ABSTRACT. We study the differentiability with respect to delays using the weakly Picard operators technique.

Consider the following Lotka-Volterra differential system with delays

Suppose that we have satisfied the following conditions:
$\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=[t_0,\tau ]$$\left({\mathrm{H}}_1\right)t_0<b,\tau ,{\tau }_1,{\tau }_2>0,{\tau }_1<{\tau }_2<\tau ,{\tau }_1,{\tau }_2\in J,J=\left[t_0,\tau \right]$(H_(1))t_(0) < b,tau,tau_(1),tau_(2) > 0,tau_(1) < tau_(2) < tau,tau_(1),tau_(2)in J,J=[t_(0),tau]\left(\mathrm{H}_{1}\right) t_{0}<b, \tau, \tau_{1}, \tau_{2}>0, \tau_{1}<\tau_{2}<\tau, \tau_{1}, \tau_{2} \in J, J=\left[t_{0}, \tau\right]$\left({\mathrm{H}}_1\right)t_0<b,\tau ,{\tau }_1,{\tau }_2>0,{\tau }_1<{\tau }_2<\tau ,{\tau }_1,{\tau }_2\in J,J=[t_0,\tau ]$ a compact interval;
$\left({\mathrm{H}}_2\right)f_i\in C^1([t_0,b]\times {\mathbb{R}}^4,\mathbb{R}),i=1,2$$\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$(H_(2))f_(i)inC^(1)([t_(0),b]xxR^(4),R),i=1,2\left(\mathrm{H}_{2}\right) f_{i} \in C^{1}\left(\left[t_{0}, b\right] \times \mathbb{R}^{4}, \mathbb{R}\right), i=1,2$\left({\mathrm{H}}_2\right)f_i\in C^1([t_0,b]\times {\mathbb{R}}^4,\mathbb{R}),i=1,2$;
$\left({\mathrm{H}}_3\right)$$\left({\mathrm{H}}_3\right)$(H_(3))\left(\mathrm{H}_{3}\right)$\left({\mathrm{H}}_3\right)$ there exists $L_f>0$$L_f>0$L_(f) > 0L_{f}>0$L_f>0$ such that

for all $t\in [t_0,b],u_j\in R,j=\overline{1,4},i=1,2$$t\in \left[t_0,b\right],u_j\in R,j=\overline{1,4},i=1,2$t in[t_(0),b],u_(j)in R,j= bar(1,4),i=1,2t \in\left[t_{0}, b\right], u_{j} \in R, j=\overline{1,4}, i=1,2$t\in [t_0,b],u_j\in R,j=\overline{1,4},i=1,2$;
$\left({\mathrm{H}}_4\right)\phi \in C([t_0-\tau ,t_0],\mathbb{R}),\psi \in C([t_0-\tau ,t_0],\mathbb{R});$$\left({\mathrm{H}}_4\right)\phi \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);$(H_(4))varphi in C([t_(0)-tau,t_(0)],R),psi in C([t_(0)-tau,t_(0)],R);\left(\mathrm{H}_{4}\right) \varphi \in C\left(\left[t_{0}-\tau, t_{0}\right], \mathbb{R}\right), \psi \in C\left(\left[t_{0}-\tau, t_{0}\right], \mathbb{R}\right) ;$\left({\mathrm{H}}_4\right)\phi \in C([t_0-\tau ,t_0],\mathbb{R}),\psi \in C([t_0-\tau ,t_0],\mathbb{R});$
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)$(x_(1)(t),x_(2)(t))\left(x_{1}(t), x_{2}(t)\right)$(x_1(t),x_2(t))$.

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,d$$X,d$X,dX, d$X,d$ ) be a metric space and $A:X\to X$$A:X\to X$A:X rarr XA: X \rightarrow X$A:X\to X$ an operator. We shall use the following notations:
$F_A:=\{x\in X\mid A(x)=x\}$$F_A:=\{x\in X\mid A(x)=x\}$F_(A):={x in X∣A(x)=x}F_{A}:=\{x \in X \mid A(x)=x\}$F_A:=\{x\in X\mid A(x)=x\}$ - the fixed point set of $A$$A$AA$A$;
$I(A):=\{Y\in P(X)\mid A(Y)\subset Y\}$$I(A):=\{Y\in P(X)\mid A(Y)\subset Y\}$I(A):={Y in P(X)∣A(Y)sub Y}I(A):=\{Y \in P(X) \mid A(Y) \subset Y\}$I(A):=\{Y\in P(X)\mid A(Y)\subset Y\}$ - the family of the nonempty invariant subset of $A$$A$AA$A$;
$A^{n+1}:=A\circ A^n,A^0=1_X,A^1=A,n\in \mathbb{N}$$A^{n+1}:=A\circ A^n,A^0=1_X,A^1=A,n\in \mathbb{N}$A^(n+1):=A@A^(n),A^(0)=1_(X),A^(1)=A,n inNA^{n+1}:=A \circ A^{n}, A^{0}=1_{X}, A^{1}=A, n \in \mathbb{N}$A^{n+1}:=A\circ A^n,A^0=1_X,A^1=A,n\in \mathbb{N}$ - the iterant operators of $A$$A$AA$A$, where $1_X$$1_X$1_(X)1_{X}$1_X$ is the identity operator;
$P(X):=\{Y\subset X\mid Y\ne \mathrm{\varnothing }\}$$P(X):=\{Y\subset X\mid Y\ne \mathrm{\varnothing }\}$P(X):={Y sub X∣Y!=O/}P(X):=\{Y \subset X \mid Y \neq \emptyset\}$P(X):=\{Y\subset X\mid Y\ne \mathrm{\varnothing }\}$ - the set of the parts of $X$$X$XX$X$.

