On a Volterra Integral Equation with Delay, via w-Distances

Abstract

The paper deals with a Volterra integral equation with delay. In order to apply the w-weak generalized contraction theorem for the study of existence and uniqueness of solutions, we rewrite the equation as a fixed point problem. The assumptions take into account the support of w-distance and the complexity of the delay equation. Gronwall-type theorem and comparison theorem are also discussed using a weak Picard operator technique. In the end, an example is provided to support our results.

1. Introduction

The study of Volterra integral equations is an interesting area of research because of their applications in physics, biology, control theory and in other fields of sciences. In the last decades, they have been extensively and intensively studied. Numerous results on existence and uniqueness, monotonicity, stability, as well as numerical solutions have been obtained. To name a few, we refer the reader to [1,2,3,4,5] and the references therein. On the other hand, the results regarding the blow-up of the solutions is among the most attractive topics in qualitative theory of Volterra integral equations due to their applications, especially in biology, economics and physics (see, e.g., [5,6,7,8]). The theory of Volterra integral equations with delay have been studied by many authors (see, e.g., [1,2,9,10,11]). The most common approach in studying the existence of solutions for a Volterra integral equation with delay, is to rewrite (1) as a fixed point problem. Then, one can apply different fixed point principles to the above equation and establish the existence of solutions (see, e.g., [9,10,12,13,14]).

Very recently, many results related to mappings satisfying various contractive conditions and underlying distance spaces were obtained in [9,15,16,17,18,19,20,21] and the references contained therein.

In this paper, we consider a Volterra integral equation with delay of the form

$$y\left(x\right)=\phi \left(x\right)+{\displaystyle {\int }_{a_1}^x}F(x,s,y\left(g\left(s\right)\right))ds$$

(1)

where $a_0,a_1,a_2\in \mathbb{R},a_0<a_1<a_2,$ and $\phi \in C\left([a_1,a_2],\mathbb{R}\right),F\in C\left([a_1,a_2]\times [a_1,a_2]\times \mathbb{R},\mathbb{R}\right)$, $g\in C([a_1,a_2],[a_0,a_2])$ with $g\left(x\right)\le x,y\left(x\right)=\tilde{y}\left(x\right)$ for $x\in [a_0,a_1],$ are given.

In this paper, the motivation of the work has been started from the results of T. Wongyat and W. Sintunavarat [20]. Using w-weak generalized contractions theorem, we give some results in the case of Volterra integral equations with delay. In the end a Gronwall-type theorem and a comparison theorem are also obtained.

2. Preliminaries

For the convenience of the reader we recall here some definitions and preliminary results, for details, see [17,20].

Let $(\mathcal{Y},\rho )$ be a metric space. First we present the notion of w-distance on $\mathcal{Y}$ and $w^0$-distance on $\mathcal{Y}.$

Definition 1

([17]). Let $(\mathcal{Y},\rho )$ be a metric space. A function $d:\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ is called w-distance on $\mathcal{Y},$ if the following conditions are satisfied:

(1) 

$d(y_1,y_2)\le d(y_1,y_3)+d(y_3,y_2),$∀$y_1,y_2,y_3\in \mathcal{Y};$

(2) 

$d(y,·):\mathcal{Y}\to [0,\infty )$ is lower semicontinuous, $y\in \mathcal{Y};$

(3) 

for each $\epsilon >0,$ there exists $\delta >0$ such that $d(y_1,y_2)\le \delta$ and $d(y_1,y_3)\le \delta$ imply $\rho (y_2,y_3)\le \epsilon ,$ ∀ $y_1,y_2,y_3\in \mathcal{Y}$.

It is well known that each metric on a nonempty set $\mathcal{Y}$ is a w-distance on $\mathcal{Y}$.

Definition 2

([20]). A function $\psi :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ is called $w^0$-distance on $\mathcal{Y},$ if it is a w-distance on $\mathcal{Y}$ with $d(y,y)=0$ for all $y\in \mathcal{Y}.$

Next, we give the definitions of an altering distance function, ceiling distance of $\rho$ and w-generalized weak contraction mapping used in the paper [20].

Definition 3

([20]). A function $\psi :[0,\infty )\to [0,+\infty )$ is called an altering distance function, if the following conditions are satisfied:

(1) 

ψ is a continuous and nondecreasing function;

(2) 

$\psi \left(x\right)=0$ if and only if $x=0$.

Definition 4

([20]). A w-distance q on a metric space $(\mathcal{Y},\rho )$ is called a ceiling distance of ρ if and only if $d(y_1,y_2)\ge \rho (y_1,y_2),$ ∀ $y_1,y_2\in \mathcal{Y}.$

Definition 5

([20]). Let d be a w-distance on a metric space $(\mathcal{Y},\rho ).$ An operator $V:\mathcal{Y}\to \mathcal{Y}$ is called a w-generalized weak contraction mapping if

$$\psi \left(d(V\left(y_1\right),V\left(y_2\right))\right)\le \psi \left(m(y_1,y_2)\right)-\varphi \left(d(y_1,y_2)\right),\forall y_1,y_2\in \mathcal{Y},$$

(2)

where

$$m(y_1,y_2):=\max \left\{d(y_1,y_2),\frac{1}{2}\left[d\left(y_1,V\left(y_2\right)\right)+d\left(V\left(y_1\right),y_2\right)\right]\right\},$$

(3)

$\psi :[0,\infty )\to [0,\infty )$ is an altering distance function, and $\varphi :[0,\infty )\to [0,\infty )$ is a continuous function with $\varphi \left(x\right)=0$ if and only if $x=0$. If $d=\rho ,$ then the mapping V is called generalized weak contraction mapping.

Let $(\mathcal{Y},\rho )$ be a complete metric space. We present below some results of fixed point of the operatorial equation $V\left(y\right)=y,y\in \mathcal{Y}$ via w-distances.

Theorem 1

([20]). Let $d:\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ be a $w^0$-distance on $\mathcal{Y}$ and a ceiling distance of ρ. Suppose that $V:\mathcal{Y}\to \mathcal{Y}$ is a continuous w-generalized weak contraction. Then, the operator V has a unique fixed point in $\mathcal{Y}$. Moreover, for each $y_0\in \mathcal{Y}$, the successive approximation sequence ${\left\{y_n\right\}}_{n\in \mathbb{N}},$ defined by $y_n=V^n\left(y_0\right),$ for all $n\in \mathbb{N}$ converges to the unique fixed point of the operator V.

Theorem 2

([20]). Let $d:\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ be a continuous $w^0$-distance on $\mathcal{Y}$ and a ceiling distance of ρ. Suppose that $V:\mathcal{Y}\to \mathcal{Y}$ is a w-generalized weak contraction. Then, the operator V has a unique fixed point in $\mathcal{Y}$. Moreover, for each $y_0\in \mathcal{Y}$, the successive approximation sequence ${\left\{y_n\right\}}_{n\in \mathbb{N}},$ defined by $y_n=V^n\left(y_0\right),$ for all $n\in \mathbb{N}$ converges to the unique fixed point of the operator V.

Theorem 3

([20]). Let $d:\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ be a continuous w-distance on $\mathcal{Y}$ and a ceiling distance of ρ. Suppose that $V:\mathcal{Y}\to \mathcal{Y}$ is a continuous operator such that, for all $y_1,y_2\in \mathcal{Y}$

$$\psi \left(d\left(V\left(y_1\right),V\left(y_2\right)\right)\right)\le \psi \left(m(y_1,y_2)\right)-\varphi \left(d\left(y_1,y_2\right)\right),$$

(4)

where $\psi :[0,\infty )\to [0,\infty )$ is an altering distance function, and $\varphi :[0,\infty )\to [0,\infty )$ is a continuous function with $\varphi \left(x\right)=0$ if and only if $x=0$. Then, the operator V has a unique fixed point in $\mathcal{Y}$. Moreover, for each $y_0\in \mathcal{Y}$, the successive approximation sequence ${\left\{y_n\right\}}_{n\in \mathbb{N}},$ defined by $y_n=V^n\left(y_0\right),$ for all $n\in \mathbb{N}$ converges to the unique fixed point of the operator V.

We now give some definitions and lemmas (see [22,23,24]), which are needed in advance.

Let $(\mathcal{Y},\rho )$ be a metric space. Let us consider a given operator $V:\mathcal{Y}\to \mathcal{Y}$. In this setting, V is called weakly Picard operator (briefly WPO) if, for all $y\in \mathcal{Y}$, the sequence of Picard iterations, ${\left\{V^n\left(y\right)\right\}}_{n\in \mathbb{N}}$, converges in $(\mathcal{Y},\rho )$ and its limit (which generally depend on y) is a fixed point of V. We denote by $F_V$ the fixed point set of V, i.e., $F_V=\{y\in V:V\left(y\right)=y\}$. If an operator V is WPO with $F_V=\left\{y^{*}\right\}$, then, by definition, V is called a Picard operator (briefly PO).

If $V:\mathcal{Y}\to \mathcal{Y}$ is a WPO, we can define the operator $V^{\infty }:\mathcal{Y}\to F_V$, by $V^{\infty }\left(y\right):=\underset{n\to +\infty }{\lim }$$V^n\left(y\right).$

If $\mathcal{Y}$ is a nonempty set, then the triple $(\mathcal{Y},\rho ,\le )$ is an ordered metric space, where ≤ is a partial order relation on $\mathcal{Y}.$

In the setting of ordered metric spaces, we have some properties related to WPOs and POs.

Theorem 4

(Rus [22,23] Characterization theorem). Let $(\mathcal{Y},\rho )$ be a metric space and $V:\mathcal{Y}\to \mathcal{Y}$ an operator. Then V is WPO if and only if there exists a partition of $\mathcal{Y}$, $\mathcal{Y}=$$\underset{\lambda \in \Lambda }{\cup }{\mathcal{Y}}_{\lambda }$, such that

(i) 

${\mathcal{Y}}_{\lambda }\in I\left(V\right)$, for all $\lambda \in \Lambda ;$

(ii) 

${V|}_{{\mathcal{Y}}_{\lambda }}:{\mathcal{Y}}_{\lambda }\to {\mathcal{Y}}_{\lambda }$ is PO, for all $\lambda \in \Lambda .$

Theorem 5

([23] Abstract Gronwall Lemma). Let $(\mathcal{Y},\rho ,\le )$ be an ordered metric space and $V:\mathcal{Y}\to \mathcal{Y}$ be an increasing WPO. Then we have the following:

(j) 

for $y\in \mathcal{Y},y\le V\left(y\right)\Rightarrow y\le V^{\infty }\left(y\right);$

(jj) 

for $y\in \mathcal{Y},y\ge V\left(y\right)\Rightarrow y\ge V^{\infty }\left(y\right).$

Theorem 6

([23] Abstract Comparison Lemma). Let $(\mathcal{Y},\rho ,\le )$ be an ordered metric space and $V_1,V_2,V_3:\mathcal{Y}\to \mathcal{Y}$ be such that:

(h) 

$V_1\le V_2\le V_3;$

(hh) 

the operators $V_1,V_2,V_3$ are WPO;

(hhh) 

the operator $V_2$ is increasing.

Then, for $y_1,y_2,y_3\in \mathcal{Y},y_1\le y_2\le y_3\Rightarrow V_1^{\infty }\left(y_1\right)\le V_2^{\infty }\left(y_2\right)\le V_3^{\infty }\left(y_3\right)$.

For the theory of weakly Picard operators, its generalization and applications, see [9,11,12,14,22,23,24,25,26,27,28].

3. Main Result

Throughout this paper it will be assumed that:

(C₁)

$y\left(x\right)=\tilde{y}\left(x\right),x\in [a_0,a_1];$

(C₂)

$\tilde{y}\left(a_1\right)=\phi \left(a_1\right);$

(C₃)

$\underset{x\in [a_0,a_2]}{sup}y\left(x\right)=\underset{x\in [a_1,a_2]}{sup}y\left(x\right).$

With respect to the Equation (1) we consider the equation (in $\alpha \in \mathbb{R}$)

$$\alpha =\phi \left(a_1\right).$$

(5)

Let $S_{\phi }$ be the solution set of the Equation (5).

