EXISTENCE THEORY AND QUALITATIVE PROPERTIES OF SOLUTIONS TO DOUBLE DELAY INTEGRAL EQUATIONS

In this work, we are concerned with nonlinear integral equations with two constant delays. According to the behavior of the data functions, existence and uniqueness results of a measurable solution, an exponentially stable solution, a bounded solution and an integrable solution are provided.

Equations of the type (1.1) are typical in the mathematical modeling of age structured populations in which, for example, the growth of two sizes of the same population is considered (see [1,2,5,6]).In this case τ 1 and τ 2 represent the maturation and the maximal age, respectively.

B(t) =
We refer to [5,8] for the meaning of all the data functions.
The unknown S in (1.2) can be transformed, under some initial conditions (see [8]) to a solution of the following double delay integral equation µ(a − t + σ, S(σ))da φ (S(σ)) S(σ)dσ, (1.3) where φ is a nonnegative decreasing function which is responsible for the reduction of fertility by crowding effect, a + and a m are the maximum and the maturation age of the considered population.
(1.4) EJQTDE, 2013 No. 56, p. 2 We note that if y is a solution of (1.4), then x(t) = y(t) − y(t − (τ − δ)) is a solution of the following double delay integral equation Also, if β is periodic with period τ − δ, then y is a solution of (1.5).
To our knowledge, there are a few papers concerning the existence and the uniqueness of the solution of (1.1).E. Messina et al. (see [7,8,9]) studied the existence and the uniqueness of the continuous solution of the following integral equation where the functions g and k are continuous and the function h satisfies the Lipschitz condition.
However, many physical and biological models include data functions, which are discontinuous.For this reason, we devote our investigations, here, to extend the theory developed in [7,8,9] to study the existence and the uniqueness of a solution of (1.1), under simple and convenient conditions on the data functions, in more general spaces.
The paper is organized as follows.In Section 3, we prove a general existence principle.Section 4 is devoted to proving existence and uniqueness of a locally bounded solution, an exponentially stable solution and a bounded solution.In Section 5, we show existence and uniqueness of a locally integrable solution and an integrable solution.Finally, existence and uniqueness results of the solution of double delay convolution integral equations are discussed in Section 6.

Notations and some auxiliary facts
In this section, we provide some notations, definitions and auxiliary facts which will be needed for stating our results.
Denote by L 1 (R + ) the set of all Lebesgue integrable functions on R + , endowed with the standard norm Let F(R + , R) be the set of all measurable functions from a subset of R + to R.
Let f : R + × R −→ R be a measurable function.We define the operator The operator N f is said to be the Nemytskii operator associated to the function f .
Let k : [τ 2 , +∞) × R + −→ R be a given measurable function.We define the linear operator K on F(R + , R) by the formula Let E ⊂ F(R + , R) be a vectorial space satisfying the following property: belongs to E.
We note that the spaces Loc (R + ) satisfy the property ( * ).

Existence of a measurable Solution
Let E ⊂ F(R + , R) be a vectorial space satisfying the property ( * ).
Theorem 3.1.Suppose that the following conditions are satisfied: (ii) f : R + × R −→ R is a measurable function such that the Nemytskii operator Then Problem (1.1) has a unique measurable solution defined on R + .
Proof.It is clear that there exists a unique integer r ≥ 1 such that rτ 1 ≤ τ 2 < (r + 1)τ 1 .We define the function x : R + −→ R as follows: x = x n on the interval EJQTDE, 2013 No. 56, p. 5 [0, (r + n)τ 1 ) for n ≥ 1 such that and for n ≥ 2 We will prove that the sequence (x n ) is well defined and x n ∈ E for all n ≥ 1.
2) Assume that x n−1 ∈ E for n ≥ 2, hence by the definition of x n , we get . Moreover, by the assumptions of Theorem 3.1, we deduce that Then, by the property ( * ), we get x n ∈ E.
Thus the sequence (x n ) is well defined and x n ∈ E for all n ≥ 1, therefore the function x is measurable and defined on R + .Now, we will prove that x is a solution of (1.1).
For the uniqueness, let y be a solution of (1.1) on R + , we will prove that x = y by the following induction.
Then Problem (1.1) has a unique measurable solution defined on R + .
Remark 3.2.Under the conditions of Theorem 3.1, the solution x need not be in the space E as in the following counterexample.

Existence of an Exponentially Stable Solution
We will need the following lemma.
Lemma 4.1.Suppose that the following conditions are satisfied: where the sequence (x n ) is defined by (3.1) and (3.2).Moreover, the sequence Thus Problem (1.1) has a unique solution in L ∞ Loc (R + ).
The following result gives a sufficient condition on k so that the operator K transforms the space L ∞ Loc (R + ) into itself.
In the sequel, we will utilize the following definition.The following result gives the existence of an exponentially stable solution of Problem (1.1).
Remark 4.7.If we replace the expression "exponentially stable" by "bounded" in the assumptions (i) and (ii) of Theorem 4.5 and by setting γ = γ 1 = γ 2 = 0 in the proof, we obtain a unique bounded solution of (1.1).
Before state the second result, we need the following lemma.
[3](Discrete Gronwall's inequality) Assume that (α n ) n≥1 and (q n ) n≥1 are given non-negative sequences and the sequence (ε n ) n≥1 satisfies Theorem 4.9.Suppose that the following conditions are satisfied: (ii) f : R + × R −→ R is a measurable function and there exist a constant b and an exponentially stable function a : R + −→ R such that |f (t, x)| ≤ a(t)+b|x| for all t ∈ R + and x ∈ R.
Then Problem (1.1) has a unique solution x ∈ L ∞ Loc (R + ).Moreover, there exist γ > 0, λ ≥ 0 and β ≥ 0 such that for all t ∈ R + , Proof.We have, by the assumption (iii), for all α ≥ τ 2 and for all t ∈ [τ 2 , α] Then, by Proposition 4.2, the operator K transforms L ∞ Loc (R + ) into itself, hence from the above assumptions, we deduce by Lemma 4.1, that Problem (1.1) has a unique solution x ∈ L ∞ Loc (R + ).Moreover, the solution is given by the following iteration: x = x n on the interval [0, (r + n)τ 1 ), n ≥ 1 such that and for n ≥ 2 On the other hand, there exist γ 1 , γ 2 > 0 such that |g(t)|e γ1t ∈ L ∞ (R + ) and where h(s)ds.
Remark 4.10.1) If h ∈ L 1 (R + ) we deduce, by the inequality (4.1), that the solution is exponentially stable.
2) If we replace the expression "exponentially stable" by "bounded" in the assumptions (i) and (ii) of Theorem 4.9, then, by setting γ = γ 1 = γ 2 = 0 in the proof, we obtain the inequality (4. then, by Theorem 4.9, Problem (1.1) has a unique exponentially stable solution.

Existence of an integrable Solution
Arguing as in Lemma 4.1, we deduce the following result.
Lemma 5.1.Suppose that the following conditions are satisfied: The following result gives a sufficient condition on k so that the operator K transforms the space L 1 Loc (R + ) into itself.If the function k ∈ L ∞ Loc (R + ), then the operator K transforms the space L  EJQTDE, 2013 No. 56, p. 17
is clear that E satisfies the property ( * ) and contains the functions Φ and g.Moreover, the operators K and N f transform the space E into itself.Then, by Theorem 3.1, Problem (3.3) has a unique measurable solution x defined on R + by (3.1) and (3.2).Hence, for all t ∈ [τ 2 , (r + 1)τ 1 ), x(t) = x 1 (t) = 1 2013 No. 56, p. 8 (ii) f : R + × R −→ R is a measurable function and there exist a constant b anda function a ∈ L ∞ Loc (R + ) such that |f (t, x)| ≤ a(t) + b|x| for all t ∈ R + and x ∈ R. (iii) k : [τ 2 , +∞) × R + −→ R is a measurable function and the linear integral operator K transforms the space L ∞ Loc (R + ) into itself.Then Problem (1.1) has a unique solution in L ∞ Loc (R + ).Proof.We have the vectorial space L ∞ Loc (R + ) verifies the property ( * ) and the functions Φ, g ∈ L ∞ Loc (R + ).Moreover, the assumption (ii) guarantees that the Nemytskii operator N f transforms the space L ∞ Loc (R + ) into itself.Additionally to the assumption (iii), we deduce, by Theorem 3.1, that Problem (1.1) has a unique measurable solution x on R + defined by x = x n on [0, (r + n)τ 1 ) for n ≥ 1,
|dt and by L ∞ (R + ) the set of all bounded functions on R + , endowed with the normx L ∞ (R + ) = ess sup{|x(t)|, t ∈ R + }.Also, denote by L 1Loc (R + ) the set of all Lebesgue integrable functions on any compact set of R + and by L ∞ Loc (R + ) the set of all bounded functions on any compact set of R + .