Abstract
Authors
D.Otrocol
Tiberiu Popoviciu In stitute of Numerical Analysis Romanian Academy
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D. Otrocol, Differentiability with respect to delays for a Lotka-Volterra system, Creative Math. and Inf., 2007, Vol. 16, pp. 36-41
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Creative Math. and Inf
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1584 – 286X
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1843 – 441X
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Differentiability with respect to delays for a Lotka-Volterra system*
Diana Otrocol
ABSTRACT. We study the differentiability with respect to delays using the weakly Picard operators technique.
1. Introduction
Consider the following Lotka-Volterra differential system with delays
{:[(1.1)x_(i)^(')(t)=f_(i)(t,x_(1)(t),x_(2)(t),x_(1)(t-tau_(1)),x_(2)(t-tau_(2)))","i=1","2","t in[t_(0),b]],[(1.2){[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)]]:}]:}\begin{gather*}
x_{i}^{\prime}(t)=f_{i}\left(t, x_{1}(t), x_{2}(t), x_{1}\left(t-\tau_{1}\right), x_{2}\left(t-\tau_{2}\right)\right), i=1,2, t \in\left[t_{0}, b\right] \tag{1.1}\\
\left\{\begin{array}{l}
x_{1}(t)=\varphi(t), t \in\left[t_{0}-\tau_{1}, t_{0}\right] \\
x_{2}(t)=\psi(t), t \in\left[t_{0}-\tau_{2}, t_{0}\right]
\end{array}\right. \tag{1.2}
\end{gather*}
Suppose that we have satisfied the following conditions:
(H_(1))t_(0) < b,tau,tau_(1),tau_(2) > 0,tau_(1) < tau_(2) < tau,tau_(1),tau_(2)in J,J=[t_(0),tau]\left(\mathrm{H}_{1}\right) t_{0}<b, \tau, \tau_{1}, \tau_{2}>0, \tau_{1}<\tau_{2}<\tau, \tau_{1}, \tau_{2} \in J, J=\left[t_{0}, \tau\right] a compact interval;
(H_(2))f_(i)inC^(1)([t_(0),b]xxR^(4),R),i=1,2\left(\mathrm{H}_{2}\right) f_{i} \in C^{1}\left(\left[t_{0}, b\right] \times \mathbb{R}^{4}, \mathbb{R}\right), i=1,2;
(H_(3))\left(\mathrm{H}_{3}\right) there exists L_(f) > 0L_{f}>0 such that
(H_(1))t_(0) < b,tau,tau_(1),tau_(2) > 0,tau_(1) < tau_(2) < tau,tau_(1),tau_(2)in J,J=[t_(0),tau]\left(\mathrm{H}_{1}\right) t_{0}<b, \tau, \tau_{1}, \tau_{2}>0, \tau_{1}<\tau_{2}<\tau, \tau_{1}, \tau_{2} \in J, J=\left[t_{0}, \tau\right] a compact interval;
(H_(2))f_(i)inC^(1)([t_(0),b]xxR^(4),R),i=1,2\left(\mathrm{H}_{2}\right) f_{i} \in C^{1}\left(\left[t_{0}, b\right] \times \mathbb{R}^{4}, \mathbb{R}\right), i=1,2;
(H_(3))\left(\mathrm{H}_{3}\right) there exists L_(f) > 0L_{f}>0 such that
||(delf_(i))/(delu_(j))(t,u_(1),u_(2),u_(3),u_(4))||_(R) <= L_(f)\left\|\frac{\partial f_{i}}{\partial u_{j}}\left(t, u_{1}, u_{2}, u_{3}, u_{4}\right)\right\|_{\mathbb{R}} \leq L_{f}
for all t in[t_(0),b],u_(j)in R,j= bar(1,4),i=1,2t \in\left[t_{0}, b\right], u_{j} \in R, j=\overline{1,4}, i=1,2;
(H_(4))varphi in C([t_(0)-tau,t_(0)],R),psi in C([t_(0)-tau,t_(0)],R);\left(\mathrm{H}_{4}\right) \varphi \in C\left(\left[t_{0}-\tau, t_{0}\right], \mathbb{R}\right), \psi \in C\left(\left[t_{0}-\tau, t_{0}\right], \mathbb{R}\right) ;
In the above conditions, from the Theorem 1, in [4], we have that the problem (1.1)-(1.2) has a unique solution, (x_(1)(t),x_(2)(t))\left(x_{1}(t), x_{2}(t)\right).
(H_(4))varphi in C([t_(0)-tau,t_(0)],R),psi in C([t_(0)-tau,t_(0)],R);\left(\mathrm{H}_{4}\right) \varphi \in C\left(\left[t_{0}-\tau, t_{0}\right], \mathbb{R}\right), \psi \in C\left(\left[t_{0}-\tau, t_{0}\right], \mathbb{R}\right) ;
In the above conditions, from the Theorem 1, in [4], we have that the problem (1.1)-(1.2) has a unique solution, (x_(1)(t),x_(2)(t))\left(x_{1}(t), x_{2}(t)\right).
2. Weakly Picard operators
In this paper we need some notions and results from the weakly Picard operator theory (for more details see I. A. Rus [9], [8], M. Şerban [14]).
