Monotonic Positive Solutions of Nonlocal Boundary Value Problems for a Second-Order Functional Differential Equation

and Applied Analysis 3 i f : 0, 1 × R → R is measurable in t ∈ 0, 1 for all x ∈ R and continuous in x ∈ R for almost all t ∈ 0, 1 and there exists an integrable function a ∈ L1 0, 1 , and a constant b > 0 such that ∣ ∣f t, x ∣ ∣ ≤ |a t | b|x|, ∀ t, x ∈ 0, 1 ×D. 2.1 ii φ : 0, 1 → 0, 1 is continuous. iii b < 1/ 3 − B , B ∑nj 1 bj 1 −1. iv m ∑ k 1 ak > 0, ∀k 1, 2, . . . , m, n ∑ j 1 bj > 0, ∀j 1, 2, . . . , n. 2.2 Now, we have the following Lemma. Lemma 2.1. The solution of the nonlocal problem 1.3 1.4 can be expressed by the integral equation


Introduction
The nonlocal boundary value problems of ordinary differential equations arise in a variety of different areas of applied mathematics and physics.
The study of nonlocal boundary value problems was initiated by Il'in and Moiseev 1, 2 .Since then, the non-local boundary value problems have been studied by several authors.The reader is referred to 3-22 and references therein.
In most of all these papers, the authors assume that the function f : 0, 1 × R → R is continuous.They all assume that lim

1.1
These assumptions are restrictive, and there are many functions that do not satisfy these assumptions.
Here we assume that the function f : 0, 1 × R → R is measurable in t ∈ 0, 1 for all x ∈ R and continuous in x ∈ R for almost all t ∈ 0, 1 is and there exists an integrable function a ∈ L 1 0, 1 and a constant b > 0 such that Our aim here is to study the existence of at least one monotonic positive solution for the nonlocal problem of the second-order functional differential equation where τ k ∈ a, d ⊂ 0, 1 , η j ∈ c, e ⊂ 0, 1 , and x 0 , x 1 > 0.
As an application, the problem with the integral and nonlocal conditions d a x t dt x 0 , x 0 x e − x c x 1 , 1.5 is studied.
It must be noticed that the nonlocal conditions 1.6 are special cases of our the nonlocal and integral conditions.

Integral Equation Representation
i f : 0, 1 × R → R is measurable in t ∈ 0, 1 for all x ∈ R and continuous in x ∈ R for almost all t ∈ 0, 1 and there exists an integrable function a ∈ L 1 0, 1 , and a constant b > 0 such that Now, we have the following Lemma.
Lemma 2.1.The solution of the nonlocal problem 1.3 -1.4 can be expressed by the integral equation
and we deduce that Substitute from 2.7 into 2.5 , we obtain Let t η j , in 2.4 , we obtain 2.9 and we deduce that

2.10
Substitute from 2.10 into 2.8 , we obtain which proves that the solution of the nonlocal problem 1.3 -1.4 can be expressed by the integral equation 2.3 .

Existence of Solution
We study here the existence of at least one monotonic nondecreasing solution x ∈ C 0, 1 for the integral equation 2. Proof.Define the subset Clear the set Q r which is nonempty, closed, and convex.
Let H be an operator defined by

3.3
Then a ds br t 2 − t 1 .

3.4
The above inequality shows that Therefore {Hx t } is equicontinuous.By the Arzelà-Ascoli theorem, {Hx t } is relatively compact.Since all conditions of the Schauder theorem hold, then H has a fixed point in Q r which proves the existence of at least one solution x ∈ C 0, 1 of the integral equation 2.3 , where lim

3.6
To complete the proof, we prove that the integral equation 2.3 satisfies nonlocal problem 1.3 -1.4 .Differentiating 2.3 , we get x t f t, x φ t .

3.8
Let t τ k in 2.3 , we obtain Also let t η j in 3.7 , we obtain f s, x φ s ds.

3.12
Let t 0 in 3.7 , we obtain Adding 3.12 and 3.13 , we obtain

3.14
This implies that there exists at least one solution x ∈ C 0, 1 of the nonlocal problem 1.3 and 1.4 .This completes the proof.
Proof.Let t 1 < t 2 , we deduce from 2.3 that

3.15
which proves that the solution x of the problem 1.3 -1.4 is monotonic nondecreasing.

Positive Solution
Let b j 0, j 1, 2, . . .n and x 1 0, then the nonlocal problem condition 1.4 will be

3.18
Multiplying by

Nonlocal Integral Condition
Let x ∈ C 0, 1 be the solution of the nonlocal problem 1.3 and 1.4 . Let

Theorem 3 . 3 . 0 tt
Let the assumptions (i)-(iv) of Theorem 3.1 be satisfied.Then the solution of thenonlocal problem 1.3 -3.16 is positive t ∈ d, 1 .Proof.Let b j 0, j 1, 2, . . .n and x 1 0 in the integral equation 2.3 and the nonlocal condition 1.4 , then the solution of the nonlocal problem 1.3 -3.16 will be given by the integral equation s f s, x φ s ds ≤ t − s f s, x φ s ds, τ k ≤ t, − s f s, x φ s ds.

1 , then m k 1 t, x 0 n j 1 ξ 1 From 0 t
k − t k−1 x τ k x 0 j − ξ j−1 x η j x 1.4.thecontinuity of the solution x of the nonlocal problem 1.3 and 1.4 , we obtain lim of the integral equation 2.3 will bex t d − a −1 x 0 − d a t − s f s, x φ s ds dt b − c 1 −1 t − 1 x 1 −

4 . 4 Now, we have the following theorem. Theorem 4 . 1 . 1 4. 5 has
Let the assumptions (i)-(iv) of Theorem 3.1 be satisfied.Then the nonlocal problemx t f t, x φ t , t ∈ 0, 1 , d a x s ds x 0 , x 0 x e − x cx at least one monotonic nondecreasing solution x ∈ C 0, 1 represented by 4.4 .