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

: 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.


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]).
In this paper, we consider a Volterra integral equation with delay of the form F(x, s, y(g(s)))ds where a 0 , a 1 , a 2 ∈ R, a 0 < a 1 < a 2 , and ϕ ∈ C([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.

Preliminaries
For the convenience of the reader we recall here some definitions and preliminary results, for details, see [17,20].
Let (Y, ρ) be a metric space. First we present the notion of w-distance on Y and w 0 -distance on Y.
if the following conditions are satisfied: It is well known that each metric on a nonempty set Y is a w-distance on Y.
Next, we give the definitions of an altering distance function, ceiling distance of ρ and w-generalized weak contraction mapping used in the paper [20].
Suppose that V : Y → Y is a continuous w-generalized weak contraction. Then, the operator V has a unique fixed point in Y. Moreover, for each y 0 ∈ Y, the successive approximation sequence {y n } n∈N , defined by y n = V n (y 0 ), for all n ∈ N converges to the unique fixed point of the operator V.
be a continuous w 0 -distance on Y and a ceiling distance of ρ. Suppose that V : Y → Y is a w-generalized weak contraction. Then, the operator V has a unique fixed point in Y. Moreover, for each y 0 ∈ Y, the successive approximation sequence {y n } n∈N , defined by y n = V n (y 0 ), for all n ∈ N converges to the unique fixed point of the operator V.
Then, the operator V has a unique fixed point in Y. Moreover, for each y 0 ∈ Y, the successive approximation sequence {y n } n∈N , defined by y n = V n (y 0 ), for all n ∈ 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 (Y, ρ) be a metric space. Let us consider a given operator V : Y → Y. In this setting, V is called weakly Picard operator (briefly WPO) if, for all y ∈ Y, the sequence of Picard iterations, {V n (y)} n∈N , converges in (Y, ρ) 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., In the setting of ordered metric spaces, we have some properties related to WPOs and POs.
Let S ϕ be the solution set of the Equation (5). Now we consider the operator V : F(x, s, y(g(s)))ds (6) for all y ∈ C([a 0 , a 2 ], R) and x ∈ [a 1 , Y y is a partition of Y.

Lemma 1.
We suppose that the conditions (C 1 ), (C 2 ) and (C 3 ) are satisfied. Then it is obvious that V(Y ) ⊂ Y y and V(Y y ) ⊂ Y 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 Gronwalltype lemma for the Equation (1). (1) where a 1 , a 2 ∈ R, a 1 < a 2 , and the functions 1 , a 2 ], [a 0 , a 2 ]) with g(x) ≤ x, are given. We assume that the conditions (i)-(iii) from Theorem 7 hold. Furthermore, we suppose that (iv) F(x, s, ·) : R → R is an increasing function with respect to the last argument, for all
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 ∈ R, a 1 < a 2 , and the functions , [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 Let y i ∈ C([a 1 , b], R) be a solution of the equation 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.
Hence, by Theorem 7, V has a unique fixed point and we conclude that Equation (8) has a unique solution.

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.