On the existence of positive solutions of a state-dependent neutral functional differential equation with two state-delay functions

In this paper, we study the existence of positive solutions for an initial value problem of a state-dependent neutral functional differential equation with two state-delay functions. The continuous dependence of the unique solution will be proved. Some especial cases and examples will be given.


Introduction
T he differential and integral equations with deviating arguments usually involve the deviation of the argument only the time itself, however, another case, in which the deviating arguments depend on both the state variable x and the time t is important in theory and practice. This kind of equations is called self-reference or state dependent equations. Differential equations with state-dependent delays attract interests of specialists since they widely arise from application models, such as two-body problem of classical electrodynamics also have may applications in the class of problems that have past memories, for example in hereditary phenomena see [1][2][3][4]. For papers studying such kind of equations, see for example [5][6][7][8][9][10][11][12][13][14][15][16] and references therein.
One of the first papers studying this class of equations was introduced by Buicá [7], the author proved the existence and the uniqueness of the solution of the initial value problem x(x(t))), t ∈ [a, b], where f ∈ C([a, b] × [a, b]) and satisfied Lipshitz condition. EL-Sayed and Ebead [13] relaxed the assumptions of Buicá and generalized their results, they studied the functional integral equation of the more general form where g satisfies Carathéodory condition.
El-Sayed and Ebead [11,12] studied the existence of solution and its continuous dependence of the initial value problem of the delay-refereed differential equation x(t)))), a.e. t ∈ (0, T], with the two cases of state-delay functions Here we shall study the initial value problem of state-dependent neutral functional differential equation with two state-delay functions with the initial data where g i , i = 1, 2 are continuous and Our aim in this work is to study the existence of at least one and exactly one positive solution of the Problem (1)- (2). The continuous dependence of the unique solution on the two functions g 1 and g 2 will be proved. To illustrate our results some examples will be given.
In order to achieve our goal, we study the existence of positive solutions x ∈ C[0; T] for the state-dependent functional integral equation and we will show later that this integral equation is equivalent to the initial value Problem (1)-(2).

Existence of solutions
Here we study the existence of solutions x ∈ C[0, T] for the integral Equation (3) under the following assumptions (1) f 1 : [0, T] × [0, T] → R + is continuous and there exist two positive constants k 1 and k 2 such that and continuous in x for almost all t ∈ [0, T].  which implies that φ(t) ≤ t.
(7) There exists a real positive solution L ∈ (0, 1) of the equation Now we are in a position to prove the following existence theorem.
It is clear that S L is nonempty, closed, bounded and convex subset of C[0, T]. Now define the operator F associated with Equation (3) by It is clear that F makes sense and well-defined. Now, first we prove that the class of functions {Fx} is uniformly bounded on the set S L . Let x ∈ X, then for t ∈ [0, T], we obtain Using assumptions (1), (2) and (6) we can get Using assumptions (3) and (4) we can get Now from (4) and (5) and by assumption (8), we get This proves that the class of functions { Fx } is uniformly bounded on the set S L . Next, we prove that F : S L → S L and the class of functions {Fx} is equi-continuous on the set S L . Let x ∈ S L and t 1 , Using assumptions (2) and x ∈ S L , we can get Hence, we proved that F : S L → S L and the class of functions {Fx} is equi-continuous on the set S L . Applying Arzela-Ascoli Theorem [17], we deduce that F is compact operator. Now we show that F is continuous. Let {x n } ⊂ S L , x n → x on [0, T], i.e. |x n (φ(t)) − x(φ(t))| ≤ 1 this implies that |x n (g i (t, x(φ(t)))) − x(g i (t, x(φ(t))))| ≤ 2 for arbitrary 1 , 2 ≥ 0, i = 1, 2, then Then x n (g i (t, x n (φ(t)))) → x(g i (t, x(φ(t)))) in S L , i = 1, 2 and by using the continuity of the functions f 1 , we obtain f 1 t, x n (g 1 (t, x n (φ(t)))) → f 1 t, x(g 1 (t, x(φ(t)))) . Now by using the continuity of the functions f 2 , assumption (5) and Lebesgues dominated convergence theorem [17], we obtain t 0 f 2 t, x n (g 2 (t, x n (φ(t)))) ds → t 0 f 2 t, x(g 2 (t, x(φ(t)))) ds, and lim n→∞ Fx n (t) = lim n→∞ f 1 t, x n (g 1 (t, x n (φ(t)))) + lim n→∞ t 0 f 2 t, x n (g 2 (t, x n (φ(t)))) ds This proves that the operator F is continuous. Now all conditions of Schauder fixed point theorem [17] are satisfied, then the operator F has at least one fixed point x ∈ S L . Consequently there exists at leat one solution x ∈ C[0, T] of Equation (3). This completes the proof. Now, we introduce the following equivalence theorem. Proof. Let x be a solution of the Problem (1)-(2). Integrate (1) and substitute by (2), we obtain the integral Equation (3). Let x be a solution of (3) differentiate (3) we obtain (1) and the initial value (2). This proves the equivalence between the Problem (1)-(2) and the integral Equation (3). Then the Problem (1)-(2) has at least one positive solution x ∈ C[0, T].
Then we deduce that and from the assumptions k 2 (L k 4 + 1) + k 5 T (L k 6 + 1) < 1 we can obtain x = y and the solution of (3) is unique. Consequently, the initial value Problem (1)-(2) has a unique positive solution x ∈ C[0, T].

Continuous dependence on the function g 1
Here we prove that the solution of the Problem (1)-(2) depends continuously on the function g 1 .
Theorem 4. Let the assumptions of Theorem 3 be satisfied, then the solution of initial value Problem (1)-(2) depends continuously on the function g 1 .

Continuous dependence on the function g 2
Here we prove that the solution of the Problem (1)-(2) depends continuously on the function g 2 .
Theorem 5. Let the assumptions of Theorem 3 be satisfied, then the solution of initial value Problem (1)-(2) depends continuously on the function g 2 .
Proof. The proof follow similarly as the proof of Theorem 4.