Now we consider the operator $V:C([a_0,a_2],\mathbb{R})\to C([a_0,a_2],\mathbb{R})$ defined by

$$V\left(y\right)\left(x\right):=\phi \left(x\right)+{\displaystyle {\int }_{a_1}^x}F(x,s,y\left(g\left(s\right)\right))ds$$

(6)

for all $y\in C([a_0,a_2],\mathbb{R})$ and $x\in [a_1,a_2].$

Let $\mathcal{Y}:=C([a_0,a_2],\mathbb{R})$ and ${\mathcal{Y}}_{\tilde{y}}:=\{y\in \mathcal{Y}|{\left.y\right|}_{[a_0,a_1]}=\tilde{y}\}$. Then

$$\mathcal{Y}=\underset{\tilde{y}\in C[a_0,a_1]}{\cup }{\mathcal{Y}}_{\tilde{y}}$$

is a partition of $\mathcal{Y}$.

Lemma 1.

We suppose that the conditions (C${}_1$), (C${}_2$) and (C${}_3$) are satisfied. Then it is obvious that $V\left(\mathcal{Y}\right)\subset {\mathcal{Y}}_{\tilde{y}}$ and $V\left({\mathcal{Y}}_{\tilde{y}}\right)\subset {\mathcal{Y}}_{\tilde{y}}$.

The main purpose of this section is to prove a new result of the existence, uniqueness and approximation of the solution for nonlinear Volterra integral equation with delay by using Theorem 3.

Theorem 7.

We consider the integral Equation (1) where $a_0,a_1,a_2\in \mathbb{R},a_0<a_1<a_2,$ and $\phi \in C\left([a_1,a_2],\mathbb{R}\right),F\in C\left([a_1,a_2]\times [a_1,a_2]\times \mathbb{R},\mathbb{R}\right),g\in C([a_1,a_2],[a_0,a_2])$ with $g\left(x\right)\le x,$ are given functions. We suppose the following:

(i) 

the mapping $V:C([a_0,a_2],\mathbb{R})\to C([a_0,a_2],\mathbb{R})$ defined by (6) is continuous;

(ii) 

the altering distance function $\psi :[0,\infty )\to [0,\infty )$ satisfies $\psi \left(x\right)<x,$ for all $x>0,$ and the continuous function $\varphi :[0,\infty )\to [0,\infty )$ satisfies $\varphi \left(x\right)=0,$ if and only if $x=0;$

(iii) 

$$\begin{array}{cc}\hfill & \left|F(x,s,y_1\left(g\left(s\right)\right))\right|+\left|F(x,s,y_2\left(g\left(s\right)\right))\right|\le \hfill \\ \hfill & \le \frac{1}{a_2-a_1}\left[\psi \left(\left|y_1\left(g\left(s\right)\right)\right|+\left|y_2\left(g\left(s\right)\right)\right|\right)-\varphi \left(\underset{t\in [a_0,a_2]}{\sup }\left|y_1\left(g\left(t\right)\right)\right|+\underset{t\in [a_0,a_2]}{\sup }\left|y_2\left(g\left(t\right)\right)\right|\right)-2\left|\phi \left(x\right)\right|\right],\hfill \end{array}$$

for all $y_1,y_2\in C([a_0,a_2],\mathbb{R}),x,s\in [a_1,a_2].$

Then the integral Equation (1) has a unique solution;

Moreover, for each $y_0\in C([a_0,a_2],\mathbb{R}),$ the sequence of Picard iterations ${\left\{y_n\right\}}_{n\in \mathbb{N}}$, defined by $y_n=V^n\left(y_0\right),$ for all $n\in \mathbb{N}$, converges to the unique solution of the integral Equation (1).

Proof. 

Let $\mathcal{Y}=C([a_0,a_2],\mathbb{R})$ and we consider the metric $\rho :\mathcal{Y}\times \mathcal{Y}\to [0,\infty )$ given by $\rho (y_1,y_2):=\underset{x\in [a_1,b]}{\sup }\left|y_1\left(x\right)-y_2\left(x\right)\right|$, for all $y_1,y_2\in C([a_0,a_2],\mathbb{R}).$ It is clear that $(\mathcal{Y},\rho )$ is a complete metric space. Now, we define the function $d:C([a_0,a_2],\mathbb{R})\times C([a_0,a_2],\mathbb{R})\to [0,\infty )$ by the relation:

$$d(y_1,y_2):=\underset{x\in [a_0,a_2]}{\sup }\left|y_1\left(x\right)\right|+\underset{x\in [a_0,a_2]}{\sup }\left|y_2\left(x\right)\right|,\mathrm{for}\mathrm{all}{y}_1,y_2\in C([a_0,a_2],\mathbb{R})$$

(7)

and it is easy to see that d is a w-distance on $C([a_0,a_2],\mathbb{R})$ and a ceiling distance of $\rho$.

We intend to show that the operator V satisfies the condition (4). We have

$$\begin{array}{cc}\hfill & \left|V\left(y_1\right)\left(x\right)\right|+\left|V\left(y_2\right)\left(x\right)\right|\hfill \\ \hfill & =\left|\phi \left(x\right)+{\displaystyle {\int }_{a_1}^x}F(x,s,y_1\left(g\left(s\right)\right))ds\right|+\left|\phi \left(x\right)+{\displaystyle {\int }_{a_1}^x}F(x,s,y_2\left(g\left(s\right)\right))ds\right|\hfill \\ \hfill & \le \left|\phi \left(x\right)\right|+\left|{\displaystyle {\int }_{a_1}^x}F(x,s,y_1\left(g\left(s\right)\right))ds\right|+\left|\phi \left(x\right)\right|+\hfill \\ \hfill & +\left|{\displaystyle {\int }_{a_1}^x}F(x,s,y_2\left(g\left(s\right)\right))ds\right|\hfill \\ \hfill & \le 2\left|\phi \left(x\right)\right|+{\displaystyle {\int }_{a_1}^x}\left(\left|F(x,s,y_1\left(g\left(s\right)\right))\right|+\left|F(x,s,y_2\left(g\left(s\right)\right))\right|\right)ds\hfill \\ \hfill & \le 2\left|\phi \left(x\right)\right|+\frac{1}{a_2-a_1}{\displaystyle {\int }_{a_1}^x}\left[\psi \left(\left|y_1\left(g\left(s\right)\right)\right|+\left|y_2\left(g\left(s\right)\right)\right|\right)\right.\hfill \\ \hfill & \left.-\varphi \left(\underset{t\in [a_0,a_2]}{\sup }\left|y_1\left(g\left(t\right)\right)\right|+\underset{t\in [a_0,a_2]}{\sup }\left|y_2\left(g\left(t\right)\right)\right|\right)-2\left|\phi \left(x\right)\right|\right]ds\hfill \\ \hfill & \le \psi \left(d(y_1,y_2)\right)-\varphi \left(d(y_1,y_2)\right).\hfill \end{array}$$

This implies that

$$\underset{x\in [a_0,a_2]}{\sup }\left|V{y}_1\left(x\right)\right|+\underset{x\in [a_0,a_2]}{\sup }\left|V{y}_2\left(x\right)\right|\le \psi \left(d(y_1,y_2)\right)-\varphi \left(d(y_1,y_2)\right).$$

This leads to

$$d(V{y}_1,V{y}_2)\le \psi \left(m(y_1,y_2)\right)-\varphi \left(d(y_1,y_2)\right),\mathrm{for}\mathrm{all}{y}_1,y_2\in \mathcal{Y}.$$

Furthermore, we have

$$\psi \left(d(V{y}_1,V{y}_2)\right)\le d(V{y}_1,V{y}_2)\le \psi \left(d(y_1,y_2)\right)-\varphi \left(d(y_1,y_2)\right),\mathrm{for}\mathrm{all}{y}_1,y_2\in \mathcal{Y}.$$

We obtain that V satisfies the condition (4) and thus V is a Picard operator. This implies that there exists a unique solution of the integral Equation (1). □

Since the operator V defined in (6) is a PO, we can establish the following Gronwall-type lemma for the Equation (1).

Theorem 8.

