This paper is concerned with the following integro-differential equation

where $g$ and $K$ are continuous functions on a Banach space and satisfy certain conditions to be specified later.

Regarding the earlier works on existence, uniqueness and convergence of the sequence of the successive approximation to integro-differential equations with delays and functional-differential equations with delays under different conditions, we refer to Guo et al [2], Kolmanowskii and Mishkis [5], Precup [7], Precup and Kirr [8], I.A. Rus [12] and the references therein. The related results for the existence and uniqueness, convergence of the sequence of the successive approximation, lower and upper solutions to the differential equations with delays can be found in Dobritioiu et al [1], Ilea [3] and Otrocol [6].

The authors Rus, Şerban, Trif [13] have considered the following integral equation

in a Banach space and proved that the sequence of the successive approximation generated by the step method converges to the solution of this integral equation using the results of Rus [9].

Sakata and Hara [14] have considered the linear differential equation with two kinds of time lags

where $\tau>0,h>0$ and $a,b$ are both real and they have studied the dependence on delays towards stability regions.

In the present work we use the ideas of Rus, Şerban, Trif [13] to establish the convergence of the sequence of successive approximation to equation (1.1). Regarding the two delays we have the following cases: $h>0,\tau>0,\tau>h$ and $h>0,\tau<0,|\tau|>h$. Here, the authors study the first case, while the second case is studied in [4].

The aim of this paper is to obtain existence and uniqueness theorems using contraction principle and step method. Such kind of results have been proved in [13]. The approach proposed in the present paper is different to the ones in [13] and [1] and it is based on the different time lags. Also, we present here some lower and upper solution result, and a numerical example concerning equation (1.1).

We note that Sakata and Hara study in [14] the stability regions for similar integrodifferential equation with two time lags.

Let $\tau>0,h>0,h<\tau$ and

Relative to (2.1)-(2.2) we consider the following conditions:
$\left(C_{1}\right)(\mathbb{B},\|\cdot\|)$ is a Banach space, $g\in C\left([0,T]\times\mathbb{B}^{2},\mathbb{B}\right),K\in C([0,T]\times\mathbb{B},\mathbb{B}),\varphi\in C([-\tau,0],\mathbb{B})$;
$\left(C_{2}\right)$ there exists $L_{g}>0$ such that

$\left(C_{2}^{\prime}\right)$ there exists $L_{g}^{\prime}>0$ such that

$\left(C_{3}\right)$ there exists $L_{K}>0$ such that

$\left(C_{4}\right)\varphi^{\prime}(0)=g(0,\varphi(0),\varphi(-\tau))+\int_{-h}^{0}K(s,\varphi(s))ds$.
We consider the space $X=C([-\tau,T],\mathbb{B})$ endowed with the norms $\|\cdot\|_{\infty}$ and $\|\cdot\|_{\lambda}$ where

The following relation between the Cebyshev and Bielecki norms holds

The problem (2.1)-(2.2) is equivalent with the following fixed point problem:

Let ( $X,d$ ) be a metric space and $A:X\rightarrow X$ an operator. In this paper we shall use the terminologies and notations from [13]. For the convenience of the reader we shall recall some of them.

Denote by $A_{0}:=1_{X},A^{1}:=A,A^{n+1}:=A\circ A^{n},n\in\mathbb{N}$, the iterate operators of the operator $A$. Also

Definition 3.1. $A:X\rightarrow X$ is called a Picard operator (briefly PO) if:
(i) $F_{A}=\left\{x^{*}\right\}$;
(ii) $A^{n}(x)\rightarrow x^{*}$ as $n\rightarrow\infty$, for all $x\in X$.

Definition 3.2. $A:X\rightarrow X$ is said to be a weakly Picard operator (briefly WPO) if the sequence $\left(A^{n}(x)\right)_{n\in N}$ converges for all $x\in X$ and the limit (which may depend on $x$ ) is a fixed point of $A$.