Definition 2.1. Let ( $X,d$$X,d$X,dX, d$X,d$ ) be a metric space. An operator $A:X\to X$$A:X\to X$A:X rarr XA: X \rightarrow X$A:X\to X$ is a Picard operator (PO) if there exists $x^{\ast }\in X$$x^{\ast }\in X$x^(**)in Xx^{*} \in X$x^{\ast }\in X$ such that:
(i) $F_A=\left\{x^{\ast }\right\}$$F_A=\left\{x^{\ast }\right\}$F_(A)={x^(**)}F_{A}=\left\{x^{*}\right\}$F_A=\left\{x^{\ast }\right\}$;
(ii) the sequence ${\left(A^n\left(x_0\right)\right)}_{n\in \mathbb{N}}$${\left(A^n\left(x_0\right)\right)}_{n\in \mathbb{N}}$(A^(n)(x_(0)))_(n inN)\left(A^{n}\left(x_{0}\right)\right)_{n \in \mathbb{N}}${\left(A^n\left(x_0\right)\right)}_{n\in \mathbb{N}}$ converges to $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$ for all $x_0\in X$$x_0\in X$x_(0)in Xx_{0} \in X$x_0\in X$.

Definition 2.2. Let ( $X,d$$X,d$X,dX, d$X,d$ ) be a metric space. An operator $A:X\to X$$A:X\to X$A:X rarr XA: X \rightarrow X$A:X\to X$ is a weakly Picard operator (WPO) if the sequence ${(A^n(x))}_{n\in \mathbb{N}}$${\left(A^n(x)\right)}_{n\in \mathbb{N}}$(A^(n)(x))_(n inN)\left(A^{n}(x)\right)_{n \in \mathbb{N}}${(A^n(x))}_{n\in \mathbb{N}}$ converges for all $x\in X$$x\in X$x in Xx \in X$x\in X$, and its limit ( which may depend on $x$$x$xx$x$ ) is a fixed point of $A$$A$AA$A$.
Theorem 2.1. Let ( $X,d$$X,d$X,dX, d$X,d$ ) be a metric space and $A:X\to X$$A:X\to X$A:X rarr XA: X \rightarrow X$A:X\to X$ an operator. The operator $A$$A$AA$A$ is WPO ( $c$$c$cc$c$-WPO) if and only if there exists a partition of $X$$X$XX$X$,

such that:
(a) $X_{\lambda }\in I(A),\lambda \in \mathrm{\Lambda },I(A)$$X_{\lambda }\in I(A),\lambda \in \mathrm{\Lambda },I(A)$X_(lambda)in I(A),lambda in Lambda,I(A)X_{\lambda} \in I(A), \lambda \in \Lambda, I(A)$X_{\lambda }\in I(A),\lambda \in \mathrm{\Lambda },I(A)$-the family of nonempty invariant subsets of $A$$A$AA$A$;
(b) ${A|}_{X_{\lambda }}:X_{\lambda }\to X_{\lambda }$${\left.A\right|}_{X_{\lambda }}:X_{\lambda }\to X_{\lambda }$A|_(X_(lambda)):X_(lambda)rarrX_(lambda)\left.A\right|_{X_{\lambda}}: X_{\lambda} \rightarrow X_{\lambda}${A|}_{X_{\lambda }}:X_{\lambda }\to X_{\lambda }$ is a Picard (c-Picard) operator for all $\lambda \in \mathrm{\Lambda }$$\lambda \in \mathrm{\Lambda }$lambda in Lambda\lambda \in \Lambda$\lambda \in \mathrm{\Lambda }$.