We consider the integral Equation (1) where $a_1,a_2\in \mathbb{R},a_1<a_2,$ and the functions $\phi \in C\left([a_1,a_2],\mathbb{R}\right),F\in C\left([a_1,a_2]\times [a_1,a_2]\times \mathbb{R},\mathbb{R}\right),g\in C([a_1,a_2],[a_0,a_2])$ with $g\left(x\right)\le x$, are given. We assume that the conditions (i)–(iii) from Theorem 7 hold. Furthermore, we suppose that

(iv) 

$F(x,s,·):\mathbb{R}\to \mathbb{R}$ is an increasing function with respect to the last argument, for all $x,s\in [a_1,a_2].$

Let $y^{*}\in C([a_0,a_2],\mathbb{R})$ be the unique solution of the system. Then, the following implications hold:

(1) 

for all $y\in C([a_0,a_2],\mathbb{R})$ with

$$y\left(x\right)\le \phi \left(x\right)+{\displaystyle {\int }_{a_1}^x}F(x,s,y\left(g\left(s\right)\right))ds,$$

for all $y\in [a_1,a_2]$, we have $y\le y^{*};$

(2) 

for all $y\in C([a_0,a_2],\mathbb{R})$ with

$$y\left(x\right)\ge \phi \left(x\right)+{\displaystyle {\int }_{a_1}^x}F(x,s,y\left(g\left(s\right)\right))ds,$$

for all $x\in [a_1,a_2]$, we have $y\ge y^{*}.$

Proof. 

From (iv), we have that the operator V defined in (6) is increasing with respect to the partial order.

By the proof of Theorem 7, it follows that V is a Picard operator. The conclusion of the theorem follows from Theorem 5. □

In a similar way, a comparison theorem for Equation (1) can be obtained, using the abstract comparison theorem given in Section 2 of this paper.

Theorem 9.

We consider the integral Equation (1) where $a_1,a_2\in \mathbb{R},a_1<a_2,$ and the functions ${\phi }_i\in C\left([a_1,a_2],\mathbb{R}\right),F_i\in C\left([a_1,a_2]\times [a_1,a_2]\times \mathbb{R},\mathbb{R}\right)$ and $g_i\in C([a_1,a_2],[a_0,a_2]),i=1,2,3$ are given. We assume that the conditions (i)–(iii) from Theorem 7 hold. Furthermore, we suppose that

(i) 

${\phi }_1\le {\phi }_2\le {\phi }_3,F_1\le F_2\le F_3,g_1\le g_2\le g_3$;

(ii) 

${\phi }_2,F_2,g_2$ are increasing;

(iii) 

$S_{{\phi }_1}=S_{{\phi }_2}=S_{{\phi }_3}.$

Let $y_i\in C([a_1,b],\mathbb{R})$ be a solution of the equation

$$y_i\left(x\right)={\phi }_i\left(x\right)+{\displaystyle {\int }_{a_1}^x}{F}_i(x,s,y\left(g_i\left(s\right)\right))ds,x\in [a,b],i=1,2,3.$$

If $y_1\left(x\right)\le y_2\left(x\right)\le y_3\left(x\right)$, $x\in [a_1,a]$ then $y_1\left(x\right)\le y_2\left(x\right)\le y_3\left(x\right),x\in [a_1,a_2].$

Proof. 

The proof follows from the Theorem 6. □

Next we study the existence and uniqueness of solutions of the following integral equation using Theorem 7.

Example

We consider the integral equation

$$y\left(x\right)=\frac{x}{4}+{\displaystyle {\int }_0^x}x{s}^{2}y\left(\lambda s\right)ds,$$

(8)

where $x\in [0,1],\lambda \in [0,1]$, and the following condition

$$\begin{array}{cc}\hfill \left|x{s}^2{y}_1\left(\lambda x\right)\right|+\left|x{s}^2{y}_2\left(\lambda x\right)\right|& \le \frac{1}{2}\left(\left|y_1\left(\lambda x\right)\right|+\left|y_2\left(\lambda x\right)\right|\right)-\frac{1}{2}x.\hfill \end{array}$$

Now let $\mathcal{Y}=C[0,1]$ with the metric $\rho :X\times X\to [0,\infty )$ given by

$$\rho (y_1,y_2)=\underset{x\in [0,1]}{\sup }\left|y_1\left(x\right)-y_2\left(x\right)\right|,$$

for all $y_1,y_2\in \mathcal{Y}$.

