We consider the system

where

with initial conditions

This kind of problems has been the subject of many works, from which we quote here first of all Ch.J. de la Vallèe Poussin’s memorial [15]. From the research upon the polylocal problem in relation with differential equations, research more related to the present approach, we mention the following: improved formulation of Ch.J. de la Vallèe Poussin’s theorem regarding second order differential equations have been obtained by Ph . Hartman and A. Wintner [3] and later by D. Ripianu [9], evaluations of differentiable functions with applications on the study of the polylocal problem have been provided by O . Arama [1], the connection between the study of the polylocal problem and the theory of superior order convex functions has been provided by T. Popoviciu, in the case of linear systems of functions [8].

We suppose that:
$\left(\mathrm{C}_{1}\right)a=t_{1}<t_{2}<\ldots<t_{m-1}<t_{m}=b;$
$\left(\mathrm{C}_{2}\right)f\in C\left([a,b]\times\mathbb{R}^{2m},\mathbb{R}^{m}\right),\omega_{i}\in C([a,b],[a,b]),i=\overline{1,m}$;
$\left(\mathrm{C}_{3}\right)$ there exists $S_{i1},\ldots,S_{i,2m}\in M_{m,2m}\left(\mathbb{R}_{+}\right)$such that
$\left|f_{i}\left(t,u_{1},\ldots,u_{2m}\right)-f_{i}\left(t,v_{1},\ldots,v_{2m}\right)\right|\leq S_{i1}\left|u_{1}-v_{1}\right|+\ldots+S_{i,2m}\left|u_{2m}-v_{2m}\right|$ for all $t\in[a,b],u_{j},v_{j}\in\mathbb{R}^{2m},i=\overline{1,m}$.

We consider the problem

with the conditions (1.2), where $g:[a,b]\rightarrow\mathbb{R}^{m},g:=\left(g_{1},\ldots,g_{m}\right)$. The unique solution of this problem has the form

The problem (1.1)-(1.2) is equivalent with the fixed point problem (1.4) where $\mathbf{K}:=\left(K_{ij}\right)_{n}^{n}$ has the property that the operator defined by

is from $C\left([a,b],\mathbb{R}^{m}\right)$ to $C\left([a,b],\mathbb{R}^{m}\right)$.
$K_{ij}$ is given by the below relations

The problem (1.1)-(1.2) is equivalent with the system

where $\mathbf{K}:=\left(K_{ij}\right)_{n}^{n}$ is given by the relations (1.5)-(1.12).

Consider the Banach space $X:=\left(C\left([a,b],\mathbb{R}^{m}\right),\|\cdot\|\right)$ where $\|\cdot\|$ is the generalized Chebyshev norm,

and the operator

defined by

In this paper we shall use the Perov’s fixed point theorem and the weakly Picard operator theory in the study of existence and uniqueness and data dependence of the solution for the problem (1.1)-(1.2). For a better understanding we need some notions and results from WPO theory, see [10]-[14].

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 $A$;

Definition 2.1. 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 $\left(A^{n}\left(x_{0}\right)\right)_{n\in\mathbb{N}}$ converges to $x^{*}$ for all $x_{0}\in X$.

Definition 2.2. 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 $A$.

Definition 2.3. If $A$ is weakly Picard operator then we consider the operator $A^{\infty}$ defined by

Remark 2.4. It is clear that $A^{\infty}(X)=F_{A}$.
Lemma 2.5. Let $(X,d,\leq)$ be an ordered metric space and $A:X\rightarrow X$ an operator. We suppose that:
(i) $A$ is $WPO$;
(ii) $A$ is increasing.

Then, the operator $A^{\infty}$ is increasing.
Lemma 2.6. Let ( $X,d,\leq$ ) an ordered metric space and $A,B,C:X\rightarrow X$ be such that:
(i) the operator $A,B,C$ are WPOs;
(ii) $A\leq B\leq C$;
(iii) the operator $B$ is increasing.

Then $x\leq y\leq z$ implies that $A^{\infty}(x)\leq B^{\infty}(y)\leq C^{\infty}(z)$.
Theorem 2.7 (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

Theorem 2.8 (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.
For more details on WPOs theory see [10], [12], [13].

In what follows we consider the problem (1.1)-(1.2) in the conditions ( $\mathrm{C}_{1}$ )$\left(\mathrm{C}_{3}\right)$.

The problem (1.1)-(1.2) is equivalent with the fixed point equation

where $B_{f}=$ the second part of (1.13).
From the condition ( $C_{3}$ ) we have, for $t\in[a,b]$

for all $x,y\in X$ and
$Q=\max_{a\leq t\leq b}\int_{a}^{b}\mathbf{K}(t,s)ds\cdot\left(\begin{array}[]{cccc}S_{11}+S_{1,m+1}&S_{12}+S_{1,m+2}&\cdots&S_{1,m}+S_{1,2m}\\ \vdots&\vdots&\cdots&\vdots\\ S_{m,1}+S_{m,m+1}&S_{m,2}+S_{m,m+2}&\cdots&S_{m,m}+S_{m,2m}\end{array}\right)$.
Then

and if $Q^{n}\rightarrow 0$ as $n\rightarrow\infty$, the operator $B_{f}$ is $Q$-contraction. From the Perov’s fixed point theorem we have that the operator $B_{f}$ is PO and has a unique
fixed point

Since $f$ is continuous, we have that $\stackrel{{\scriptstyle*}}\in C\left([a,b],\mathbb{R}^{m}\right)$ is the unique solution for the problem (1.1)-(1.2).

So, we have the following existence and uniqueness theorem
Theorem 3.1. We suppose that:
(i) the conditions $\left(C_{1}\right)-\left(C_{3}\right)$ are satisfied;
(ii) $Q^{n}\rightarrow 0$ as $n\rightarrow\infty$.

Then:
(a) the problem (1.1)-(1.2) has in $C\left([a,b],\mathbb{R}^{m}\right)$ a unique solution $\stackrel{{\scriptstyle*}}=\left(\stackrel{{\scriptstyle*}}{{x_{1}}},\ldots,\stackrel{{\scriptstyle*}}{{x_{m}}}\right)$
$\in C\left([a,b],\mathbb{R}^{m}\right);$
(b) for all $x^{0}\in C\left([a,b],\mathbb{R}^{m}\right)$, the sequence $\left(x^{n}\right)_{n\in\mathbb{N}}$ defined by

converges uniformly to $\stackrel{{\scriptstyle*}}$, for all $t\in[a,b]$, and

In this section we shall study the relation between the solution of the problem (1.1)-(1.2) and the subsolution of the same problem.

Let $\stackrel{{\scriptstyle*}}$ the unique solution of the problem (1.1)-(1.2) and $y$ the subsolution of the same problem, i.e.

where $y:=\left(y_{1},y_{2},\ldots,y_{m}\right),y(\omega):=\left(y_{1}\left(\omega_{1}\right),y_{2}\left(\omega_{2}\right),\ldots,y_{m}\left(\omega_{m}\right)\right)$ and $f:=\left(f_{1},f_{2},\ldots\right.$,
$\left.f_{m}\right)$ ) satisfy the conditions $\left(\mathrm{C}_{1}\right)-\left(\mathrm{C}_{3}\right)$ and

In this section we consider the operator $B_{f}=$ the second part of (1.13) on the ordered Banach space $X=\left(\left(C[a,b],\mathbb{R}^{m}\right),\|\cdot\|,\leq\right)$, where on $\mathbb{R}^{m}$ we have the ordered relation:

We have the following theorem
Theorem 4.1. We suppose that:
(a) the conditions $\left(C_{1}\right)-\left(C_{3}\right)$ are satisfied;
(b) $Q^{n}\rightarrow 0$ as $n\rightarrow\infty$;
(c) $f(t,\cdot,\cdot):\mathbb{R}^{2m}\rightarrow\mathbb{R}^{m}$ is increasing, for all $t\in[a,b]$.

Let $x$ be a solution of the system (1.1) and $y$ be a solution of the inequality problem (4.1)-(4.2).

Then $y\leq x$ for all $t\in[a,b]$.
Proof. In terms of the operator $B_{f}$ defined by the relation (1.13), we have

On the other hand from condition (c) and Lemma 2.5, we have that the operator $B_{f}^{\infty}$ is increasing. Hence
$y\leq B_{f}(y)\leq B_{f}^{2}(y)\leq\cdots\leq B_{f}^{\infty}(y)\leq B_{f}^{\infty}(x)=x$.
So, $y\leq x$.

In this section we study the monotony of the system (1.1)-(1.2) with respect to $f$. For this we use the abstract comparison Lemma from section 2.

Consider the following equations

with the polylocal conditions (1.2) for each problem and let $\stackrel{{\scriptstyle*}},\stackrel{{\scriptstyle*}}$ and $\stackrel{{\scriptstyle*}}$ the unique solutions of these problems. Then we need the operators $B_{f},B_{g}$ and $B_{h}$ corresponding to the second part of the problems (5.1), (5.2) and (5.3).

Theorem 5.1. Let $f,g,h\in C\left([a,b]\times\mathbb{R}^{2m},\mathbb{R}\right)$, that satisfy the conditions $\left(C_{1}\right)-\left(C_{3}\right)$ from section 1.

We suppose that we have
(i) $f\leq g\leq h$;
(ii) $g(t,\cdot,\cdot):\mathbb{R}^{2m}\rightarrow\mathbb{R}$ is increasing.

Let $\stackrel{{\scriptstyle*}},\stackrel{{\scriptstyle*}}$ and $\stackrel{{\scriptstyle*}}$ the solutions of the equations (5.1), (5.2) and (5.3).
Then $\stackrel{{\scriptstyle*}}(t)\leq\stackrel{{\scriptstyle*}}(t)\leq\stackrel{{\scriptstyle*}}(t)$ for all $t\in[a,b]$, meaning that the unique solution of the system (1.1)-(1.2) is increasing with respect to the right hand.

Proof. From Theorem 3.1 the operators $B_{f},B_{g},B_{h}$ are POs.
From the condition (ii) it follows that the operator $B_{g}$ is monotone increasing and from condition (i) we have $B_{f}\leq B_{g}\leq B_{h}$.

But $\stackrel{{\scriptstyle*}}=B_{f}{}^{\infty}(\stackrel{{\scriptstyle*}}),\stackrel{{\scriptstyle*}}=B_{g}{}^{\infty}(\stackrel{{\scriptstyle*}})$ and $\stackrel{{\scriptstyle*}}=B_{h}{}^{\infty}(\stackrel{{\scriptstyle*}})$.
By applying the abstract comparison Lemma 2.6 follows that the unique solution of the problem (1.1)-(1.2) is increasing with respect to $B_{f}$.

Consider the problems (1.1)-(1.2) with the dates $f,g$ and suppose that the conditions from Theorem 3.1 are satisfied.

Let $f,g\in C\left([a,b]\times\mathbb{R}^{2m},\mathbb{R}^{m}\right)$ and

as in condition $\left(\mathrm{C}_{3}\right)$.
Consider $S_{ij}\in M_{m,2m}\left(\mathbb{R}_{+}\right),i=\overline{1,m},j=\overline{1,2m}$ with

Let
$Q_{f}=\max_{a\leq\pm\leq b}\int_{a}^{b}\mathbf{K}(t,s)ds\left(\begin{array}[]{cccc}S_{11}^{f}+S_{1,m+1}^{f}&S_{12}^{f}+S_{1,m+2}^{f}&\cdots&S_{1,m}^{f}+S_{1,2m}^{f}\\ \vdots&\vdots&\cdots&\vdots\\ S_{m,1}^{f}+S_{m,m+1}^{f}&S_{m,2}^{f}+S_{m,m+2}^{f}&\cdots&S_{m,m}^{f}+S_{m,2m}^{f}\end{array}\right)$,
$Q_{g}$ analogously and
$Q=\max_{a\leq t\leq b}\int_{a}^{b}\mathbf{K}(t,s)ds\cdot\left(\begin{array}[]{cccc}S_{11}+S_{1,m+1}&S_{12}+S_{1,m+2}&\cdots&S_{1,m}+S_{1,2m}\\ \vdots&\vdots&\cdots&\vdots\\ S_{m,1}+S_{m,m+1}&S_{m,2}+S_{m,m+2}&\cdots&S_{m,m}+S_{m,2m}\end{array}\right)$,
Denote by $\stackrel{{\scriptstyle*}}(\cdot;f)$ the solution of the problem (1.1)-(1.2).
Theorem 6.1. Let $f,g$ satisfy the conditions $\left(C_{1}\right)-\left(C_{3}\right)$. Furthermore, we suppose that there exist $\eta\in\mathbb{R}_{+}^{m}$ 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 $g$.

Proof. Consider the operators $B_{f}$ and $B_{g}$. From Theorem 3.1 it follows that

Additionally

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).

Consider the following differential system with parameter:

where $x:=\left(x_{1},\ldots,x_{m}\right)$ and $f:=\left(f_{1},\ldots,f_{m}\right)$.
We suppose that:
$\left(\mathrm{C}_{1}\right)a=t_{1}<t_{2}<\ldots<t_{m-1}<t_{m}=b;\lambda\in J\subset\mathbb{R}$ a compact interval;
$\left(\mathrm{C}_{2}\right)f\in C^{1}\left([a,b]\times\mathbb{R}^{2m}\times J,\mathbb{R}^{m}\right),\omega_{i}\in C([a,b],[a,b])$;
$\left(\mathrm{C}_{3}\right)$ there exists $S_{ij}\in M_{m,2m}\left(\mathbb{R}_{+}\right)$such that

for all $t\in[a,b],u_{j}\in\mathbb{R}^{2m},i=\overline{1,m},j=\overline{1,2m}$;
$\left(\mathrm{C}_{4}\right)$ for

we have $Q^{n}\rightarrow 0$ as $n\rightarrow\infty$.
In the above conditions, from Theorem 3.1 we have that the problem (1.1)(1.2) has a unique solution, $\stackrel{{\scriptstyle*}}(\cdot;\lambda)$, for any $\lambda\in\mathbb{R}$.

We prove that $\stackrel{{\scriptstyle*}}(t;\cdot)\in C^{1}\left(J,\mathbb{R}^{m}\right),\forall t\in[a,b]$.
For this we consider the system

$t\in[a,b],\lambda\in J,x\in C\left([a,b]\times J,\mathbb{R}^{m}\right)$.

The system (7.3) is equivalent with
$x_{i}(t;\lambda)=\int_{a}^{b}\mathbf{K}(t,s)f_{i}\left(s,x_{1}(s;\lambda),\ldots,x_{m}(s;\lambda),x_{1}\left(\omega_{1}(s);\lambda\right),\ldots,x_{m}\left(\omega_{m}(s);\lambda\right);\lambda\right)ds$,
where $i=\overline{1,m}$.
Let $X:=\left(C\left([a,b]\times J,\mathbb{R}^{m}\right),\|\cdot\|\right)$ with the Chebyshev norm,

Now we consider the operator

where