If $A:X\rightarrow X$ is a WPO, then we may define the operator $A^{\infty}:X\rightarrow X$ by

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

Lemma 3.3. 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 3.4. (Fibre contraction principle, Rus [10]) Let ( $X,d$ ) be a metric space and $(Y,\rho)$ be a complete metric space. Let $B:X\rightarrow X$ and $C:X\times Y\rightarrow Y$ be two operators. We suppose that:
(i) $B$ is a $WPO$;
(ii) $C(x,\cdot):Y\rightarrow Y$ is $\alpha$-contraction, for all $x\in X$;
(iii) if $\left(x^{*},y^{*}\right)\in F_{A}$, where $A:X\times Y\rightarrow X\times Y,A(x,y)=(B(x),C(x,y))$, then $C\left(\cdot,y^{*}\right)$ is continuous in $x^{*}$.
Then $A$ is a WPO. Moreover, if $B$ is PO then $A$ is PO.
By induction, from the above result we have:
Theorem 3.5. (Rus [11]) Let ( $X_{i},d_{i}$ ), $i=\overline{0,m},m\geq 1$, be some metric spaces. Let

be some operator. We suppose that:
(i) $\left(X_{i},d_{i}\right),i=\overline{1,m}$, are complete metric spaces;
(ii) the operator $A_{0}$ is WPO;
(iii) there exists $\alpha_{i}\in(0;1)$ such that:

are $\alpha_{i}$-contractions;
(iv) the operator $A_{i},i=\overline{1,m}$, are continuous.

Then the operator $A:X_{0}\times\cdots\times X_{m}\rightarrow X_{0}\times\cdots\times X_{m}$,

is $WPO$. If $A_{0}$ is PO, then $A$ is PO.

In this section we give an existence theorem for the solution of the problem (2.1)(2.2).

Theorem 4.1. In the condition $\left(C_{1}\right),\left(C_{2}\right),\left(C_{3}\right)$ and $\left(C_{4}\right)$, the problem (2.1)-(2.2) has in $C([-\tau,T],\mathbb{B})$ a unique solution $x^{*}$ and the sequence of successive approximation, $\left(x^{n}\right)_{n\in\mathbb{N}}$

converges uniformly to $x^{*}$, for every $x^{0}\in C([-\tau,T],\mathbb{B})$, with $\left.x^{0}\right|_{[-\tau,0]}=\varphi$.
Proof. Let $X_{\varphi}\subset X,X_{\varphi}=\{x\in X\mid x(t)=\varphi(t),t\in[-\tau,0]\}$ and $A:X_{\varphi}\rightarrow X_{\varphi}$ defined by

Note that $X_{\varphi}$ is a closed subset of $X$, so $\left(X_{\varphi},d_{\|\cdot\|_{\lambda}}\right)$ is a complete metric space.

In a standard way we have

which proves that $A$ is Lipschitz with $L_{A}=\frac{1}{\lambda}\left(L_{g}+L_{K}h\right)$. We can choose $\lambda$ sufficiently large such that $L_{A}=\frac{1}{\lambda}\left(L_{g}+L_{K}h\right)<1$, so $A$ is contraction. Applying the contraction principle we get the conclusion.

Remark 4.2. From the proof of Theorem 4.1, it follows that the operator $A$ is PO.

Using step method and contraction principle on each step for the problem (2.1)(2.2), in this section we obtain a better result by replacing the condition ( $C2$ ) from Theorem 4.1 with ( $C2^{\prime}$ ).

Let $m\in\mathbb{N}^{*}$ such that $(m-1)h\leq T,mh>T$. To simplify our presentation we suppose that $h<\tau\leq 2h$. In the conditions $\left(C_{1}\right),\left(C_{2}^{\prime}\right),\left(C_{3}\right)$ and $\left(C_{4}\right)$ the step method for (2.1)-(2.2) consist in the following:
$\left(p_{0}\right)\quad x_{0}(t)=\varphi(t),\quad t\in[-\tau,0]$
$\left(p_{1}\right)\quad x_{1}(t)=\varphi(0)+\int_{0}^{t}g\left(\xi,x_{1}(\xi),\varphi(\xi-\tau)\right)d\xi+\int_{0}^{t}\int_{\xi-h}^{0}K(s,\varphi(s))dsd\xi++\int_{0}^{t}\int_{0}^{\xi}K\left(s,x_{1}(s)\right)dsd\xi,t\in[0,h]$
$\left.p_{2}\right)\quad x_{2}(t)=x_{1}^{*}(h)+\int_{h}^{\tau}g\left(\xi,x_{2}(\xi),\varphi(\xi-\tau)\right)d\xi+\int_{\tau}^{t}g\left(\xi,x_{2}(\xi),x_{1}^{*}(\xi-\tau)\right)d\xi++\int_{h}^{t}\int_{\xi-h}^{h}K\left(s,x_{1}^{*}(s)\right)dsd\xi+\int_{h}^{t}\int_{h}^{\xi}K\left(s,x_{2}(s)\right)dsd\xi,t\in[h,2h]$
$\left(p_{i}\right)\quad x_{i}(t)=x_{i-1}^{*}((i-1)h)+\int_{(i-1)h}^{(i-2)h+\tau}g\left(\xi,x_{i}(\xi),x_{i-2}^{*}(\xi-\tau)\right)d\xi++\int_{(i-2)h+\tau}^{t}g\left(\xi,x_{i}(\xi),x_{i-1}^{*}(\xi-\tau)\right)d\xi++\int_{(i-1)h}^{t}\int_{\xi-h}^{(i-1)h}K\left(s,x_{i-1}^{*}(s)\right)dsd\xi++\int_{(i-1)h}^{t}\int_{(i-1)h}^{\xi}K\left(s,x_{i}(s)\right)dsd\xi,t\in[(i-1)h,ih]$

