Let there be given real numbers $a<c<b$ and two functions $f_{1},f_{2}:[a,b]\times\mathbb{R}^{4}\rightarrow\mathbb{R}$. We consider the boundary value problem for the system of functional-differential equations

with polylocal conditions

Boundary value problems that arise from different areas of applied mathematics and physics have received a lot of attention in the literature in the last decades (see for example [2], [3], [5], [6], [7] and references therein). In [9], D.V. Ionescu study the problem (1.1)-(1.2) using the successive approximation method. Several results of D.V. Ionescu have been cited and extended by: O Aramă [1], Gh. Coman [4], V. Ilea and D. Otrocol [8], G. Micula [10], A. Petrusel and I.A. Rus [12], etc. Our approach is based on the Perov fixed point theorem [11] and weakly Picard operator theory [15]-[17] in the following conditions
$\left(\mathrm{C}_{1}\right)f_{1},f_{2}\in C^{1}\left([a,b]\times\mathbb{R}^{4},\mathbb{R}\right);$
$\left(\mathrm{C}_{2}\right)$ there exists $L_{i}>0$ such that

for all $t\in[a,b],u_{j},v_{j}\in\mathbb{R},i=1,2,j=\overline{1,4}$.

In this section we study the existence and uniqueness theorem for the problem

where $\chi_{i}:[a,b]\rightarrow\mathbb{R},i=1,2$.
Theorem 2.1. We suppose that $\chi_{i}\in C([a,b],\mathbb{R}),i=1,2$. Then the problem (2.1)-(2.2) has a unique solution $\stackrel{{\scriptstyle*}}=\left(\stackrel{{\scriptstyle*}}{{x_{1}}},\stackrel{{\scriptstyle*}}{{x_{2}}}\right)\in C^{1}\left([a,b],\mathbb{R}^{2}\right)$ and

where $\mathbf{G}(t,s)$ is the Green function of the problem

$\mathbf{G}(t,s)$ has the following form

where for $t\in(a,c)$

and for $t\in(c,b)$

From Theorem 2.1 it follows that the problem (1.1)-(1.2) is equivalent with the system

In order to study the system (2.3), we shall use the weakly Picard operator technique. In the next section we present some notions and results from this theory.

In this section, we introduce notation, definitions, and preliminary facts which are used throughout this paper (see [15]-[17]). 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 $A$;
$A^{n+1}:=A\circ A^{n},A^{0}=1_{X},A^{1}=A,n\in\mathbb{N}$.
Definition 3.1. Let ( $X,d$ ) be a metric space. An operator $A:X\rightarrow X$ is a Picard operator $(\mathrm{PO})$ if there exists $x^{*}\in X$ such that:
(i) $F_{A}=\left\{x^{*}\right\}$;
(ii) the sequence $\left(A^{n}\left(x_{0}\right)\right)_{n\in\mathbb{N}}$ converges to $x^{*}$ for all $x_{0}\in X$.

Definition 3.2. Let ( $X,d$ ) be a metric space. An operator $A:X\rightarrow X$ is a weakly Picard operator (WPO) if the sequence $\left(A^{n}(x)\right)_{n\in\mathbb{N}}$ converges for all $x\in X$, and its limit (which may depend on $x$ ) is a fixed point of $A$.

Throughout this paper we denote by $M_{mm}\left(\mathbb{R}_{+}\right)$the set of all $m\times m$ matrices with positive elements and by $I$ the identity $m\times m$ matrix. A square matrix $Q$ with nonnegative elements is said to be convergent to zero if $Q^{k}\rightarrow 0$ as $k\rightarrow\infty$. It is known that the property of being convergent to zero is equivalent to each of the following three conditions (see [13], [14]):
(a) $I-Q$ is nonsingular and $(I-Q)^{-1}=I+Q+Q^{2}+\cdots$ (where $I$ stands for the unit matrix of the same order as $Q$ );
(b) the eigenvalues of $Q$ are located inside the unit disc of the complex plane;
(c) $I-Q$ is nonsingular and $(I-Q)^{-1}$ has nonnegative elements.

We finish this section by recalling the following fundamental result
Theorem 3.3. (Perov’s fixed point theorem). Let $(X,d)$ with $d(x,y)\in\mathbb{R}^{m}$, be a complete generalized metric space and $A:X\rightarrow X$ an operator. We suppose that there exists a matrix $Q\in M_{mm}\left(\mathbb{R}_{+}\right)$, such that
(i) $d(A(x),A(y))\leq Qd(x,y)$, for all $x,y\in X$;
(ii) $Q^{n}\rightarrow 0$ as $n\rightarrow\infty$.

Then
(a) $F_{A}=\left\{x^{*}\right\}$,
(b) $A^{n}(x)=x^{*}$ as $n\rightarrow\infty$ and

In this section we use Perov’s fixed point theorem to obtain existence and uniqueness theorem for the solution of the problem (1.1)-(1.2).

We consider the Banach space $X=\left(C^{1}\left([a,b],\mathbb{R}^{2}\right),\|\cdot\|_{C^{1}}\right)$ where $\|\cdot\|_{C^{1}}$, is the Chebyshev norm defined by

and the operator $A:X\rightarrow X$ defined by

We consider the problem (1.1)-(1.2) in the conditions $\left(\mathrm{C}_{1}\right)$ and $\left(\mathrm{C}_{2}\right)$. The problem (1.1)-(1.2) is equivalent with the fixed point equation

For $t\in[a,b]$ we have

At the same time we have
$\left\|A_{2}\left(x_{1},x_{2}\right)(t)-A_{2}\left(y_{1},y_{2}\right)(t)\right\|_{\infty}\leq\frac{3}{2}(b-a)^{2}\left(\begin{array}[]{ll}1&1\\ 1&1\end{array}\right)L_{2}\left\|\left(x_{1},x_{2}\right)-\left(y_{1},y_{2}\right)\right\|_{C^{1}}$.
For $t\in[a,b]$ we have

Analogous we have

Then

with $Q:=\left(\frac{3}{2}(b-a)^{2}\left(\begin{array}[]{ll}L_{1}&L_{1}\\ L_{2}&L_{2}\end{array}\right)+\frac{3}{2}(b-a)\left(\begin{array}[]{cc}L_{1}&L_{1}\\ 0&L_{2}\end{array}\right)\right)$.
So, we have the following existence and uniqueness theorem
Theorem 4.1. We suppose that:
(i) the conditions ( $C_{1}$ ) and ( $C_{2}$ ) are satisfied;
(ii) $Q^{n}\rightarrow 0$ as $n\rightarrow\infty$.

Then
(a) the problem (1.1)-(1.2) has a unique solution

(b) for all $\left(x_{1}^{0},x_{2}^{0}\right)\in C^{1}\left([a,b],\mathbb{R}^{2}\right)$, the sequence $\left(x_{1}^{n},x_{2}^{n}\right)_{n\in\mathbb{N}}$ defined by

converges uniformly to $\left(\stackrel{{\scriptstyle*}}_{1},\stackrel{{\scriptstyle*}}_{2}\right)$, for all $t\in[a,b]$, and

Consider the problem (1.1)-(1.2) with the dates $f=\left(f_{1},f_{2}\right),g=\left(g_{1},g_{2}\right)$ and suppose that the conditions from Theorem 4.1 are satisfied.

Let $f,g\in C^{2}\left([a,b]\times\mathbb{R}^{4},\mathbb{R}^{2}\right)$. For simplicity we denote

and

where $\max$ is taken w.r.t. the ordered relation of $M_{22}(\mathbb{R})$.
Theorem 5.1. Let $f=\left(f_{1},f_{2}\right)$ and $g=\left(g_{1},g_{2}\right)$ satisfy the condition $\left(C_{1}\right)$. Furthermore, we suppose that there exist $\eta:=\left(\eta_{1},\eta_{2}\right)\in\mathbb{R}_{+}^{2}$, such that

Then

where $\stackrel{{\scriptstyle*}}(t;f)$ and $\stackrel{{\scriptstyle*}}(t;g)$ are the solution of the problem (1.1)-(1.2) with respect to $f$ and $g$, with $f_{i}=\left(f_{1},f_{2}\right),g_{i}=\left(g_{1},g_{2}\right)$.

Proof. Consider the operators $A_{f}$ and $A_{g}$. From Theorem 4.1 it follows that $\left\|A_{f}\left(x_{1},x_{2}\right)-A_{g}\left(y_{1},y_{2}\right)\right\|_{C^{1}}\leq Q\left\|\left(x_{1},x_{2}\right)-\left(y_{1},y_{2}\right)\right\|_{C^{1}},\left(x_{1},x_{2}\right),\left(y_{1},y_{2}\right)\in X$.
We have now

Because $Q^{n}\rightarrow 0$ as $n\rightarrow\infty$ imply that

so we have

In this section we present the dependence by parameter $\lambda$ of the solution of the problem (1.1)-(1.2).

We shall use the following theorem
Theorem 6.1. (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.
Consider the following differential system with parameter:

where $x=\left(x_{1},x_{2}\right)$ and $f=\left(f_{1},f_{2}\right)$.
We suppose that
( $\mathrm{C}_{1}$ ) $a,b\in\mathbb{R},c\in(a,b)$ given, $\lambda\in J\subset\mathbb{R}$ a compact interval;
$\left(\mathrm{C}_{2}\right)f=\left(f_{1},f_{2}\right)\in C^{2}\left([a,b]\times\mathbb{R}^{4}\times J,\mathbb{R}^{2}\right);$
$\left(\mathrm{C}_{3}\right)$ there exists $L_{i}>0$ such that

for all $t\in[a,b],u_{j}\in\mathbb{R},i=1,2,j=\overline{1,4}$;
$\left(\mathrm{C}_{4}\right)$ for $Q=\max\left\{Q_{f},Q_{g}\right\}$ with

and $Q_{g}$ analogous, we have $Q^{n}\rightarrow 0$ as $n\rightarrow\infty$.
In the above conditions, from Theorem 4.1 we have that the problem (1.1)(1.2) has a unique solution, $\stackrel{{\scriptstyle*}}(\cdot;\lambda)=\left(\stackrel{{\scriptstyle*}}{{x_{1}}}(\cdot;\lambda),\stackrel{{\scriptstyle*}}{{x_{2}}}(\cdot;\lambda)\right)$, for any $\lambda\in\mathbb{R}$. In what follows we shall prove that $\stackrel{{\scriptstyle*}}(t;\cdot)\in C^{2}\left(J,\mathbb{R}^{2}\right)$, for all $t\in[a,b]$.

For this we consider the system

for all $t\in[a,b],\lambda\in J,x=\left(x_{1},x_{2}\right)\in C^{1}\left([a,b]\times J,\mathbb{R}^{2}\right),f=\left(f_{1},f_{2}\right)$.
The system (6.3) is equivalent with

Let $X:=\left(C^{1}\left([a,b]\times J,\mathbb{R}^{2}\right),\|\cdot\|_{C^{1}}\right)$ with the Chebyshev norm

Now we consider the operator $B:C^{1}\left([a,b]\times J,\mathbb{R}^{2}\right)\rightarrow C^{1}\left([a,b]\times J,\mathbb{R}^{2}\right)$ where

It is clear, from the proof of the Theorem 4.1, that in the conditions ( $\mathrm{C}_{1}$ )$\left(\mathrm{C}_{4}\right)$, the operator $B$ is Picard operator, since

Let $\stackrel{{\scriptstyle*}}=\left(\stackrel{{\scriptstyle*}}{{x_{1}}},\stackrel{{\scriptstyle*}}{{x_{2}}}\right)$ be the unique fixed point of $B$.
We suppose that there exists $\frac{\partial\stackrel{{\scriptstyle\mathcal{x}}}^{*}}{\partial\lambda},i=1,2$. From condition ( $\mathrm{C}_{3}$ ) we have