrow\infty.$

By Weierstrass’ theorem it follows that there exists $\dfrac{\partial x_{i}^{*}}{\partial\lambda},\;i=1,2$ and

\frac{\partial x_{1}^{*}}{\partial\lambda}=u^{*}\text{ ,\ }\frac{\partial x_{2}^{*}}{\partial\lambda}=v^{*}\text{ .}

From the above consideration, we have the theorem proved. ∎

References

[1] Mureşan, V., Functional-Integral Equations, Editura Mediamira, Cluj-Napoca, 2003.
[2] Otrocol, D., Data dependence for the solution of a Lotka-Volterra system with two delays, Mathematica, vol. 48, 71, 1, 61-68, 2006.
[3] Otrocol, D., Lotka-Volterra system with two delays via weakly Picard operators, Nonlinear Analysis Forum, 10, 2, 193-199, 2005.
[4] Rus, I.A., Generalized contractions, Seminar of Fixed Point Theory, ”Babeş-Bolyai” University, 1-130, 1983.
[5] Rus, I.A., Functional-differential equation of mixed type, via weakly Picard operators, Seminar of Fixed Point Theory, Cluj-Napoca, vol. 3, 335-346, 2002.
[6] Rus, I.A., Generalized Contractions and Applications, Cluj University Press, 2001.
[7] Rus, I.A., Principii si aplicaţii ale teoriei punctului fix, Editura Dacia, Cluj-Napoca, 1979.
[8] Şerban, M.A., Fiber $\varphi$ -contractions, studia Univ. ”Babeş-Bolyai”, Mathematica, 44, 3, 99-108, 1999.

Smooth dependence on parameters for a differential equation with delay from population dynamics

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SMOOTH DEPENDENCE ON PARAMETERS FOR A DIFFERENTIAL SYSTEM WITH DELAYS FROM POPULATION DYNAMICS

Abstract.

1. Introduction

2. Weakly Picard operators

Definition 1.

Remark 1.

Definition 2.

Theorem 1.

Definition 3.

Definition 4.

Example 1.

Theorem 2.

3. Main results

Theorem 3.

Theorem 4.

Proof.

References

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