where $x_{i}^{*}$ is the unique solution of ( $p_{i}$ ), $i=\overline{1,m}$.
So, we have the following result:
Theorem 5.1. In the conditions $\left(C_{1}\right),\left(C_{2}^{\prime}\right),\left(C_{3}\right)$ and $\left(C_{4}\right)$ we have:
a) the problem (2.1)-(2.2) has in $C([-\tau,T],\mathbb{B})$ a unique solution $x^{*}$,

b) for each $x_{i}^{0}\in C([(i-1)h,ih],\mathbb{B}),i=\overline{1,m-1}$, $x_{m}^{0}\in C([(m-1)h,T],\mathbb{B})$, the sequence defined by:

Proof. In order to proof this theorem we apply the contraction principle for each step: $[(i-1)h,ih],[(m-1)h,T]$, where $i=\overline{1,m-1}$.

For the first step we consider the Banach space $X_{1}:=\left(C([0,h],\mathbb{B}),\|\cdot\|_{\lambda_{1}}\right)$, where $\|x\|_{\lambda_{1}}=\max_{t\in[0,h]}\left\{\|x(t)\|e^{-\lambda_{1}t}\right\}$ and the operator $A_{1}:X_{1}\rightarrow X_{1}$ defined by

For $x,y\in X_{1}$, we have

We can choose a $\lambda_{1}>0$ such that $\frac{1}{\lambda_{1}}\left(L_{g}^{\prime}+L_{K}h\right)<1$, so $A_{1}$ is a contraction, therefore $F_{A_{1}}=\left\{x_{1}^{*}\right\}$.

For the next steps let us consider the following Banach spaces: for $i=\overline{2,m-1}$ given by

and
$X_{m}:=\left(C([(m-1)h,T];\mathbb{B}),\|\cdot\|_{\lambda_{m}}\right)$, with $\|x\|_{\lambda_{m}}:=\max_{t\in[(m-1)h,T]}\left\{\|x(t)\|e^{-\lambda_{m}(t-(m-1)h)}\right\}$
and the operators $A_{i}:X_{i}\rightarrow X_{i},i=\overline{2,m}$ defined by

For $x,y\in X_{i}$ we have $\left\|A_{i}(x)-A_{i}(y)\right\|_{\lambda_{i}}\leq\frac{1}{\lambda_{i}}\left(L_{g}^{\prime}+L_{K}h\right)\|x-y\|_{\lambda_{i}}$, so $A_{i}$ is a contraction for a suitable choice of $\lambda_{i}$ such that $\frac{1}{\lambda_{i}}\left(L_{g}^{\prime}+L_{K}h\right)<1$. Therefore, we get that $F_{A_{i}}=\left\{x_{i}^{*}\right\},i=\overline{2,m}$.

From condition ( $C_{4}$ ) we get $\varphi(0)=x_{1}^{*}(0)$ and from definition of $A_{i},i=\overline{1,m}$, we have

therefore

is the unique solution in $C([-h,T],\mathbb{B})$.
Now the question is: Can we put an approximation of $x_{i}^{n},i=\overline{1,m}$ instead of $x_{i}^{*}$, $i=\overline{1,m}$ ?

The answer of this question is given by the following theorem:
Theorem 5.2. In the condition of Theorem 5.1, for each $x_{i}^{0}\in C([(i-1)h,ih],\mathbb{B}),i=\overline{1,m-1},x_{m}^{0}\in C([(m-1)h,T],\mathbb{B})$, the sequences defined by: