Qualitative properties of a functional differential equation

The aim of this paper is to discuss some basic problems (existence and uniqueness, data dependence) of the fixed point theory for a functional differential equation with an abstract Volterra operator. In the end an application is given.


Introduction
It is well-known that differential equations appear in mathematical models of various phenomena in physics, economy, biology, engineering, and other fields of science.Many illustrative examples of such models can be found in the literature (see, e.g., [1,[5][6][7][8] and the references therein).
The present paper is motivated by a recent paper [9] where the author studied a differential equation with abstract Volterra operator of the form The aim of our paper is to apply the technique from [2-4, 13, 14] to a functional differential equation that includes an abstract Volterra operator.
The equation involving abstract Volterra operators have been investigated by many authors.The results on the existence and uniqueness, continuous dependence of solutions of Cauchy's problem and even more specialized topics can be found in [2,9,14] and the references therein.
The novelty of our paper consist in applying the weakly Picard operators technique for an equation written as a sum of two operators.
The paper is organized as follows.In Section 2, we recall some definitions and results concerning the weakly Picard operator theory.In Section 3 we prove first the existence and uniqueness theorem and then we obtain some properties regarding the data dependence of the solution.In the last section an example is given.

Preliminaries
In this section we will use the terminologies and notations extracted from [10][11][12].For the convenience of the reader some of them are recalled below.
Let (X, d) be a metric space and A : X → X an operator.We denote by: Definition 2.1.Let (X, d) be a metric space.An operator A : X → X is a Picard operator (PO) if there exists x * ∈ X such that: (i) F A = {x * }; (ii) the sequence (A n (x 0 )) n∈N converges to x * for all x 0 ∈ X.
Definition 2.2.Let (X, d) be a metric space.An operator A : X → X is a weakly Picard operator (WPO) if the sequence (A n (x)) n∈N converges for all x ∈ X, and its limit (which may depend on x) is a fixed point of A.

Definition 2.3.
If A is weakly Picard operator then we consider the operator A ∞ defined by The following results are very useful in the sequel.
Lemma 2.5.Let (X, d, ≤) be an ordered metric space and A : X → X an operator.We suppose that: (i) A is WPO; (ii) A is increasing.
Then, the operator A ∞ is increasing.
Lemma 2.6 (Abstract Gronwall lemma).Let (X, d, ≤) be an ordered metric space and A : X → X an operator.We suppose that: (i) A is WPO; (ii) A is increasing.

If we denote by x *
A the unique fixed point of A, then: Lemma 2.7 (Abstract comparison lemma).Let (X, d, ≤) an ordered metric space and A, B, C : X → X be such that: (i) the operator A, B, C are WPOs; (ii) A ≤ B ≤ C; (iii) the operator B is increasing.
Another important notion is the following.
Definition 2.8.Let (X, d) be a metric space, A : X → X be a weakly Picard operator and For the c-PO s and c-WPO s we have the following lemma.
Lemma 2.9.Let (X, d) be a metric space and A, B : X → X be two operators.We suppose that: Lemma 2.10.Let (X, d) be a metric space and A, B : X → X be two operators.We suppose that: (i) the operators A and B are c-WPO s ; (ii) there exists η ∈ R * + such that d(A(x), B(x)) ≤ η, ∀x ∈ X.
Then H d (F A , F B ) ≤ cη, where H d stands for the Pompeiu-Hausdorff functional with respect to d.
The following result is the characterization theorem of weakly Picard operators.
Theorem 2.11.An operator A is a weakly Picard operator if and only if there exists a partition of X, For some examples of WPOs see [10][11][12].
For x 0 ∈ R, we consider On the other hand, for a suitable choice of τ > 0 such that 1 τ (L g + L f ) < 1, we have that B f is a contraction in (X, • τ ).So, we obtain (a) and (b).Moreover the operator E f | X x 0 : X x 0 → X x 0 is a contraction and from the characterization theorem of WPO (Theorem 2.11) we have that Next we study the relation between the solution of the problem (1.1)-(1.2) and the subsolution of the same problem.We have the following theorem.Theorem 3.2 (Theorem of Čaplygin type).We suppose that: (a) the conditions (C 1 ), (C 2 ) and (C 3 ) are satisfied; Let x be a solution of equation (1.1) and y a solution of the inequality Then y(a) ≤ x(a) implies that y ≤ x.
Proof.We have the following two relations x = E f (x) and y ≤ E f (y).
From the conditions (C 1 ), (C 2 ), and (C 3 ) follows that the operator E f is WPO.Also, from conditions (b) and (c) we have that E f is an increasing operator.Applying Lemma 2.5 we obtain that E ∞ f is increasing.Let x 0 ∈ R, then we denote by x 0 the following function So the proof is completed.Now we study the monotony of the system (1.1)-(1.2) with respect to f .For this we use Lemma 2.7.

Theorem 3.3 (Comparison theorem).
We suppose that f i ∈ C([a, b] × R, R), i = 1, 2, 3 satisfy the conditions (C 1 ), (C 2 ), and (C 3 ).Furthermore, we suppose that: Proof.From Theorem 3.1 we have that the operators E f i , i = 1, 2, 3, are WPO s .From the condition (ii) the operator E f 2 is monotone increasing.From the condition (i) it follows that From Lemma 2.7 we have that ), i = 1, 2, 3 and therefore applying Lemma 2.7 we get that Consider the Cauchy problem (1.1)-(1.2) and suppose the conditions of Theorem 3.1 are satisfied.Denote by x * (•; x 0 , g, f ), the solution of this problem.We have the following result.Theorem 3.4 (Data dependence theorem).We suppose that x 0i , g i , f i , i = 1, 2 satisfy the conditions (C 1 ), (C 2 ), and (C 3 ).Furthermore, we suppose that there exist , where x * i (t; x 0i , g i , f i ), i = 1, 2 are the solution of the problem (1.1)-(1.2) with respect to x 0i , g i , f i , Proof.Consider the operators B x 0i ,g i , f i = x 0i + t a g i (x)(s)ds+ t a f i (s, x(s)) ds, i = 1, 2. From Theorem 3.1 these operators are c i -PO s with c i = 1 − 1 τ (L g + L f ) −1 .On the other hand Now the proof follows from Lemma 2.9.
Applying Lemma 2.10 we have the theorem: Theorem 3.5.We suppose that f 1 and f 2 satisfy the conditions (C 1 ), (C 2 ), and (C 3 ).Let S E f 1 , S E f 2 be the solution set of system (1.1) corresponding to f 1 and f 2 .Suppose that there exist

Application
Next we give an application concerning the results from the main section.

Theorem 3 . 1 .
) and if x ∈ C([a, b], R) is a solution of (3.2), then x ∈ C 1 ([a, b], R) and is a solution of (1.1).Let us consider the following operators B f , E f : C([a, b], R) → C([a, b], R) defined by B f (x)(t) := the right-hand side of (3.1) and E f (x)(t) := the right-hand side of (3.2).The first result of the paper is the following: We suppose that the conditions (C 1 ), (C 2 ), and (C 3 ) are satisfied.Then (a) the problem (1.1)-(1.2) has in C([a, b], R) a unique solution; (b) the operator B f is PO in C([a, b], R); (c) the operator E f is WPO in C([a, b], R).Proof.Consider on X = C([a, b], R) the Bielecki norm • τ defined by x τ = sup t∈[a,b] and H • C denotes the Pompeiu-Hausdorff functional with respect to • C on C[a, b].