Recently, the interest in differential equations with “maxima” has increased exponentially. Such equations model real world problems whose present state depends significantly on its maximum value on a past time interval. For example, many problems in the control theory correspond to the maximal deviation of the regulated quantity. Some qualitative properties of the solutions of ordinary differential equations with “maxima” can be found in [1, 2, 5], [16, 17] and the references therein.

The main goal of the presented paper is to study a hybrid differential equation with maxima, using the theory of weakly Picard operators. The theory of Picard operators was introduced by I. A. Rus (see [12], [14] and their references) to study problems related to fixed point theory. This abstract approach is used by many mathematicians and it seemed to be a very useful and powerful method in the study of integral equations and inequalities, ordinary and partial differential equations (existence, uniqueness, differentiability of the solutions), etc.

In this paper we consider the following hybrid differential equation with maxima

with initial condition

where $t\in[a,b],\ a,b\in\mathbb{R},x_{0}\in\mathbb{R}^{m},\ f,g:[a,b]\times\mathbb{R}^{m}\mathbb{\rightarrow R}^{m}.$

We use the terminologies and notations from [12] and [14]. For the convenience of the reader we recall some of them.

Let $(X,d)$ be a metric space and $A:X\rightarrow X$ an operator. We denote by $A^{0}:=1_{X},\;A^{1}:=A,\ A^{n+1}:=A^{n}\circ A,\;\;n\in\mathbb{N}$, the iterate operators of the operator $A$. We also have:

Obviously, $A^{\infty}(X)=F_{A}$. Moreover, if $A$ is a PO and we denote by $x^{\ast}$ its unique fixed point, then $A^{\infty}(x)=x^{\ast}$, for each $x\in X$.

We prove the existence and uniqueness for the solution of the problem (1.1)-(1.2) using the Perov’s Theorem as in [7]. For standard techniques, when it is used the Banach contraction principle, see [13], [9] and [10].

We consider on $\mathbb{R}^{m}$ the following vectorial norm

We have the following result:

The equation (1.1) is equivalent with

$x\in C([a,b],\mathbb{R}^{m}).$

In what follows we consider the operator $B:X\rightarrow X$ defined by $B(x)(t):=$the right hand side of (2.5). For $x_{0}\in\mathbb{R}^{m}$, we consider $X_{x_{0}}:=\{x\in C([a,b],\mathbb{R}^{m})|\ x(a)=x_{0}\}$. It is clear that

is a partition of $X.\$We have

Now we consider the operators $A$ and $B$ on the ordered Banach space $(C([a,b],\mathbb{R}^{m}),\left\|\cdot\right\|,\leq)$ where the order relation on $\mathbb{R}^{m}$ is given by: $x\leq y\Leftrightarrow x_{i}\leq y_{i},\ i=\overline{1,m}$.

In order to establish the Čaplygin type inequalities we need the following abstract result.

We have the following result

In order to study the monotony of the solution of the problem (1.1)–(1.2) with respect to $x_{0},f,g$ we need the following result from WPOs theory.

From this abstract result we obtain the following result:

In this section we prove the continuous dependence of the solution for equation (1.1) and suppose the conditions of the Theorem 2.2 are satisfied.

In this section we emphasize some special cases of (1.1).

Let $\tau>0$ be a given number and we define the operator $G:C([-\tau,\infty),\mathbb{R}^{m})\rightarrow\mathbb{R}^{m}$ such that for any function $x\in C([-\tau,\infty),\mathbb{R}^{m})$ and any point $t\in\mathbb{R}_{+}$ there exists a point $\xi\in[t-\tau,t]$ such that $G(x)(t)=a(t)x(\xi)$ where $a\in C(\mathbb{R}_{+},\mathbb{R}).$

Consider the nonlinear delay functional differential equation

for $t\geq t_{0}$ with initial condition

where $x\in\mathbb{R}^{m},\ f:\mathbb{R}_{+}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{m},\ t_{0}\in\mathbb{R}_{+},\ \varphi:[-\tau,0]\rightarrow\mathbb{R}^{m}.$

Particular cases of (1.1):

• (i)

  For $G(x)(t)=x(t-\tau),\ t\in\mathbb{R}_{+}$, then (5.1) reduces to a delay differential equation (see [6], [12], [14], [15]);

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• (ii)

  For $G(x)(t)=\underset{s\in[t-\tau,t]}{\max}x(s),\ t\in\mathbb{R}_{+}$, then (5.1) reduces to a differential equation with maxima (see [16], [17], [9], [10], [1]);

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• (iii)

  For $G(x)(t)=\int\nolimits_{t-\tau}^{t}x(s)ds,\ t\in\mathbb{R}_{+},\tau>0$, then (5.1) reduces to a differential equation with distributed delay (see [11], [4]);

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• (iv)

  For $g(t,G(x)(t))=h(x)(t)$, where $h:C([a,b],\mathbb{R})\rightarrow C([a,b],\mathbb{R})$ is an abstract Volterra operator, then (5.1) reduces to a differential equation with abstract Volterra operator (see [8]);

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• (v)

  If $x^{\prime}(t)-f(t,x(t)):=\frac{d}{dt}\left[\frac{x(t)}{f(t,x(t))}\right],\ G(x)(t)=x(t),\ t\geq t_{0}$, then (5.1) reduces to a quadratic differential equation (see [3]).

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