Theorem 2.2. (Fibre contraction principle). Let ( $X,d$$X,d$X,dX, d$X,d$ ) and ( $Y,\rho$$Y,\rho$Y,rhoY, \rho$Y,\rho$ ) be two metric spaces and $A:X\times X\to X\times X,A=(B,C),(B:X\to X,C:X\times Y\to Y)$$A:X\times X\to X\times X,A=(B,C),(B:X\to X,C:X\times Y\to Y)$A:X xx X rarr X xx X,A=(B,C),(B:X rarr X,C:X xx Y rarr Y)A: X \times X \rightarrow X \times X, A=(B, C),(B: X \rightarrow X, C: X \times Y \rightarrow Y)$A:X\times X\to X\times X,A=(B,C),(B:X\to X,C:X\times Y\to Y)$ a triangular operator. We suppose that
(i) $(Y,\rho )$$(Y,\rho )$(Y,rho)(Y, \rho)$(Y,\rho )$ is a complete metric space;
(ii) the operator $B$$B$BB$B$ is $PO$$PO$POP O$PO$;
(iii) there exists $L\in [0,1)$$L\in [0,1)$L in[0,1)L \in[0,1)$L\in [0,1)$ such that $C(x,\cdot ):Y\to Y$$C(x,\cdot ):Y\to Y$C(x,*):Y rarr YC(x, \cdot): Y \rightarrow Y$C(x,\cdot ):Y\to Y$ is a $L$$L$LL$L$-contraction, for all $x\in X$$x\in X$x in Xx \in X$x\in X$;
(iv) if $(x^{\ast },y^{\ast })\in F_A$$\left(x^{\ast },y^{\ast }\right)\in F_A$(x^(**),y^(**))inF_(A)\left(x^{*}, y^{*}\right) \in F_{A}$(x^{\ast },y^{\ast })\in F_A$, then $C(\cdot ,y^{\ast })$$C\left(\cdot ,y^{\ast }\right)$C(*,y^(**))C\left(\cdot, y^{*}\right)$C(\cdot ,y^{\ast })$ is continuous in $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$.

Then the operator $A$$A$AA$A$ is $PO$$PO$POP O$PO$.

Now we prove that

For this we consider the system

where $t\in [t_0,b],x_1\in C[t_0-{\tau }_1,b]\cap C^1[t_0,b],x_2\in C[t_0-{\tau }_2,b]\cap C^1[t_0,b]$$t\in \left[t_0,b\right],x_1\in C\left[t_0-{\tau }_1,b\right]\cap C^1\left[t_0,b\right],x_2\in C\left[t_0-{\tau }_2,b\right]\cap C^1\left[t_0,b\right]$t in[t_(0),b],x_(1)in C[t_(0)-tau_(1),b]nnC^(1)[t_(0),b],x_(2)in C[t_(0)-tau_(2),b]nnC^(1)[t_(0),b]t \in\left[t_{0}, b\right], x_{1} \in C\left[t_{0}-\tau_{1}, b\right] \cap C^{1}\left[t_{0}, b\right], x_{2} \in C\left[t_{0}-\tau_{2}, b\right] \cap C^{1}\left[t_{0}, b\right]$t\in [t_0,b],x_1\in C[t_0-{\tau }_1,b]\cap C^1[t_0,b],x_2\in C[t_0-{\tau }_2,b]\cap C^1[t_0,b]$.
From the above considerations, we can formulate the following theorem
Theorem 3.3. Consider the problem (3.3)-(1.2), in the conditions $\left(H_1\right)-\left(H_4\right)$$\left(H_1\right)-\left(H_4\right)$(H_(1))-(H_(4))\left(H_{1}\right)-\left(H_{4}\right)$\left(H_1\right)-\left(H_4\right)$. Then the problem (3.3)-(1.2) has a unique solution $(x_1^{\ast },x_2^{\ast }),x_1^{\ast }\in C[t_0-{\tau }_1,b]\cap C^1[t_0,b],x_2^{\ast }\in C[t_0-{\tau }_2,b]\cap C^1[t_0,b]$$\left(x_1^{\ast },x_2^{\ast }\right),x_1^{\ast }\in C\left[t_0-{\tau }_1,b\right]\cap C^1\left[t_0,b\right],x_2^{\ast }\in C\left[t_0-{\tau }_2,b\right]\cap C^1\left[t_0,b\right]$(x_(1)^(**),x_(2)^(**)),x_(1)^(**)in C[t_(0)-tau_(1),b]nnC^(1)[t_(0),b],x_(2)^(**)in C[t_(0)-tau_(2),b]nnC^(1)[t_(0),b]\left(x_{1}^{*}, x_{2}^{*}\right), x_{1}^{*} \in C\left[t_{0}-\tau_{1}, b\right] \cap C^{1}\left[t_{0}, b\right], x_{2}^{*} \in C\left[t_{0}-\tau_{2}, b\right] \cap C^{1}\left[t_{0}, b\right]$(x_1^{\ast },x_2^{\ast }),x_1^{\ast }\in C[t_0-{\tau }_1,b]\cap C^1[t_0,b],x_2^{\ast }\in C[t_0-{\tau }_2,b]\cap C^1[t_0,b]$ and the solution is differentiable on ${\tau }_1$${\tau }_1$tau_(1)\tau_{1}${\tau }_1$ and ${\tau }_2$${\tau }_2$tau_(2)\tau_{2}${\tau }_2$.
Proof. In what follows we consider the following integral equations:

