Differential equations with maxima are often met in applications, for instance in the theory of automatic control. The existence and uniqueness of
solutions of the equation with maxima is considered in [3], [4], [9]. The asymptotic stability of the solution of this equations and other problems concerning equations with maxima are investigated in [2], [3], [6], [15], [16].

The main goal of the presented paper is to study a second order functionaldifferential equations with maxima, of mixed type, using the theory of weakly Picard operators ([10]-[14]).

We consider the following functional-differential equation

with the "boundary" conditions

We suppose that:
$\left(\mathrm{C}_{1}\right)h_{1},h_{2},a$ and $b\in\mathbb{R},a<b,h_{1}>0,h_{2}>0$;
$\left(\mathrm{C}_{2}\right)f\in C\left([a,b]\times\mathbb{R}^{3}\right)$;
( $\mathrm{C}_{3}$ ) there exists $L_{f}>0$ such that

for all $t\in[a,b]$ and $u_{i},v_{i}\in\mathbb{R},i=1,2,3;$
$\left(\mathrm{C}_{4}\right)\varphi\in C\left[a-h_{1},a\right]$ and $\psi\in C\left[b,b+h_{2}\right]$.
Let $G$ be the Green function of the following problem

The problem (1.1)-(1.2), $x\in C\left[a-h_{1},b+h_{2}\right]\cap C^{2}[a,b]$ is equivalent with the following fixed point equation

$x\in C\left[a-h_{1},b+h_{2}\right]$, where

The equation (1.1) is equivalent with

$x\in C\left[a-h_{1},b+h_{2}\right]$.
In what follow we consider the operators:

defined by

and

Let $X:=C\left[a-h_{1},b+h_{2}\right]$ and $X_{\varphi,\psi}:=\left\{x\in X|x|_{\left[a-h_{1},a\right]}=\varphi,\left.x\right|_{\left[b,b+h_{2}\right]}=\psi\right\}$. It is clear that

is a partition of $X$.
We have
Lemma 1.1. We suppose that the conditions $\left(C_{1}\right),\left(C_{2}\right)$ and $\left(C_{4}\right)$ are satisfied. Then
(a) $B_{f}(X)\subset X_{\varphi,\psi}$ and $B_{f}\left(X_{\varphi,\psi}\right)\subset X_{\varphi,\psi}$;
(b) $\left.B_{f}\right|_{X_{\varphi,\psi}}=\left.E_{f}\right|_{X_{\varphi,\psi}}$.

In this paper we shall prove that, if $L_{f}$ is small enough, then the operator $E_{f}$ is weakly Picard operator and we shall study the equation (1.1) in the terms of this operator.

Let ( $X,d$ ) be a metric space and $A:X\rightarrow X$ an operator. We shall use the following notations:

$I(A):=\{Y\subset X\mid A(Y)\subset Y,Y\neq\emptyset\}$ - the family of the nonempty invariant subset of $A$;

Definition 2.1. ([10], [13]) 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. ([10], [13]) 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. ([10], [13]) If $A$ is weakly Picard operator then we consider the operator $A^{\infty}$ defined by

Remark 2.4. It is clear that $A^{\infty}(X)=F_{A}=\{x\in X\mid A(x)=x\}$.
Definition 2.5. ([10], [13]) Let $A$ be a weakly Picard operator and $c>0$. The operator $A$ is $c$-weakly Picard operator if

Theorem 2.6. ([10], [13]) 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 $\lambda\in\Lambda$.

Example 2.7. ([10], [13]) Let ( $X,d$ ) be a complete metric space and $A:X\rightarrow X$ an $\alpha$-contraction. Then $A$ is $\frac{1}{1-\alpha}-PO$.

Example 2.8. Let $(X,d)$ be a complete metric space and $A:X\rightarrow X$ continuous and $\alpha$-graphic contraction. Then $A$ is $\frac{1}{1-\alpha}-WPO$.

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

Let $I:\mathbb{R}\rightarrow P_{cp,cv}(\mathbb{R}):=\{Y\subset\mathbb{R}\mid Y$ compact and convex $\}$ be a multivalued operator. We suppose that $I(t)=[\alpha(t),\beta(t)]$ where $\alpha\leq\beta$ and $\alpha,\beta\in C(\mathbb{R})$.

For $x\in C(\mathbb{R})$ we consider the function $\max_{I}x$ defined by $\left(\max_{I}x\right)(t):=\max_{\xi\in I(t)}x(\xi)$. We remark that $\max_{I}x\in C(\mathbb{R})$. So, we have the operator

Some properties of the operator $\max_{I}$ are given by
Lemma 3.1. We have
(i) $x\leq y\Rightarrow\max_{I}x\leq\max_{I}y$, i.e. the operator $\max_{I}$ is increasing;
(ii) $\left|\max_{\xi\in I(t)}x(\xi)-\max_{\xi\in I(t)}y(\xi)\right|\leq\max_{\xi\in I(t)}|x(\xi)-y(\xi)|$, for all $t\in\mathbb{R},\quad x,y\in C(\mathbb{R});$
(iii) $\max_{t\in K}\left|\max_{\xi\in I(t)}x(\xi)-\max_{\xi\in I(t)}y(\xi)\right|\leq\max_{\xi\in\cup_{t\in K}I(t)}|x(\xi)-y(\xi)|$, for all $t\in\mathbb{R},x,y\in C(\mathbb{R})$.

(ii) Let $\xi\in I(t)$. We have

Then

and

(iii) Follows from (ii).

Our first result is the following
Theorem 4.1. We suppose that:
(a) the conditions $\left(C_{1}\right)-\left(C_{4}\right)$ are satisfied;
$\left(\mathrm{C}_{5}\right)\frac{L_{f}}{8}(b-a)^{2}<1$.
Then the problem (1.1)-(1.2) has a unique solution which is the uniform limit of the successive approximations.

Proof. Consider the Banach space ( $C\left[a-h_{1},b+h_{2}\right],\|\cdot\|$ ) where $\|\cdot\|$ is the Chebyshev norm, $\|\cdot\|:=\max_{a-h_{1}\leq t\leq b+h_{2}}|x(t)|$.

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

From the condition ( $C_{3}$ ) we have, for $t\in[a,b]$

This implies that $B_{f}$ is an $\alpha$-contraction, with $\alpha=\frac{L_{f}}{8}(b-a)^{2}$. The proof follows from the contraction principle.

Remark 4.2. From the proof of Theorem 4.1, it follows that the operator $B_{f}$ is PO. Since

and

hence, the operator $E_{f}$ is WPO and

where $x_{\varphi,\psi}^{*}$ is the unique solution of the problem (1.1)-(1.2).
Remark 4.3. $E_{f}$ is $\alpha$-graphic contraction, i.e.

In this section we need the following abstract result
Lemma 5.1. (see[12]) 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.
Now we consider the operators $E_{f}$ and $B_{f}$ on the ordered Banach space $\left(C\left[a-h_{1},b+h_{2}\right],\|\cdot\|,\leq\right)$.

We have
Theorem 5.2. We suppose that:
(a) the conditions $\left(C_{1}\right)-\left(C_{4}\right)$ are satisfied;
(b) $\frac{L_{f}}{8}(b-a)^{2}<1$;
(c) $f(t,\cdot,\cdot,\cdot):\mathbb{R}^{3}\rightarrow\mathbb{R}$ is increasing, $\forall t\in[a,b]$.

Let $x$ be a solution of equation (1.1) and $y$ a solution of the inequality

Then

Proof. Let us consider the operator $\widetilde{w}:C\left[a-h_{1},b+h_{2}\right]\rightarrow C\left[a-h_{1},b+h_{2}\right]$ defined by

First of all we remark that

and

In the terms of the operator $E_{f}$, we have

On the other hand, from the condition (c) and Lemma 5.1, we have that the operator $E_{f}^{\infty}$ is increasing. Hence

So, $y\leq x$.

In this section we study the monotony of the solution of the problem (1.1)(1.2) with respect to $\varphi,\psi$ and $f$. For this we need the following result from the WPOs theory.

Lemma 6.1. (Abstract comparison lemma, [13]) 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 $A^{\infty}(x)\leq B^{\infty}(y)\leq C^{\infty}(z)$.
From this abstract result we have
Theorem 6.2. Let $f_{i}\in C\left([a,b]\times\mathbb{R}^{3}\right),i=1,2,3$, be as in Theorem 4.1. We suppose that:
(i) $f_{1}\leq f_{2}\leq f_{3}$;
(ii) $f_{2}(t,\cdot,\cdot,\cdot):\mathbb{R}^{3}\rightarrow\mathbb{R}$ is monotone increasing;

Let $x_{i}$ be a solution of the equation

Then, $x_{1}(t)\leq x_{2}(t)\leq x_{3}(t),\forall t\in\left[a-h_{1},a\right]\cup\left[b,b+h_{2}\right]$, implies that $x_{1}\leq x_{2}\leq x_{3}$, i.e. the unique solution of the problem (1.1)-(1.2) is increasing with respect to $f,\varphi$ and $\psi$.

Proof. From Theorem 4.1, the operators $E_{f_{i}},i=1,2,3$, are WPOs. From the condition (ii) the operator $E_{f_{2}}$ is monotone increasing. From the condition (i) it follows that

On the other hand we remark that

and

So, the proof follows from Lemma 6.1.

Consider the boundary value problem (1.1)-(1.2) and suppose the conditions of the Theorem 4.1 are satisfied. Denote by $x^{*}(\cdot;\varphi,\psi,f)$, the solution of this problem.

We need the following well known result (see [12]).
Theorem 7.1. 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

We state the following result:
Theorem 7.2. Let $\varphi_{i},\psi_{i},f_{i},i=1,2$ be as in the Theorem 4.1. Furthermore, we suppose that there exists $\eta_{i}>0,i=1,2$ such that
(i) $\left|\varphi_{1}(t)-\varphi_{2}(t)\right|\leq\eta_{1},\forall t\in\left[a-h_{1},a\right]$ and $\left|\psi_{1}(t)-\psi_{2}(t)\right|\leq\eta_{1},\forall t\in\left[b,b+h_{2}\right];$
(ii) $\left|f_{1}\left(t,u_{1},u_{2},u_{3}\right)-f_{2}\left(t,u_{1},u_{2},u_{3}\right)\right|\leq\eta_{2},\forall t\in C[a,b],u_{i}\in\mathbb{R},i=1,2,3$.

Then

where $x_{i}^{*}\left(t;\varphi_{i},\psi_{i},f_{i}\right)$ are the solution of the problem (1.1)-(1.2) with respect to $\varphi_{i},\psi_{i},f_{i},i=1,2$, and $L_{f}=\max\left(L_{f_{1}},L_{f_{2}}\right)$.

Proof. Consider the operators $B_{\varphi_{i},\psi_{i},f_{i}},i=1,2$. From Theorem 4.1 these operators are contractions.

Additionally

$\forall x\in C\left[a-h_{1},b+h_{2}\right]$.
Now the proof follows from the Theorem 7.1, with $A:=B_{\varphi_{1},\psi_{1},f_{1}},B=B_{\varphi_{2},\psi_{2},f_{2}},\eta=\eta_{1}+\eta_{2}\frac{(b-a)^{2}}{8}$ and $\alpha:=\frac{L_{f}}{8}(b-a)^{2}$ where $L_{f}=\max\left(L_{f_{1}},L_{f_{2}}\right)$.

We have
Theorem 7.3. ([13]) 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 $H\left(F_{A_{1}},F_{A_{2}}\right)\leq\eta\max\left(c_{1},c_{2}\right)$.
In what follow we shall use the $c$-WPOs techniques to give some data dependence results using Theorem 7.3.

Theorem 7.4. Let $f_{1}$ and $f_{2}$ be as in the Theorem 4.1. Let $S_{E_{f_{1}}},S_{E_{f_{2}}}$ be the solution sets of system (1.1) corresponding to $f_{1}$ and $f_{2}$. Suppose that there exists $\eta>0$, such that

for all $t\in[a,b],u_{i}\in\mathbb{R},i=1,2,3$.
Then

where $L_{f}=\max\left(L_{f_{1}},L_{f_{2}}\right)$ and $H_{\|\cdot\|_{C}}$ denotes the Pompeiu-Hausdorff functional with respect to $\|\cdot\|_{C}$ on $C[a,b]$.

Proof. In the condition of Theorem 4.1, the operators $E_{f_{1}}$ and $E_{f_{2}}$ are $c_{1}$ WPO and $c_{2}$-weakly Picard operators.

Let

It is clear that $\left.E_{f_{1}}\right|_{X_{\varphi,\psi}}=B_{f_{1}},\left.E_{f_{2}}\right|_{X_{\varphi,\psi}}=B_{f_{2}}$. Therefore,

for all $x\in C\left[a-h_{1},b+h_{2}\right]$.
Now, choosing

we get that $E_{f_{1}}$ and $E_{f_{2}}$ 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 (7.1) we obtain that

$\forall x\in C\left[a-h_{1},b+h_{2}\right]$. Applying Theorem 7.3 we have that

where $L_{f}=\max\left(L_{f_{1}},L_{f_{2}}\right)$ and $H_{\|\cdot\|_{C}}$ is the Pompeiu-Hausdorff functional with respect to $\|\cdot\|_{C}$ on $C\left[a-h_{1},b+h_{2}\right]$.

Let $p,q,r,g\in C[a,b]$. We consider the following boundary value problem

with the "boundary" conditions

In this case $f\left(t,u_{1},u_{2},u_{3}\right)=p(t)u_{1}+q(t)u_{2}+r(t)u_{3}+g(t),t\in[a,b],u_{i}\in\mathbb{R},i=1,2,3$, and $L_{f}=\max_{t\in[a,b]}(|p(t)|+|q(t)|+|r(t)|)$.

We suppose that:
$\left(\mathrm{C}_{1}^{\prime}\right)h_{1},h_{2},a$ and $b\in\mathbb{R},a<b,h_{1}>0,h_{2}>0$;
$\left(\mathrm{C}_{2}^{\prime}\right)p,q,r,g\in C[a,b]$;
$\left(\mathrm{C}_{3}^{\prime}\right)\varphi\in C\left[a-h_{1},a\right]$ and $\psi\in C\left[b,b+h_{2}\right]$.
From this conditions and the above results we have
Theorem 8.1. We suppose that:
(a) the conditions $\left(C_{1}^{\prime}\right)-\left(C_{3}^{\prime}\right)$ are satisfied;
$\left(\mathrm{C}_{4}^{\prime}\right)\frac{L_{f}}{8}(b-a)^{2}<1$.
Then the problem (8.1)-(8.2) has a unique solution which is the uniform limit of the successive approximations.