It is clear that $(\mathcal{Y},\rho )$ is a complete metric space. For all $y_1,y_2\in \mathcal{Y},$ the function

$$d(y_1,y_2)=\underset{x\in [0,1]}{\sup }\left|y_1\left(x\right)\right|+\underset{x\in [0,1]}{\sup }\left|y_2\left(x\right)\right|$$

is a w-distance on $\mathcal{Y}$ and a ceiling distance of $\rho$. Next, we define the operator $V:\mathcal{Y}\to \mathcal{Y}$, defined by

$$V\left(y\right)\left(x\right)=\frac{x}{4}+{\displaystyle {\int }_0^x}x{s}^{2}y\left(\lambda s\right)ds,\mathrm{for}\mathrm{all}y\in \mathcal{Y}.$$

The functions $\psi$,$\varphi :[0,\infty )\to [0,\infty )$ defined by $\psi \left(x\right)={\displaystyle \frac{x}{2}}$ and $\varphi \left(x\right)={\displaystyle \frac{x}{4}}$ verify that $\psi \left(x\right)<x$ for all $x>0$ and $\varphi \left(x\right)<\psi \left(x\right),$ for all $x>0.$

Thus

$$\begin{array}{cc}\hfill \left|V\left(y_1\right)\left(x\right)\right|+\left|V\left(y_2\right)\left(x\right)\right|& =\left|\frac{x}{4}+{\displaystyle {\int }_0^x}x{s}^2{y}_1\left(\lambda s\right)ds\right|+\left|\frac{x}{4}+{\displaystyle {\int }_0^x}x{s}^2{y}_2\left(\lambda s\right)ds\right|\hfill \\ \hfill & \le \frac{x}{2}+{\displaystyle {\int }_0^x}\left|x{s}^2\right|\left[\left|y_1\left(\lambda s\right)\right|+\left|y_2\left(\lambda s\right)\right|\right]ds\hfill \\ \hfill & \le \frac{x}{2}+\frac{1}{2}\left(\left|y_1\left(\lambda x\right)\right|+\left|y_2\left(\lambda x\right)\right|\right)-\frac{x}{2}\hfill \\ \hfill & \le \psi \left(d(y_1,y_2)\right)-\varphi \left(d(y_1,y_2)\right).\hfill \end{array}$$

From this we get

$$\underset{x\in [0,1]}{\sup }\left|V\left(y_1\right)\left(x\right)\right|+\underset{x\in [0,1]}{\sup }\left|V\left(y_2\right)\left(x\right)\right|\le \psi \left(d(y_1,y_2)\right)-\varphi \left(d(y_1,y_2)\right).$$

We obtain that

$$d(V\left(y_1\right),V\left(y_2\right))\le \psi \left(d(y_1,y_2)\right)-\varphi \left(d(y_1,y_2)\right),\mathrm{for}\mathrm{all}{y}_1,y_2\in \mathcal{Y}.$$

Finally,

$$\psi \left(d(V\left(y_1\right),V\left(y_2\right))\right)\le d(V\left(y_1\right),V\left(y_2\right))\le \psi \left(d(y_1,y_2)\right)-\varphi \left(d(y_1,y_2)\right),\mathrm{for}\mathrm{all}{y}_1,y_2\in \mathcal{Y}.$$

Hence, by Theorem 7, V has a unique fixed point and we conclude that Equation (8) has a unique solution.

4. Conclusions

In this paper, we have investigated a Volterra integral equation with delay. Using w-weak generalized contractions theorem and the assumptions (C${}_1$)–(C${}_3$), we obtain an existence and uniqueness result, a Gronwall-type theorem and a comparison theorem for Equation (1). We employed the Picard operator method, fixed point theorems and abstract Gronwall lemma, to obtain our results. In the end, an example is presented. The theorems obtained in this paper are also applicable to systems of integral equations with delay. As for a future study, several numerical examples can be taken and a comparative study with previously published results or theory can be done.