In recent years the theory of excitable medium has rapidly developed and its results have been applied in various areas: chemistry, biology, ecology, electric engineering, populations dynamics, cardiology, neurology. At present, different approaches for the mathematical description of biological excitable medium by means of partial-differential equation, functional-differential, functional and discrete equations are applied. The papers [2], [3], [10] has offered the opportunity for understanding the normal regulation of living systems as well as its anomalies.

The activity of the $i$-th element of the excitable medium can be described by the following equation:

where $x_{i}(t)$ is the activity of the $i$-th element; $a_{i}$ is the functional parameter of the $i$-th element; $f_{i}(\cdot)$ is the feedback function; $b_{i}$ is the decay constant, $i=\overline{1,m}$.

The aim of this paper is to study the following problem

$t\in\left[t_{0},b\right],i=\overline{1,m}$, with initial conditions

where
$\left(\mathrm{H}_{1}\right)t_{0}<b,h>0,t_{0},b,h\in R;$
$\left(\mathrm{H}_{2}\right)f_{i}\in C\left(R^{m},R\right),i=\overline{1,m}$;
$\left(\mathrm{H}_{3}\right)\varphi_{i}\in C\left(\left[t_{0}-h,t_{0}\right],R\right),i=\overline{1,m};$
$\left(\mathrm{H}_{4}\right)$ there exists $L_{f}>0$, such as:

for all $u_{i},v_{i}\in R,i=\overline{1,m}$.
By a solution of the problem (2)-(3) we understand the function $x=\left(x_{1},\ldots,x_{m}\right)\in R^{m}$ with $x_{i}\in C\left(\left[t_{0}-h,b\right],R\right)\cap C^{1}\left(\left[t_{0},b\right],R\right),i=\overline{1,m}$ which satisfies (2)-(3).

The problem (2)-(3) is equivalent with the following fixed point system:

where $x_{i}\in C\left(\left[t_{0}-h,b\right],R\right),i=\overline{1,m}$.
On the other hand, the system (2) is equivalent with

where $x_{i}\in C\left(\left[t_{0}-h,b\right],R\right),i=\overline{1,m}$.
In this paper we apply the weakly Picard operators technique to study the systems (4) and (5).

I.A. Rus introduced the Picard operators class $(\mathrm{PO})$ 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 [7]-[9]. Some problems concerning this techniques were study in [4], [11], [5], [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 $A$;
$A^{n+1}:=A\circ A^{n},A^{0}=1_{X},A^{1}=A,n\in N;$
$P(X):=\{Y\subset X\mid Y\neq\emptyset\}$ - the set of the parts of $X$;
$H(Y,Z):=\max\left\{\sup_{y\in Y}\inf_{z\in Z}d(y,z),\sup_{z\in Z}\inf_{y\in Y}d(y,z)\right\}$-the Pompeiu-Housdorff functional on $P(X)\times P(X)$.

Definition 1. ([7], [9]) Let ( $X,d$ ) be a metric space. An operator $A:X\rightarrow X$ is a Picard operator $(PO)$ if there exists $x^{*}\in X$ such that:
(i) $F_{A}=\left\{x^{*}\right\}$;
(ii) the sequence $\left(A^{n}\left(x_{0}\right)\right)_{n\in N}$ converges to $x^{*}$ for all $x_{0}\in X$.

Remark 1. ([7], [9]) 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.

Theorem 1. ([7], [9]) (Data dependence theorem). Let ( $X,d$ ) be a complete metric space and $A,B:X\rightarrow X$ two operators. We suppose that
(i) the operator $A$ is a $\alpha$-contraction;
(ii) $F_{B}\neq\emptyset$;
(iii) there exists $\eta>0$ such that

Then, if $F_{A}=\left\{x_{A}^{*}\right\}$ and $x_{B}^{*}\in F_{B}$, we have

Definition 2. ([7], [9]) Let ( $X,d$ ) be a metric space. An operator $A:X\rightarrow X$ is a weakly Picard operator (WPO) if the sequence $\left(A^{n}(x)\right)_{n\in N}$ converges for all $x\in X$, and its limit (which may depend on $x$ ) is a fixed point of $A$.

Theorem 2. ([7], [9]) 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 $X$,

where $\Lambda$ is the indices set of partition, such that:
(a) $X_{\lambda}\in I(A),\lambda\in\Lambda$;
(b) $\left.A\right|_{X_{\lambda}}:X_{\lambda}\rightarrow X_{\lambda}$ is a Picard operator for all $\lambda\in\Lambda$.

Definition 3. ([7], [9]) If $A$ is weakly Picard operator then we consider the operator $A^{\infty}$ defined by

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

Definition 4. ([7], [9]) Let $A$ be a weakly Picard operator and $c>0$. The operator $A$ is $c$-weakly Picard operator if

Example 1. ([7], [9]) 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

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

Theorem 3. ([7], [9]) Let ( $X,d$ ) be a metric space and $A_{i}:X\rightarrow X,i=1,2$. Suppose that
(i) the operator $A_{i}$ is $c_{i}$-weakly Picard operator, $i=1,2$;
(ii) there exists $\eta>0$ such that

Then $H\left(F_{A_{1}},F_{A_{2}}\right)\leq\eta\max\left(c_{1},c_{2}\right)$.

Theorem 4. ([7], [9]) (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 $\left(x^{*},y^{*}\right)\in F_{A}$, then $C\left(\cdot,y^{*}\right)$ is continuous in $x^{*}$.

Then the operator $A$ is Picard operator.

We consider the fixed point system (4).
Let $A_{f}:C\left(\left[t_{0}-h,b\right],R^{m}\right)\rightarrow C\left(\left[t_{0}-h,b\right],R^{m}\right)$ given by the relation

where

We consider the Banach space $C\left(\left[t_{0}-h,b\right],R^{m}\right)$ with the Chebyshev norm $\|\cdot\|_{C}$. Let $X=\left(C\left(\left[t_{0}-h,b\right],R^{m}\right),\|\cdot\|_{C}\right)$.

We have the following result

Theorem 5. We suppose that
(i) the conditions $\left(\mathrm{H}_{1}\right)-\left(\mathrm{H}_{4}\right)$ are satisfied;
(ii) $L_{f}\left(b-t_{0}\right)\sum_{i=1}^{m}\frac{a_{i}}{b_{i}}e^{-b_{i}t_{0}}<1$.

Then the Cauchy problem (2)-(3) has in $C\left(\left[t_{0}-h,b\right],R^{m}\right)$ a unique solution. Moreover, the operator $A_{f}:C\left(\left[t_{0}-h,b\right],R^{m}\right)\rightarrow C\left(\left[t_{0}-h,b\right],R^{m}\right)$ is c-Picard with

Proof. For $t\in\left[t_{0}-h,t_{0}\right]$, we have
$\left|A_{f_{i}}\left(x_{1},\ldots,x_{m}\right)(t)-A_{f_{i}}\left(\bar{x}_{1},\ldots,\bar{x}_{m}\right)(t)\right|=0,i=\overline{1,m}$.
For $t\in\left[t_{0},b\right]$, we have

Then

So $A_{f}$ is $c$-Picard operator with $c=\frac{1}{1-L_{A_{f}}}$, where $L_{A_{f}}=L_{f}\left(b-t_{0}\right)\sum_{i=1}^{m}\frac{a_{i}}{b_{i}}e^{-b_{i}t_{0}}$.

In what follows, we consider the following operator

given by

where

Theorem 6. In the condition of Theorem 5, $B_{f}:C\left(\left[t_{0}-h,b\right],R^{m}\right)\rightarrow C\left(\left[t_{0}-\right.\right.\left.h,b],R^{m}\right)$ is WPO.

Proof. The operator $B_{f}$ is a continuous operator but it is not a contraction. Let take the following notation:

Then we can write

We have that $X_{\varphi_{1}}\times\ldots\times X_{\varphi_{m}}\in I\left(B_{f}\right)$ and $\left.B_{f}\right|_{X_{\varphi_{1}}\times\ldots\times X_{\varphi_{m}}}$ is a Picard operator, because it is the operator which appears in the proof of the Theorem 5. By applying the Theorem 2, we obtain that $B_{f}$ is WPO.

Consider the Cauchy problem (2)-(3) and suppose the conditions of the Theorem 5 are satisfied. Denote by $x(\cdot,\varphi,f)=\left(x_{1}\left(\cdot;\varphi_{1},f_{1}\right),\ldots,x_{m}\left(\cdot;\varphi_{m},f_{m}\right)\right)$, the solution of this problem. We can state the following result:

Theorem 9. Let $\varphi_{i}^{j},f_{i}^{j},i=\overline{1,m},j=1,2$ be as in the Theorem 5. Furthermore, we suppose that there exists $\eta_{i}^{1},\eta_{i}^{2},i=\overline{1,m}$ such that
(i) $\left|\varphi_{i}^{1}(t)-\varphi_{i}^{2}(t)\right|\leq\eta_{i}^{1},\forall t\in\left[t_{0}-h,t_{0}\right],i=\overline{1,m}$;
(ii) $\left|f_{i}^{1}\left(u_{1},\ldots,u_{m}\right)-f_{i}^{2}\left(u_{1},\ldots,u_{m}\right)\right|\leq\eta_{i}^{2},i=\overline{1,m},u_{i}\in R$.

Then

where $L_{f}=\max\left(L_{f^{1}},L_{f^{2}}\right),i=\overline{1,m}$.

Proof. Consider the operators $A_{\varphi_{i}^{j},f_{i}^{j}},i=\overline{1,m},j=1,2$. From Theorem 5 these operators are contractions.

Additionally

$\forall x=\left(x_{1},\ldots,x_{m}\right)\in C\left(\left[t_{0}-h,b\right],R^{m}\right)$.
Now the proof follows from the Theorem 1, with $A:=A_{\varphi_{i}^{1},f_{i}^{1}},B=A_{\varphi_{i}^{2},f_{i}^{2}},\eta=\sum_{i=1}^{m}\eta_{i}^{1}e^{-b_{i}\left(t-t_{0}\right)}+\left(b-t_{0}\right)\sum_{i=1}^{m}\frac{a_{i}}{b_{i}}e^{-b_{i}t_{0}}\eta_{i}^{2}$ and $\alpha:=L_{A_{f}}=L_{f}\left(b-t_{0}\right)\sum_{i=1}^{m}\frac{a_{i}}{b_{i}}e^{-b_{i}t_{0}}$ where $L_{f}=\max\left(L_{f^{1}},L_{f^{2}}\right),i=\overline{1,m}$.

From the Theorem above we have:

Theorem 10. Let $f_{i}^{1}$ and $f_{i}^{2}$ be as in the Theorem 5, $i=\overline{1,m}$. Let $S_{B_{f_{i}^{1}}},S_{B_{f_{i}^{2}}}$ be the solution set of system (2) corresponding to $f_{i}^{1}$ and $f_{i}^{2},i=\overline{1,m}$. Suppose that there exists $\eta_{i}>0,i=\overline{1,m}$ such that

for all $u_{i}\in R,i=\overline{1,m}$.
Then

where $L_{f}=\max\left(L_{f^{1}},L_{f^{2}}\right)$ and $H_{\|\cdot\|_{C}}$ denotes the Pompeiu-Housdorff functional with respect to $\|\cdot\|_{C}$ on $C\left(\left[t_{0}-h,b\right],R^{m}\right)$.

Proof. In condition of the Theorem 5, the operators $B_{f_{i}^{1}}$ and $B_{f_{i}^{2}},i=\overline{1,m}$ are $c_{1}$-WPO and $c_{2}$-weakly Picard operators.

Let

It is clear that $\left.B_{f_{i}^{1}}\right|_{X_{\varphi_{i}}}=A_{f_{i}^{1}},\left.B_{f_{i}^{2}}\right|_{X_{\varphi_{i}}}=A_{f_{i}^{2}}$. So, from Theorem 2 and Theorem 5 we have

for all $\left(x_{1},\ldots,x_{m}\right)\in C\left(\left[t_{0}-h,b\right],R^{m}\right),i=\overline{1,m}$.
Now, choosing

we get that $B_{f_{i}^{1}}$ and $B_{f_{i}^{2}}$ are $c_{1}$-weakly Picard operators and $c_{2}$-weakly Picard operators with $c_{1}=\left(1-\alpha_{1}\right)^{-1}$ and $c_{2}=\left(1-\alpha_{2}\right)^{-1}$. From (7) we obtain that

$\forall\left(x_{1},\ldots,x_{m}\right)\in C\left(\left[t_{0}-h,b\right],R^{m}\right),i=\overline{1,m}$. Applying Theorem 3 we have that

where $L_{f}=\max\left(L_{f^{1}},L_{f^{2}}\right)$ and $H_{\|\cdot\|_{C}}$ is the Pompeiu-Housdorff functional with respect to $\|\cdot\|_{C}$ on $C\left(\left[t_{0}-h,b\right],R^{m}\right)$. $\square$