Now, let take the operator

given by the relation

where

Let $X:=C[t_0-{\tau }_1,b]\times C[t_0-{\tau }_2,b]$$X:=C\left[t_0-{\tau }_1,b\right]\times C\left[t_0-{\tau }_2,b\right]$X:=C[t_(0)-tau_(1),b]xx C[t_(0)-tau_(2),b]X:=C\left[t_{0}-\tau_{1}, b\right] \times C\left[t_{0}-\tau_{2}, b\right]$X:=C[t_0-{\tau }_1,b]\times C[t_0-{\tau }_2,b]$ and $\Vert \cdot {\Vert }_C$$\Vert \cdot {\Vert }_C$||*||_(C)\|\cdot\|_{C}$\Vert \cdot {\Vert }_C$, the Chebyshev norm on $X$$X$XX$X$. It is clear, from the proof of the Theorem 1 ([4]), that in the conditions $\left({\mathrm{H}}_1\right)-\left({\mathrm{H}}_4\right)$$\left({\mathrm{H}}_1\right)-\left({\mathrm{H}}_4\right)$(H_(1))-(H_(4))\left(\mathrm{H}_{1}\right)-\left(\mathrm{H}_{4}\right)$\left({\mathrm{H}}_1\right)-\left({\mathrm{H}}_4\right)$, the operator $A_f$$A_f$A_(f)A_{f}$A_f$ is a Picard operator.

Let ( $x_1^{\ast },x_2^{\ast }$$x_1^{\ast },x_2^{\ast }$x_(1)^(**),x_(2)^(**)x_{1}^{*}, x_{2}^{*}$x_1^{\ast },x_2^{\ast }$ ) the only fixed point of $A_f$$A_f$A_(f)A_{f}$A_f$.
We consider the subset $X_1\subset X$$X_1\subset X$X_(1)sub XX_{1} \subset X$X_1\subset X$,

We remark that $(x_1^{\ast },x_2^{\ast })\in X_1,A\left(X_1\right)\subset X_1,A:(X_1,\Vert \cdot {\Vert }_C)\to (X_1,\Vert \cdot {\Vert }_C)$$\left(x_1^{\ast },x_2^{\ast }\right)\in X_1,A\left(X_1\right)\subset X_1,A:\left(X_1,\Vert \cdot {\Vert }_C\right)\to \left(X_1,\Vert \cdot {\Vert }_C\right)$(x_(1)^(**),x_(2)^(**))inX_(1),A(X_(1))subX_(1),A:(X_(1),||*||_(C))rarr(X_(1),||*||_(C))\left(x_{1}^{*}, x_{2}^{*}\right) \in X_{1}, A\left(X_{1}\right) \subset X_{1}, A:\left(X_{1},\|\cdot\|_{C}\right) \rightarrow\left(X_{1},\|\cdot\|_{C}\right)$(x_1^{\ast },x_2^{\ast })\in X_1,A\left(X_1\right)\subset X_1,A:(X_1,\Vert \cdot {\Vert }_C)\to (X_1,\Vert \cdot {\Vert }_C)$ is PO.
We suppose that there exists $\frac{\partial x_i^{\ast }}{\partial {\tau }_1},\frac{\partial x_i^{\ast }}{\partial {\tau }_2},i=1,2$$\frac{\partial x_i^{\ast }}{\partial {\tau }_1},\frac{\partial x_i^{\ast }}{\partial {\tau }_2},i=1,2$(delx_(i)^(**))/(deltau_(1)),(delx_(i)^(**))/(deltau_(2)),i=1,2\frac{\partial x_{i}^{*}}{\partial \tau_{1}}, \frac{\partial x_{i}^{*}}{\partial \tau_{2}}, i=1,2$\frac{\partial x_i^{\ast }}{\partial {\tau }_1},\frac{\partial x_i^{\ast }}{\partial {\tau }_2},i=1,2$.
Then, from (3.4) we have that: