On a Nonlocal Boundary Value Problem of a State-Dependent Differential Equation

: In this paper, the existence of absolutely continuous solutions and some properties will be studied for a nonlocal boundary value problem of a state-dependent differential equation. The inﬁnite-point boundary condition and the Riemann–Stieltjes integral condition will also be considered. Some examples will be provided to illustrate our results.


Introduction
The delay differential equations serve as an important branch of nonlinear analysis that has many applications in most fields. Usually, the deviation of the arguments depends only on the time (see [1][2][3][4][5][6]); however, when the deviation of the arguments depends upon the state variable x and also the time t is incredibly important theoretically and practically, this type of equations is known as self-reference or state-dependent equations. Equations with state-dependent delays have gained great attention to specialists since they have many application models, like the two-body problem of classical electrodynamics, even have numerous applications within the class of problems that have past memories, as an example, in hereditary phenomena, see [7,8]. Several papers studied this kind of equations, (see [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]).
Eder [12], where the author studied the problem The existence and the uniqueness of the solution of the problem were studied by Buicá [11].
In [16], the assumptions of [11], have been relaxed and generalized to the equation where f satisfies Carathéodory condition.
In [14,15], some other results have been obtained for the problem dx(t) dt = h 1 (t, x(h 2 (t, x(t)))), a.e. t ∈ (0, T], EL-Sayed and Ebead [17] studied the IVP of state-dependent hybrid functional differential equation d dt with the initial data Our aim in this work is to study the m-point boundary value problem (BVP) The existence and the uniqueness have been proved for the BVP (1) and (2). Moreover, we show that the solution of our problem depends continuously on x 0 and on the nonlocal data a k . Furthermore, we study (1) with the nonlocal integral condition where h : [0, T] → [0, T] is an increasing function. Finally, we study (1) with the infinite point boundary condition where ∑ ∞ k=1 a k is convergent.

Main Results
Consider the BVP (1) and (2) under the following hypothesis:

Integral Representation
Lemma 1. The BVP (1) and (2) and the integral equation are equivalent.

Existence of Solution
Define the set S L by Let the hypothesis (i)-(iv) be held, then (1) and (2) has a solution x ∈ S L ⊂ C[0, T].
Proof. Define the operator F by Let x ∈ S L , then we have Hence, {Fx} is uniformly bounded.
Let x ∈ S L and t 1 , This proves that F : S L → S L and {Fx} are equi-continuous. By Arzela-Ascoli Theorem ( [25] p. 54), we find that F is compact.
Now the function f is continuous in the second argument, then f t, x n (x n (φ(t))) → f t, x(x(φ(t))) .

Riemann-Stieltjes Integral
Let x ∈ AC[0, T] be a solution of BVP (1) and (2), then we can formulate the next theorem.

Theorem 2.
Let the hypothesis (i)-(iv) hold. Let h : [0, T] → [0, T] be an increasing function, then there is a solution x ∈ AC[0, T] of (1) with the Riemann-Stieltjes integral condition (3) and this solution given by Hence, and which is the solution of (1) with the Riemann-Stieltjes integral condition (3). This completes the proof.

Infinite-Point Boundary Condition
Let x ∈ AC[0, T] be a solution of the BVP (1) and (2). Then, we can formulate the next theorem.
Theorem 3. Let the assumptions (i)-(iv) be satisfied. Assume that the series: is convergent. Then, there is a solution x ∈ AC[0, T] of (1) and (4), and this solution is given by Proof. Assume that x ∈ AC[0, T] be a solution of the BVP (1) and (2), thus we have Using the comparison test, we deduce that the series ∞ ∑ k=1 a k x(τ k ) and ∞ ∑ k=1 a k τ k 0 f s, x(x(φ(s))) ds are convergent. Then as m → ∞ in (5), we get This proves that the solution of (8) satisfies (1) under infinite-point boundary condition (4). This completes the proof.

Uniqueness of the Solution
Here we prove the uniqueness of the solution of the BVP (1) and (2). Assume that  (1) and (2) is unique.

Definition 2.
The solution of the BVP (1) and (2) depends continuously on the nonlocal data a k if x * is the unique solution of the BVP Theorem 6. Let the hypothesis of Theorem 4 be hold, then the solution of BVP (1) and (2) depends continuously on a k .