We consider the system of delay differential equations

with initial conditions

where $f_{1},f_{2}\in C([a,b]\times\mathbb{R}^{4},\mathbb{R}),\ \varphi_{1},\varphi_{2}\in C([a-h,a],\mathbb{R})$ and $h>0$ is a parameter. We denote by$\ \mathbf{x=(}x_{1},x_{2}\mathbf{),\ f}=(f_{1},f_{2})$ and$\ \boldsymbol{\varphi}=(\varphi_{1},\varphi_{2}).\$By a solution of the problem (1)-(2) we mean a function $\mathbf{x}\in C([a-h,b],\mathbb{R}^{2})\cap C^{1}([a,b],\mathbb{R}^{2})$ which satisfies the system (1) and the conditions (2).

In this paper we study this problem using the ideas of I.A. Rus [14] to obtain existence, uniqueness theorems and the convergence of an iterative algorithm using Perov’s theorem, fibre contraction principle and step method. As an application, we consider a second order functional differential equation with delay and we approximate the solution using the Chebyshev spectral method (see [18, 19, 20, 21]). We compare the obtained results with Matlab dde23 procedure.

Such kind of results have been proved in [15] and [7] in the case of integro-differential equations with lags and in [2] in the case of an integral equation from biomathematics. Other results regarding efficient and rapidly convergent algorithms for solving Volterra differential and integral equations can be found in [4, 5, 8].

Let $(X,d)$ be a metric space and $A:X\rightarrow X$ an operator. In this paper we 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 $A$ and by $F_{A}:=\left\{x\in X|\ A(x)=x\right\}$ the fixed point set of $A.$

As concerns matrices which are convergent to zero, we mention the following equivalent characterizations:

The matrices convergent to zero were used by Perov [9] to generalize the contraction principle in the case of generalized metric spaces with the metric taking values in the positive cone of $\mathbb{R}^{2}.$

Finally, we recall the following result that is a generalization of the fibre contraction theorem (see I.A. Rus [12], [2]):

We begin this section with an existence theorem for the solution of the problem (1)-(2). We denote by $\left\|\cdot\right\|:\mathbb{R}^{2}\rightarrow\mathbb{R}_{+}^{2}$ the vectorial norm

Relative to the problem (1)-(2) we consider the following conditions:

• (H₁)

  $\mathbf{f}\in C([a,b]\times\mathbb{R}^{4},\mathbb{R}^{2}),\ \boldsymbol{\varphi}\in C([a-h,a],\mathbb{R}^{2}),\ h\in\mathbb{R}_{+}^{\ast},\ a,b\in\mathbb{R},\ a<b;$

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• (H₂)

  there exists $L\in\mathbb{R}_{+}^{2\times 2}$ such that

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  $$\left\|\mathbf{f}(t,\mathbf{u}^{1},\mathbf{v}^{1})-\mathbf{f}(t,\mathbf{u}^{2},\mathbf{v}^{2})\right\|\leq L(\left\|\mathbf{u}^{1}-\mathbf{u}^{2}\right\|+\left\|\mathbf{v}^{1}-\mathbf{v}^{2}\right\|),$$

  $t\in[a,b],\mathbf{u}^{1},\mathbf{u}^{2},\mathbf{v}^{1},\mathbf{v}^{2}\in\mathbb{R}^{2};$

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• (H${}_{2}^{\prime}$)

  there exists $L^{\prime}\in\mathbb{R}_{+}^{2\times 2}$ such that

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  $$\left\|\mathbf{f}(t,\mathbf{u}^{1},\mathbf{v})-\mathbf{f}(t,\mathbf{u}^{2},\mathbf{v})\right\|\leq L^{\prime}(\left\|\mathbf{u}^{1}-\mathbf{u}^{2}\right\|),$$

  $t\in[a,b],\mathbf{u}^{1},\mathbf{u}^{2},\mathbf{v}\in\mathbb{R}^{2}.$

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We consider the space $X:=C([a-h,b],\mathbb{R}^{2})$ endowed with the generalized norm $\left\|\cdot\right\|_{B}$ where $\left\|\mathbf{x}\right\|_{B}:=\binom{\left|x_{1}\right|_{B}}{\left|x_{2}\right|_{B}},\ \mathbf{x}=(x_{1},x_{2})$ and

It is clear that the space $(X,\left\|\cdot\right\|_{B})$ is a generalized Banach space. Any solution of the problem (1)-(2) is a fixed point of the operator $A_{f}:X\rightarrow X$ , defined by

Let $m\in\mathbb{N}^{\ast}$ be such that:

We denote by $t_{-1}:=a-h,\ t_{0}:=a,\ t_{i}:=a+ih,\ i=\overline{1,m-1},\ t_{m}:=b.$

The following result is well known (see [6]).

Delay differential equations may be solved as ordinary differential equations over successive intervals $[t_{m},t_{m+1}]$ by the step method (see, for example [3] or [1]).

Under the condition (H₁), the step method for the problem (1)-(2) consists of the following equations:

• $(p^{0})$

  $\mathbf{x}^{0}(t)=\boldsymbol{\varphi}(t),t\in[t_{-1},t_{0}];$

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• $(p^{1})$

  $\mathbf{x}^{1}(t)=\boldsymbol{\varphi}(t_{0})+\textstyle\int\nolimits_{t_{0}}^{t}\mathbf{f}(s,\mathbf{x}^{1}(s),\boldsymbol{\varphi}(s-h))ds,\ t\in[t_{0},t_{1}];$

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• $(p^{2})$

  $\mathbf{x}^{2}(t)=\mathbf{x}^{1,\ast}(t_{1})+\textstyle\int\nolimits_{t_{1}}^{t}\mathbf{f}(s,\mathbf{x}^{2}(s),\mathbf{x}^{1,\ast}(s-h))ds,\ t\in[t_{1},t_{2}];$

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  $\cdots$

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• $(p^{m-1})$

  $\mathbf{x}^{m-1}(t)=\mathbf{x}^{m-2,\ast}(t_{m-2})+\textstyle\int\nolimits_{t_{m-2}}^{t}\mathbf{f}(s,\mathbf{x}^{m-1}(s),\mathbf{x}^{m-2,\ast}(s-h))ds,\ t\in[t_{m-2},t_{m-1}];$

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• $(p^{m})$

  $\mathbf{x}^{m}(t)=\mathbf{x}^{m-1,\ast}(t_{m-1})+\textstyle\int\nolimits_{t_{m-1}}^{t}\mathbf{f}(s,\mathbf{x}^{m}(s),\mathbf{x}^{m-1,\ast}(s-h))ds,\ t\in[t_{m-1},t_{m}];$

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where $\mathbf{x}^{i,\ast}=(x_{1}^{i,\ast},x_{2}^{i,\ast})\in C([t_{i-1},t_{i}],\mathbb{R}^{2})$ is the unique solution of the equation $(p^{i}),\ i=\overline{1,m}.$

So, by using the step method and an idea from [14], we obtain:

Next, we will study if it is possible to replace $\mathbf{x}^{i,\ast}$ by the approximation $\mathbf{x}^{i,n},\ i=\overline{1,m}$ in the conclusion $(ii)$ of the Theorem 2.3. Applying the results from [14] we have

We consider the following second order delay differential equation

with initial conditions

where $f:[a,b]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R},\ h\in\mathbb{R}_{+}^{\ast},\ a,b\in\mathbb{R},\ a<b,$ $\varphi,\varphi^{\prime}:[a-h,a]\rightarrow\mathbb{R}.$

The problem (5)-(6) can be written in the following form

with initial conditions

where $\mathbf{y:}=\left(\begin{array}[]{c}y_{1}\\ y_{2}\end{array}\right)=\left(\begin{array}[]{c}x\\ x^{\prime}\end{array}\right)$, $\mathbf{F}\in C([a,b]\times\mathbb{R}^{2},\mathbb{R}^{2})$, $\mathbf{F:=}\left(\begin{array}[]{c}f_{1}\\ f_{2}\end{array}\right)=\left(\begin{array}[]{c}y_{2}\\ -f\end{array}\right)$, $\boldsymbol{\varphi}\in C([a-h,a],\mathbb{R}^{2})$, $\boldsymbol{\varphi}:=\left(\begin{array}[]{c}\varphi_{1}\\ \varphi_{2}\end{array}\right)=\left(\begin{array}[]{c}\varphi\\ \varphi^{\prime}\end{array}\right)$ and $h>0$ is a parameter.

Relative to the problem (7)-(8) we consider the following conditions:

• (C₁)

  $f\in C([a,b]\times\mathbb{R}^{2},\mathbb{R}),\ \varphi\in C^{1}([a-h,a],\mathbb{R});$

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• (C₂)

  there exists $l_{f}\in\mathbb{R}_{+}$ such that

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  $$\left|f(t,u_{1},v)-f(t,u_{2},v)\right|\leq l_{f}\left|u_{1}-u_{2}\right|,$$

  $t\in[a,b],u_{1},u_{2},v\in\mathbb{R}.$

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Let $m\in\mathbb{N}^{\ast}$ be such that:

We denote by $t_{-1}:=a-h$, $\ t_{0}:=a$, $\ t_{i}:=a+ih$, $\ i=\overline{1,m-1}$, $t_{m}:=b$.

Applying the results from Section 2 we have the following theorems.