Let ( X,dX, d ) be a metric space and A:X rarr XA: X \rightarrow X an operator. We shall use the following notations:
F_(A):={x in X∣A(x)=x}F_{A}:=\{x \in X \mid A(x)=x\} - the fixed point set of AA;
I(A):={Y in P(X)∣A(Y)sub Y}I(A):=\{Y \in P(X) \mid A(Y) \subset Y\} - the family of the nonempty invariant subset of AA;
A^(n+1):=A@A^(n),A^(0)=1_(X),A^(1)=A,n inNA^{n+1}:=A \circ A^{n}, A^{0}=1_{X}, A^{1}=A, n \in \mathbb{N} - the iterant operators of AA, where 1_(X)1_{X} is the identity operator;
P(X):={Y sub X∣Y!=O/}P(X):=\{Y \subset X \mid Y \neq \emptyset\} - the set of the parts of XX.
F_(A):={x in X∣A(x)=x}F_{A}:=\{x \in X \mid A(x)=x\} - the fixed point set of AA;
I(A):={Y in P(X)∣A(Y)sub Y}I(A):=\{Y \in P(X) \mid A(Y) \subset Y\} - the family of the nonempty invariant subset of AA;
A^(n+1):=A@A^(n),A^(0)=1_(X),A^(1)=A,n inNA^{n+1}:=A \circ A^{n}, A^{0}=1_{X}, A^{1}=A, n \in \mathbb{N} - the iterant operators of AA, where 1_(X)1_{X} is the identity operator;
P(X):={Y sub X∣Y!=O/}P(X):=\{Y \subset X \mid Y \neq \emptyset\} - the set of the parts of XX.
Definition 2.1. Let ( X,dX, d ) be a metric space. An operator A:X rarr XA: X \rightarrow X is a Picard operator (PO) if there exists x^(**)in Xx^{*} \in X such that:
(i) F_(A)={x^(**)}F_{A}=\left\{x^{*}\right\};
(ii) the sequence (A^(n)(x_(0)))_(n inN)\left(A^{n}\left(x_{0}\right)\right)_{n \in \mathbb{N}} converges to x^(**)x^{*} for all x_(0)in Xx_{0} \in X.
(i) F_(A)={x^(**)}F_{A}=\left\{x^{*}\right\};
(ii) the sequence (A^(n)(x_(0)))_(n inN)\left(A^{n}\left(x_{0}\right)\right)_{n \in \mathbb{N}} converges to x^(**)x^{*} for all x_(0)in Xx_{0} \in X.
Definition 2.2. Let ( X,dX, d ) be a metric space. An operator A:X rarr XA: X \rightarrow X is a weakly Picard operator (WPO) if the sequence (A^(n)(x))_(n inN)\left(A^{n}(x)\right)_{n \in \mathbb{N}} converges for all x in Xx \in X, and its limit ( which may depend on xx ) is a fixed point of AA.
Theorem 2.1. Let ( X,dX, d ) be a metric space and A:X rarr XA: X \rightarrow X an operator. The operator AA is WPO ( cc-WPO) if and only if there exists a partition of XX,
Theorem 2.1. Let ( X,dX, d ) be a metric space and A:X rarr XA: X \rightarrow X an operator. The operator AA is WPO ( cc-WPO) if and only if there exists a partition of XX,
X=uu_(lambda in Lambda)X_(lambda)X=\cup_{\lambda \in \Lambda} X_{\lambda}
such that:
(a) X_(lambda)in I(A),lambda in Lambda,I(A)X_{\lambda} \in I(A), \lambda \in \Lambda, I(A)-the family of nonempty invariant subsets of AA;
(b) A|_(X_(lambda)):X_(lambda)rarrX_(lambda)\left.A\right|_{X_{\lambda}}: X_{\lambda} \rightarrow X_{\lambda} is a Picard (c-Picard) operator for all lambda in Lambda\lambda \in \Lambda.
(a) X_(lambda)in I(A),lambda in Lambda,I(A)X_{\lambda} \in I(A), \lambda \in \Lambda, I(A)-the family of nonempty invariant subsets of AA;
(b) A|_(X_(lambda)):X_(lambda)rarrX_(lambda)\left.A\right|_{X_{\lambda}}: X_{\lambda} \rightarrow X_{\lambda} is a Picard (c-Picard) operator for all lambda in Lambda\lambda \in \Lambda.
Theorem 2.2. (Fibre contraction principle). Let ( X,dX, d ) and ( Y,rhoY, \rho ) be two metric spaces and A:X xx X rarr X xx X,A=(B,C),(B:X rarr X,C:X xx Y rarr Y)A: X \times X \rightarrow X \times X, A=(B, C),(B: X \rightarrow X, C: X \times Y \rightarrow Y) a triangular operator. We suppose that
(i) (Y,rho)(Y, \rho) is a complete metric space;
(ii) the operator BB is POP O;
(iii) there exists L in[0,1)L \in[0,1) such that C(x,*):Y rarr YC(x, \cdot): Y \rightarrow Y is a LL-contraction, for all x in Xx \in X;
(iv) if (x^(**),y^(**))inF_(A)\left(x^{*}, y^{*}\right) \in F_{A}, then C(*,y^(**))C\left(\cdot, y^{*}\right) is continuous in x^(**)x^{*}.
(i) (Y,rho)(Y, \rho) is a complete metric space;
(ii) the operator BB is POP O;
(iii) there exists L in[0,1)L \in[0,1) such that C(x,*):Y rarr YC(x, \cdot): Y \rightarrow Y is a LL-contraction, for all x in Xx \in X;
(iv) if (x^(**),y^(**))inF_(A)\left(x^{*}, y^{*}\right) \in F_{A}, then C(*,y^(**))C\left(\cdot, y^{*}\right) is continuous in x^(**)x^{*}.
Then the operator AA is POP O.
3. Main Result
Now we prove that
x_(i)(t,*)inC^(1)(J)," for all "t in[t_(0)-tau,b],i=1,2.x_{i}(t, \cdot) \in C^{1}(J), \text { for all } t \in\left[t_{0}-\tau, b\right], i=1,2 .
For this we consider the system
{:(3.3)x_(i)^(')(t)=f_(i)(t,x_(1)(t),x_(2)(t),x_(1)(t-tau_(1)),x_(2)(t-tau_(2)))","i=1","2:}\begin{equation*}
x_{i}^{\prime}(t)=f_{i}\left(t, x_{1}(t), x_{2}(t), x_{1}\left(t-\tau_{1}\right), x_{2}\left(t-\tau_{2}\right)\right), i=1,2 \tag{3.3}
\end{equation*}
where t in[t_(0),b],x_(1)in C[t_(0)-tau_(1),b]nnC^(1)[t_(0),b],x_(2)in C[t_(0)-tau_(2),b]nnC^(1)[t_(0),b]t \in\left[t_{0}, b\right], x_{1} \in C\left[t_{0}-\tau_{1}, b\right] \cap C^{1}\left[t_{0}, b\right], x_{2} \in C\left[t_{0}-\tau_{2}, b\right] \cap C^{1}\left[t_{0}, b\right].
From the above considerations, we can formulate the following theorem
Theorem 3.3. Consider the problem (3.3)-(1.2), in the conditions (H_(1))-(H_(4))\left(H_{1}\right)-\left(H_{4}\right). Then the problem (3.3)-(1.2) has a unique solution (x_(1)^(**),x_(2)^(**)),x_(1)^(**)in C[t_(0)-tau_(1),b]nnC^(1)[t_(0),b],x_(2)^(**)in C[t_(0)-tau_(2),b]nnC^(1)[t_(0),b]\left(x_{1}^{*}, x_{2}^{*}\right), x_{1}^{*} \in C\left[t_{0}-\tau_{1}, b\right] \cap C^{1}\left[t_{0}, b\right], x_{2}^{*} \in C\left[t_{0}-\tau_{2}, b\right] \cap C^{1}\left[t_{0}, b\right] and the solution is differentiable on tau_(1)\tau_{1} and tau_(2)\tau_{2}.
Proof. In what follows we consider the following integral equations:
From the above considerations, we can formulate the following theorem
Theorem 3.3. Consider the problem (3.3)-(1.2), in the conditions (H_(1))-(H_(4))\left(H_{1}\right)-\left(H_{4}\right). Then the problem (3.3)-(1.2) has a unique solution (x_(1)^(**),x_(2)^(**)),x_(1)^(**)in C[t_(0)-tau_(1),b]nnC^(1)[t_(0),b],x_(2)^(**)in C[t_(0)-tau_(2),b]nnC^(1)[t_(0),b]\left(x_{1}^{*}, x_{2}^{*}\right), x_{1}^{*} \in C\left[t_{0}-\tau_{1}, b\right] \cap C^{1}\left[t_{0}, b\right], x_{2}^{*} \in C\left[t_{0}-\tau_{2}, b\right] \cap C^{1}\left[t_{0}, b\right] and the solution is differentiable on tau_(1)\tau_{1} and tau_(2)\tau_{2}.
Proof. In what follows we consider the following integral equations:
{:[x_(1)(t,tau_(1),tau_(2))=],[={[varphi(t)","t in[t_(0)-tau_(1),t_(0)]],[varphi(t_(0))+int_(t_(0))^(t)f_(1)(s,x_(1)(s,tau_(1),tau_(2)),x_(2)(s,tau_(1),tau_(2)),x_(1)(s-tau_(1),tau_(1),tau_(2)),x_(2)(s-tau_(2),tau_(1),tau_(2)))ds","t in[t_(0),b]]:}],[x_(2)(t,tau_(1),tau_(2))=],[(3.4)={[psi(t)","t in[t_(0)-tau_(2),t_(0)]],[psi(t_(0))+int_(t_(0))^(t)f_(2)(s,x_(1)(s,tau_(1),tau_(2)),x_(2)(s,tau_(1),tau_(2)),x_(1)(s-tau_(1),tau_(1),tau_(2)),x_(2)(s-tau_(2),tau_(1),tau_(2)))ds","t in[t_(0),b]]:}]:}\begin{align*}
& x_{1}\left(t, \tau_{1}, \tau_{2}\right)= \\
& =\left\{\begin{array}{l}
\varphi(t), t \in\left[t_{0}-\tau_{1}, t_{0}\right] \\
\varphi\left(t_{0}\right)+\int_{t_{0}}^{t} f_{1}\left(s, x_{1}\left(s, \tau_{1}, \tau_{2}\right), x_{2}\left(s, \tau_{1}, \tau_{2}\right), x_{1}\left(s-\tau_{1}, \tau_{1}, \tau_{2}\right), x_{2}\left(s-\tau_{2}, \tau_{1}, \tau_{2}\right)\right) d s, t \in\left[t_{0}, b\right]
\end{array}\right. \\
& x_{2}\left(t, \tau_{1}, \tau_{2}\right)= \\
& =\left\{\begin{array}{l}
\psi(t), t \in\left[t_{0}-\tau_{2}, t_{0}\right] \\
\psi\left(t_{0}\right)+\int_{t_{0}}^{t} f_{2}\left(s, x_{1}\left(s, \tau_{1}, \tau_{2}\right), x_{2}\left(s, \tau_{1}, \tau_{2}\right), x_{1}\left(s-\tau_{1}, \tau_{1}, \tau_{2}\right), x_{2}\left(s-\tau_{2}, \tau_{1}, \tau_{2}\right)\right) d s, t \in\left[t_{0}, b\right]
\end{array}\right. \tag{3.4}
\end{align*}
Now, let take the operator
A_(f):C[t_(0)-tau_(1),b]xx C[t_(0)-tau_(2),b]rarr C[t_(0)-tau_(1),b]xx C[t_(0)-tau_(2),b]A_{f}: C\left[t_{0}-\tau_{1}, b\right] \times C\left[t_{0}-\tau_{2}, b\right] \rightarrow C\left[t_{0}-\tau_{1}, b\right] \times C\left[t_{0}-\tau_{2}, b\right]
given by the relation
A_(f)(x_(1),x_(2))=(A_(f_(1))(x_(1),x_(2)),A_(f_(2))(x_(1),x_(2)))A_{f}\left(x_{1}, x_{2}\right)=\left(A_{f_{1}}\left(x_{1}, x_{2}\right), A_{f_{2}}\left(x_{1}, x_{2}\right)\right)
where
{:[A_(f_(1))(x_(1),x_(2))(t,tau_(1),tau_(2))={[varphi(t)","t in[t_(0)-tau_(1),t_(0)]],[varphi(t_(0))+int_(t_(0))^(t)f_(1)(s,x_(1)(s,tau_(1),tau_(2)),x_(2)(s,tau_(1),tau_(2)):}],[{:x_(1)(s-tau_(1),tau_(1),tau_(2)),x_(2)(s-tau_(2),tau_(1),tau_(2)))ds","t in[t_(0),b]]:}],[A_(f_(2))(x_(1),x_(2))(t,tau_(1),tau_(2))={[psi(t)","t in[t_(0)-tau_(2),t_(0)]],[psi(t_(0))+int_(t_(0))^(t)f_(2)(s,x_(1)(s,tau_(1),tau_(2)),x_(2)(s,tau_(1),tau_(2)):}],[{:x_(1)(s-tau_(1),tau_(1),tau_(2)),x_(2)(s-tau_(2),tau_(1),tau_(2)))ds","t in[t_(0),b]]:}]:}\begin{aligned}
& A_{f_{1}}\left(x_{1}, x_{2}\right)\left(t, \tau_{1}, \tau_{2}\right)=\left\{\begin{array}{l}
\varphi(t), t \in\left[t_{0}-\tau_{1}, t_{0}\right] \\
\varphi\left(t_{0}\right)+\int_{t_{0}}^{t} f_{1}\left(s, x_{1}\left(s, \tau_{1}, \tau_{2}\right), x_{2}\left(s, \tau_{1}, \tau_{2}\right)\right. \\
\left.x_{1}\left(s-\tau_{1}, \tau_{1}, \tau_{2}\right), x_{2}\left(s-\tau_{2}, \tau_{1}, \tau_{2}\right)\right) d s, t \in\left[t_{0}, b\right]
\end{array}\right. \\
& A_{f_{2}}\left(x_{1}, x_{2}\right)\left(t, \tau_{1}, \tau_{2}\right)=\left\{\begin{array}{l}
\psi(t), t \in\left[t_{0}-\tau_{2}, t_{0}\right] \\
\psi\left(t_{0}\right)+\int_{t_{0}}^{t} f_{2}\left(s, x_{1}\left(s, \tau_{1}, \tau_{2}\right), x_{2}\left(s, \tau_{1}, \tau_{2}\right)\right. \\
\left.x_{1}\left(s-\tau_{1}, \tau_{1}, \tau_{2}\right), x_{2}\left(s-\tau_{2}, \tau_{1}, \tau_{2}\right)\right) d s, t \in\left[t_{0}, b\right]
\end{array}\right.
\end{aligned}
Let X:=C[t_(0)-tau_(1),b]xx C[t_(0)-tau_(2),b]X:=C\left[t_{0}-\tau_{1}, b\right] \times C\left[t_{0}-\tau_{2}, b\right] and ||*||_(C)\|\cdot\|_{C}, the Chebyshev norm on XX. It is clear, from the proof of the Theorem 1 ([4]), that in the conditions (H_(1))-(H_(4))\left(\mathrm{H}_{1}\right)-\left(\mathrm{H}_{4}\right), the operator A_(f)A_{f} is a Picard operator.
Let ( x_(1)^(**),x_(2)^(**)x_{1}^{*}, x_{2}^{*} ) the only fixed point of A_(f)A_{f}.
We consider the subset X_(1)sub XX_{1} \subset X,
We consider the subset X_(1)sub XX_{1} \subset X,
X_(1)={(x_(1),x_(2))in X|(delx_(i))/(del t)in C[t_(0)-tau,b],i=1,2}X_{1}=\left\{\left(x_{1}, x_{2}\right) \in X \left\lvert\, \frac{\partial x_{i}}{\partial t} \in C\left[t_{0}-\tau, b\right]\right., i=1,2\right\}
We remark that (x_(1)^(**),x_(2)^(**))inX_(1),A(X_(1))subX_(1),A:(X_(1),||*||_(C))rarr(X_(1),||*||_(C))\left(x_{1}^{*}, x_{2}^{*}\right) \in X_{1}, A\left(X_{1}\right) \subset X_{1}, A:\left(X_{1},\|\cdot\|_{C}\right) \rightarrow\left(X_{1},\|\cdot\|_{C}\right) is PO.
We suppose that there exists (delx_(i)^(**))/(deltau_(1)),(delx_(i)^(**))/(deltau_(2)),i=1,2\frac{\partial x_{i}^{*}}{\partial \tau_{1}}, \frac{\partial x_{i}^{*}}{\partial \tau_{2}}, i=1,2.
Then, from (3.4) we have that:
We suppose that there exists (delx_(i)^(**))/(deltau_(1)),(delx_(i)^(**))/(deltau_(2)),i=1,2\frac{\partial x_{i}^{*}}{\partial \tau_{1}}, \frac{\partial x_{i}^{*}}{\partial \tau_{2}}, i=1,2.
Then, from (3.4) we have that:
{:[(delx_(i)^(**)(t,tau_(1)))/(deltau_(1))=],[=int_(t_(0))^(t)(delf_(i)(s,x_(1)^(**)(s,tau_(1)),x_(2)^(**)(s,tau_(1)),x_(1)^(**)(s-tau_(1),tau_(1)),x_(2)^(**)(s-tau_(2),tau_(1))))/(delu_(1))*(delx_(1)^(**)(s,tau_(1)))/(deltau_(1))ds+],[+int_(t_(0))^(t)(delf_(i)(s,x_(1)^(**)(s,tau_(1)),x_(2)^(**)(s,tau_(1)),x_(1)^(**)(s-tau_(1),tau_(1)),x_(2)^(**)(s-tau_(2),tau_(1))))/(delu_(2))*(delx_(2)^(**)(s,tau_(1)))/(deltau_(1))ds+],[+int_(t_(0))^(t)(delf_(i)(s,x_(1)^(**)(s,tau_(1)),x_(2)^(**)(s,tau_(1)),x_(1)^(**)(s-tau_(1),tau_(1)),x_(2)^(**)(s-tau_(2),tau_(1))))/(delu_(3))],[+[(delx_(1)^(**)(s-tau_(1),tau_(1)))/(del t)(-1)+(delx_(1)^(**)(s-tau_(1),tau_(1)))/(deltau_(1))]ds+],[+int_(t_(0))^(t)(delf_(i)(s,x_(1)^(**)(s,tau_(1)),x_(2)^(**)(s,tau_(1)),x_(1)^(**)(s-tau_(1),tau_(1)),x_(2)^(**)(s-tau_(2),tau_(1))))/(delu_(4))*(delx_(2)^(**)(s-tau_(2),tau_(1)))/(deltau_(1))ds]:}\begin{aligned}
& \frac{\partial x_{i}^{*}\left(t, \tau_{1}\right)}{\partial \tau_{1}}= \\
& =\int_{t_{0}}^{t} \frac{\partial f_{i}\left(s, x_{1}^{*}\left(s, \tau_{1}\right), x_{2}^{*}\left(s, \tau_{1}\right), x_{1}^{*}\left(s-\tau_{1}, \tau_{1}\right), x_{2}^{*}\left(s-\tau_{2}, \tau_{1}\right)\right)}{\partial u_{1}} \cdot \frac{\partial x_{1}^{*}\left(s, \tau_{1}\right)}{\partial \tau_{1}} d s+ \\
& +\int_{t_{0}}^{t} \frac{\partial f_{i}\left(s, x_{1}^{*}\left(s, \tau_{1}\right), x_{2}^{*}\left(s, \tau_{1}\right), x_{1}^{*}\left(s-\tau_{1}, \tau_{1}\right), x_{2}^{*}\left(s-\tau_{2}, \tau_{1}\right)\right)}{\partial u_{2}} \cdot \frac{\partial x_{2}^{*}\left(s, \tau_{1}\right)}{\partial \tau_{1}} d s+ \\
& +\int_{t_{0}}^{t} \frac{\partial f_{i}\left(s, x_{1}^{*}\left(s, \tau_{1}\right), x_{2}^{*}\left(s, \tau_{1}\right), x_{1}^{*}\left(s-\tau_{1}, \tau_{1}\right), x_{2}^{*}\left(s-\tau_{2}, \tau_{1}\right)\right)}{\partial u_{3}} \\
& +\left[\frac{\partial x_{1}^{*}\left(s-\tau_{1}, \tau_{1}\right)}{\partial t}(-1)+\frac{\partial x_{1}^{*}\left(s-\tau_{1}, \tau_{1}\right)}{\partial \tau_{1}}\right] d s+ \\
& +\int_{t_{0}}^{t} \frac{\partial f_{i}\left(s, x_{1}^{*}\left(s, \tau_{1}\right), x_{2}^{*}\left(s, \tau_{1}\right), x_{1}^{*}\left(s-\tau_{1}, \tau_{1}\right), x_{2}^{*}\left(s-\tau_{2}, \tau_{1}\right)\right)}{\partial u_{4}} \cdot \frac{\partial x_{2}^{*}\left(s-\tau_{2}, \tau_{1}\right)}{\partial \tau_{1}} d s
\end{aligned}
where t in[t_(0),b],i=1,2t \in\left[t_{0}, b\right], i=1,2.
This relation suggests us to consider the following operator
This relation suggests us to consider the following operator
C_(f):X xx X rarr XC_{f}: X \times X \rightarrow X
where
{:[C_(f)(x_(1),x_(2),u,v)(t,tau_(1))=0","" for all "t in[t_(0)-tau_(2),t_(0)]],[C_(f)(x_(1),x_(2),u,v)(t,tau_(1))=0","" for all "t in[t_(0)-tau_(1),t_(0)]]:}\begin{aligned}
& C_{f}\left(x_{1}, x_{2}, u, v\right)\left(t, \tau_{1}\right)=0, \text { for all } t \in\left[t_{0}-\tau_{2}, t_{0}\right] \\
& C_{f}\left(x_{1}, x_{2}, u, v\right)\left(t, \tau_{1}\right)=0, \text { for all } t \in\left[t_{0}-\tau_{1}, t_{0}\right]
\end{aligned}
and
{:[C_(f)(x_(1),x_(2),u,v)(t,tau_(1)):=],[=int_(t_(0))^(t)(delf_(i)(s,x_(1)(s,tau_(1)),x_(2)(s,tau_(1)),x_(1)(s-tau_(1),tau_(1)),x_(2)(s-tau_(2),tau_(1))))/(delu_(1))u(s,tau_(1))ds+],[+int_(t_(0))^(t)(delf_(i)(s,x_(1)(s,tau_(1)),x_(2)(s,tau_(1)),x_(1)(s-tau_(1),tau_(1)),x_(2)(s-tau_(2),tau_(1))))/(delu_(2))v(s,tau_(1))ds+],[+int_(t_(0))^(t)(delf_(i)(s,x_(1)(s,tau_(1)),x_(2)(s,tau_(1)),x_(1)(s-tau_(1),tau_(1)),x_(2)(s-tau_(2),tau_(1))))/(delu_(3))],[*[( bar(u))(s-tau_(1),tau_(1))*(-1)-u(s-tau_(1),tau_(1))]ds+],[+int_(t_(0))^(t)(delf_(i)(s,x_(1)(s,tau_(1)),x_(2)(s,tau_(1)),x_(1)(s-tau_(1),tau_(1)),x_(2)(s-tau_(2),tau_(1))))/(delu_(4))v(s-tau_(2),tau_(1))ds]:}\begin{aligned}
& C_{f}\left(x_{1}, x_{2}, u, v\right)\left(t, \tau_{1}\right):= \\
& =\int_{t_{0}}^{t} \frac{\partial f_{i}\left(s, x_{1}\left(s, \tau_{1}\right), x_{2}\left(s, \tau_{1}\right), x_{1}\left(s-\tau_{1}, \tau_{1}\right), x_{2}\left(s-\tau_{2}, \tau_{1}\right)\right)}{\partial u_{1}} u\left(s, \tau_{1}\right) d s+ \\
& +\int_{t_{0}}^{t} \frac{\partial f_{i}\left(s, x_{1}\left(s, \tau_{1}\right), x_{2}\left(s, \tau_{1}\right), x_{1}\left(s-\tau_{1}, \tau_{1}\right), x_{2}\left(s-\tau_{2}, \tau_{1}\right)\right)}{\partial u_{2}} v\left(s, \tau_{1}\right) d s+ \\
& +\int_{t_{0}}^{t} \frac{\partial f_{i}\left(s, x_{1}\left(s, \tau_{1}\right), x_{2}\left(s, \tau_{1}\right), x_{1}\left(s-\tau_{1}, \tau_{1}\right), x_{2}\left(s-\tau_{2}, \tau_{1}\right)\right)}{\partial u_{3}} \\
& \cdot\left[\bar{u}\left(s-\tau_{1}, \tau_{1}\right) \cdot(-1)-u\left(s-\tau_{1}, \tau_{1}\right)\right] d s+ \\
& +\int_{t_{0}}^{t} \frac{\partial f_{i}\left(s, x_{1}\left(s, \tau_{1}\right), x_{2}\left(s, \tau_{1}\right), x_{1}\left(s-\tau_{1}, \tau_{1}\right), x_{2}\left(s-\tau_{2}, \tau_{1}\right)\right)}{\partial u_{4}} v\left(s-\tau_{2}, \tau_{1}\right) d s
\end{aligned}
for all t in[t_(0),b]t \in\left[t_{0}, b\right].
We denoted here
We denoted here
{:[u(t)=(delx_(1)(t))/(deltau_(1))","v(t)=(delx_(2)(t))/(deltau_(1))"," bar(u)(t-tau_(1))=(delx_(1)(t-tau_(1)))/(del t)","],[u(t-tau_(1))=(delx_(1)(t-tau_(1)))/(deltau_(1))","v(t-tau_(2))=(delx_(2)(t-tau_(2)))/(deltau_(1)).]:}\begin{aligned}
u(t) & =\frac{\partial x_{1}(t)}{\partial \tau_{1}}, v(t)=\frac{\partial x_{2}(t)}{\partial \tau_{1}}, \bar{u}\left(t-\tau_{1}\right)=\frac{\partial x_{1}\left(t-\tau_{1}\right)}{\partial t}, \\
u\left(t-\tau_{1}\right) & =\frac{\partial x_{1}\left(t-\tau_{1}\right)}{\partial \tau_{1}}, v\left(t-\tau_{2}\right)=\frac{\partial x_{2}\left(t-\tau_{2}\right)}{\partial \tau_{1}} .
\end{aligned}
In this way we have the triangular operator
{:[D:X xx X rarr X xx X],[(x_(1),x_(2),u,v) rarr(A_(f)(x_(1),x_(2)),C_(f)(x_(1),x_(2),u,v))]:}\begin{aligned}
D & : X \times X \rightarrow X \times X \\
\left(x_{1}, x_{2}, u, v\right) & \rightarrow\left(A_{f}\left(x_{1}, x_{2}\right), C_{f}\left(x_{1}, x_{2}, u, v\right)\right)
\end{aligned}
where A_(f)A_{f} is a Picard operator and C_(f)(x_(1),x_(2),*,*):X rarr XC_{f}\left(x_{1}, x_{2}, \cdot, \cdot\right): X \rightarrow X is an LL-contraction, with L=(4L_(f))/(rho)L=\frac{4 L_{f}}{\rho}, where rho\rho is the Bielecki constant we use in [4].
From the fibre contraction theorem we have that the operator DD is Picard operator and F_(D)=(x_(1)^(**),x_(2)^(**),u^(**),v^(**))F_{D}=\left(x_{1}^{*}, x_{2}^{*}, u^{*}, v^{*}\right).
Let (x_(1)^(**),x_(2)^(**),u^(**),v^(**))\left(x_{1}^{*}, x_{2}^{*}, u^{*}, v^{*}\right) the only fixed point of the operator DD. Then the sequences
{:[(x_(1,n+1),x_(2,n+1)):=A(x_(1,n),x_(2,n))","n inN","],[(u_(n+1),v_(n+1)):=C(x_(1,n),x_(2,n),u_(n),v_(n))","n inN","]:}\begin{aligned}
& \left(x_{1, n+1}, x_{2, n+1}\right):=A\left(x_{1, n}, x_{2, n}\right), n \in \mathbb{N}, \\
& \left(u_{n+1}, v_{n+1}\right):=C\left(x_{1, n}, x_{2, n}, u_{n}, v_{n}\right), n \in \mathbb{N},
\end{aligned}
converge uniformly (with respect to t in Xt \in X ) to (x_(1)^(**),x_(2)^(**),u^(**),v^(**))inF_(D)\left(x_{1}^{*}, x_{2}^{*}, u^{*}, v^{*}\right) \in F_{D}, for all x_(1,0),x_(2,0),u_(0),v_(0)in Xx_{1,0}, x_{2,0}, u_{0}, v_{0} \in X.
If we take
{:[x_(1,0)=0","x_(2,0)=0],[u_(0)=(delx_(1,0))/(deltau_(1))=0","v_(0)=(delx_(2,0))/(deltau_(1))=0","]:}\begin{aligned}
x_{1,0} & =0, x_{2,0}=0 \\
u_{0} & =\frac{\partial x_{1,0}}{\partial \tau_{1}}=0, v_{0}=\frac{\partial x_{2,0}}{\partial \tau_{1}}=0,
\end{aligned}
then
{:[u_(1)=(delx_(1,1))/(deltau_(1))],[v_(1)=(delx_(2,1))/(deltau_(1))]:}\begin{aligned}
u_{1} & =\frac{\partial x_{1,1}}{\partial \tau_{1}} \\
v_{1} & =\frac{\partial x_{2,1}}{\partial \tau_{1}}
\end{aligned}
By induction, we obtain that
{:[u_(n)=(delx_(1,n))/(deltau_(1))","AA n inN","],[v_(n)=(delx_(2,n))/(deltau_(1))","AA n inN.]:}\begin{aligned}
u_{n} & =\frac{\partial x_{1, n}}{\partial \tau_{1}}, \forall n \in \mathbb{N}, \\
v_{n} & =\frac{\partial x_{2, n}}{\partial \tau_{1}}, \forall n \in \mathbb{N} .
\end{aligned}
So
{:[x_(1,n)rarr"" unif ""x_(1)^(**)" as "n rarr oo","],[x_(2,n)rarr"" unif ""x_(2)^(**)" as "n rarr oo","],[(delx_(1,n))/(deltau_(1))rarr"" unif ""u^(**)" as "n rarr oo","],[(delx_(2,n))/(deltau_(1))rarr"" unif ""v^(**)" as "n rarr oo.]:}\begin{gathered}
x_{1, n} \xrightarrow{\text { unif }} x_{1}^{*} \text { as } n \rightarrow \infty, \\
x_{2, n} \xrightarrow{\text { unif }} x_{2}^{*} \text { as } n \rightarrow \infty, \\
\frac{\partial x_{1, n}}{\partial \tau_{1}} \xrightarrow{\text { unif }} u^{*} \text { as } n \rightarrow \infty, \\
\frac{\partial x_{2, n}}{\partial \tau_{1}} \xrightarrow{\text { unif }} v^{*} \text { as } n \rightarrow \infty .
\end{gathered}
From the above consideration we have that there exist (delx_(i)^(**))/(deltau_(1)),i=1,2\frac{\partial x_{i}^{*}}{\partial \tau_{1}}, i=1,2 and
(delx_(1)^(**))/(deltau_(1))=u^(**),quad(delx_(2)^(**))/(deltau_(1))=v^(**)\frac{\partial x_{1}^{*}}{\partial \tau_{1}}=u^{*}, \quad \frac{\partial x_{2}^{*}}{\partial \tau_{1}}=v^{*}
Analogously we can prove the differentiability with respect to tau_(2)\tau_{2}.
References
[1] Berinde V., Generalized Contractions and Applications (in Romanian), Ph. D. Thesis, Univ. "BabeşBolyai" Cluj-Napoca, 1993.
[2] Buică A., Existence and continuous dependence of solutions of some functional-differential equations, Seminar on Fixed Point Theory, Cluj-Napoca, 1995, 1-14.
[3] Mureşan V., Functional-Integral Equations, Editura Mediamira, Cluj-Napoca, 2003.
[4] Otrocol D., Data dependence for the solution of a Lotka-Volterra system with two delays, Mathematica, Tome 48 (71), No. 1 (2006), 61-68.
[5] Otrocol D., Lotka-Volterra system with two delays via weakly Picard operators, Nonlinear Analysis Forum, 10 (2) (2005), 193-199.
[6] Rus I. A., Principles and applications of the fixed point theory (in Romanian), Editura Dacia, ClujNapoca, 1979.
[7] Rus I. A., Generalized contractions, Seminar on Fixed Point Theory, "Babeş-Bolyai" University, 1983, pp. 1-130.
[8] Rus I. A., Weakly Picard mappings, Comment. Math. Univ. Caroline, 34 (1993), 769-773.
[9] Rus I. A., Functional-differential equations of mixed type, via weakly Picard operators, Seminar of Fixed Point Theory, Cluj-Napoca, Vol. 3, 2002, 335-346.
[10] Rus I. A., Generalized Contractions and Applications, Cluj University Press, 2001.
[11] Rus I. A., Weakly Picard operators and applications, Seminar on Fixed Point Theory, Cluj-Napoca, Vol. 2 (2001), 41-58.
[12] Rus I. A. and Egri E., Boundary value problems for iterative functional-differential equations, Studia Univ. "Babeş-Bolyai", Matematica, Vol. LI, No. 2, 2006, pp. 109-126.
[13] Saito Y., Hara T. and Ma W., Necessary and sufficient conditions for permanence and global stability of a Lotka-Volterra system with two delays, J. Math. Anal. Appl., 236 (1999), 534-556.
[14] Şerban M. A., Fiber varphi\varphi-contractions, Studia Univ. "Babeş-Bolyai", Mathematica, 44 (1999), No. 3, 99-108.
"Tiberiu Popoviciu" Institute of Numerical Analysis
P.O. Box 68-1
[2] Buică A., Existence and continuous dependence of solutions of some functional-differential equations, Seminar on Fixed Point Theory, Cluj-Napoca, 1995, 1-14.
[3] Mureşan V., Functional-Integral Equations, Editura Mediamira, Cluj-Napoca, 2003.
[4] Otrocol D., Data dependence for the solution of a Lotka-Volterra system with two delays, Mathematica, Tome 48 (71), No. 1 (2006), 61-68.
[5] Otrocol D., Lotka-Volterra system with two delays via weakly Picard operators, Nonlinear Analysis Forum, 10 (2) (2005), 193-199.
[6] Rus I. A., Principles and applications of the fixed point theory (in Romanian), Editura Dacia, ClujNapoca, 1979.
[7] Rus I. A., Generalized contractions, Seminar on Fixed Point Theory, "Babeş-Bolyai" University, 1983, pp. 1-130.
[8] Rus I. A., Weakly Picard mappings, Comment. Math. Univ. Caroline, 34 (1993), 769-773.
[9] Rus I. A., Functional-differential equations of mixed type, via weakly Picard operators, Seminar of Fixed Point Theory, Cluj-Napoca, Vol. 3, 2002, 335-346.
[10] Rus I. A., Generalized Contractions and Applications, Cluj University Press, 2001.
[11] Rus I. A., Weakly Picard operators and applications, Seminar on Fixed Point Theory, Cluj-Napoca, Vol. 2 (2001), 41-58.
[12] Rus I. A. and Egri E., Boundary value problems for iterative functional-differential equations, Studia Univ. "Babeş-Bolyai", Matematica, Vol. LI, No. 2, 2006, pp. 109-126.
[13] Saito Y., Hara T. and Ma W., Necessary and sufficient conditions for permanence and global stability of a Lotka-Volterra system with two delays, J. Math. Anal. Appl., 236 (1999), 534-556.
[14] Şerban M. A., Fiber varphi\varphi-contractions, Studia Univ. "Babeş-Bolyai", Mathematica, 44 (1999), No. 3, 99-108.
"Tiberiu Popoviciu" Institute of Numerical Analysis
P.O. Box 68-1
400110 Cluj-Napoca, Romania
E-mail address: dotrocol@ictp.acad.ro
E-mail address: diana.otrocol@gmail.com
E-mail address: dotrocol@ictp.acad.ro
E-mail address: diana.otrocol@gmail.com
- Received: 20.09.2006. In revised form: 20.02.2007.
2000 Mathematics Subject Classification. 34L05, 47H10.
Key words and phrases. Lotka-Volterra system, weakly Picard operator, delay, differentiability.- This work has been supported by MEdc under Grant 2-CEx06-11-96.