In 1990, Corduneanu investigated functional differential equations involving abstract Volterra operators. In this sense, around the year 2000 Corduneanu [7] presented a general study on functional differential equations with abstract or causal Volterra operators.

On the other hand, T.A. Burton ([3]-[6]) presented in some examples of integral equations a notion of progressive contractions on $C([a,\infty[).$ In 2019, following the idea of T.A. Burton and the forward step method ([19]), I.A. Rus formalized this notion ([21]), with ”step by step” instead of ”progressive”, and give some variant of step by step contraction principle in the case of operators with Volterra property on $C([a,b],\mathbb{B)}$ and $C([a,\infty[,\mathbb{B})$ where $\mathbb{B}$ is a Banach space.

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

with the condition

where $\alpha\in\mathbb{R}$ and $f\in C([a,b]\times\mathbb{R}^{2})$ are given. To prove our results, we shall use the abstract result given by I.A. Rus [21].

In this section, we shall establish a new result of existence and uniqueness of the solution of the functional differential equation with maxima (1.1).

The problem (1.1)–(1.2), $x\in C^{1}([a,b],\mathbb{R})$ is equivalent with the fixed point equation

It is clear that equation (3.3) is equivalent with $x=V(x)$, where the operator $V:C([a,b],\mathbb{R})\rightarrow C([a,b],\mathbb{R}),$defined by

The operator $V$ has the Volterra property, i.e.,

This implies that the operator $V$ induced, for each $c$ with $a<c<b$ and, the operator $V_{c}:C[a,c]\rightarrow C[a,c],$ defined by, $V_{c}(x)(t):=V(\widetilde{x}),\$where $\widetilde{x}\in C[a,b]$ is such that, $\left.\widetilde{x}\right|_{[a,c]}=x$.

In what follows we consider the notations from Section 2.3, where $\mathbb{B}=\mathbb{R}$.

We have

The above considerations give rise to the following problem: In which conditions the operator $V$ is Picard operator?

From the Fibre contraction principle (see [21]) the answer is the following: In the conditions of the Theorem 3.2, the operator $V$ is a Picard operator with respect to the uniform convergence on $[t_{0},t_{m}]\cdot$

In order to prove this we consider the following operators induced by the operator $V.$ First of all from (3.4) we have that:

(4.1) $V(x)(t):=\alpha+\int_{t_{0}}^{t}f(s,x(s),\underset{t_{0}\leq\xi\leq s}{\max}x(\xi))ds,\ t\in[t_{0},t_{1}]$,

(4.2) $V(x)(t):=\alpha+\int_{t_{0}}^{t_{1}}f(s,x(s),\underset{t_{0}\leq\xi\leq s}{\max}x(\xi))ds+\int_{t_{1}}^{t}f(s,x(s),\underset{t_{0}\leq\xi\leq s}{\max}x(\xi))ds,\ t\in[t_{1},t_{2}]$,

$\vdots$

(4.k) $V(x)(t):=\alpha+\int_{t_{0}}^{t_{1}}f(s,x(s),\underset{t_{0}\leq\xi\leq s}{\max}x(\xi))ds+...+\int_{t_{k-1}}^{t}f(s,x(s),\underset{t_{0}\leq\xi\leq s}{\max}x(\xi))ds,\ t\in[t_{k-1},t_{k}],k=\overline{1,m}.$

Let $R:C[t_{0},t_{m}]\rightarrow C[t_{0},t_{1}]\times C[t_{1},t_{2}]\times...\times C[t_{m-1},t_{m}]$ be defined by,

We also consider the following subset:

It is clear that $R:C[t_{0},t_{m}]\rightarrow U$ is a bijection.

Let us consider the following operators induced by the operator $V:$

$T_{1}:C[t_{0},t_{1}]\rightarrow C[t_{0},t_{1}],$

$T_{2}:C[t_{0},t_{1}]\times C[t_{1},t_{2}]\rightarrow C[t_{1},t_{2}],$

$\vdots$

$T_{k}:C[t_{0},t_{1}]\times C[t_{1},t_{2}]\times...\times C[t_{k-1},t_{k}]\rightarrow C[t_{k-1},t_{k}],$

and

In the conditions of Theorem 3.2, the operators, $T_{1},T_{2}\left(x_{1},\cdot\right),...,T_{m}\left(x_{1},...,x_{m-1},\cdot\right)$ are contractions. From the Fibre contraction Principle, $T$ is a Picard operator.

Now, we observe that: $V=R^{-1}TR$ and $\ V^{n}=R^{-1}T^{n}R.$ These imply that the operator $V$ is a Picard operator.

In this section we will emphasize the importance of the above result by applying for the operator $V$ the Gronwall type inequalities and the comparison theorem.

In this section we suppose that

1. $(H)$

   there exists $L>0$ such that

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   $$\left|f(t,u_{1},u_{2})-f(t,v_{1},v_{2})\right|\leq L\max(\left|u_{1}-v_{1}\right|,\left|u_{2}-v_{2}\right|)$$

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

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We consider on $C([a,b],\mathbb{R})$ the max norm and in condition $(H),$ the operator $V$ defined by (3.4) is a Picard operator. So, in the condition $(H)$, the problem (1.1)-(1.2) has in $C([a,b],\mathbb{R})$ a unique solution $x^{\ast}.$ Moreover, for $t\in[a,b],\ x^{\ast}(t)=\underset{n\rightarrow\infty}{\lim}x_{n}(t),$ for each $x_{0}\in C([a,b],\mathbb{R}),$ where $(x_{n})_{n\in\mathbb{N}}$ is defined by

Now we can apply Abstract Gronwall Lemma (see [21]).

In a similar way, a comparison theorem for equation (1.1) can be obtained, using the Abstract Comparison Lemma.

We consider now the following functional differential equations with maxima

with the condition

where $\alpha_{i}\in\mathbb{R}$ and $f_{i}\in C([a,b]\times\mathbb{R}^{2}),\ i=1,2,3$ are given. We suppose that

1. $(H^{\prime})$

   there exists $L_{i}>0$ such that

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   $$\left|f_{i}(t,u_{1},u_{2})-f_{i}(t,v_{1},v_{2})\right|\leq L_{i}\max(\left|u_{1}-v_{1}\right|,\left|u_{2}-v_{2}\right|),$$

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

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Acknowledgement The authors would like to express their special thanks and gratitude to Professor Ioan A. Rus for the ideas and continuous support along the years.