Existence and Positivity of Solutions for a Second-Order Boundary Value Problem with Integral Condition

This work is devoted to the study of uniqueness and existence of positive solutions for a second-order boundary value problem with integral condition. The arguments are based on Banach contraction principle, Leray Schauder nonlinear alternative, and Guo-Krasnosel’skii ﬁxed point theorem in cone. Two examples are also given to illustrate the main results.


Introduction
Boundary value problem with integral boundary conditions is a mathematical model for of various phenomena of physics, ecology, biology, chemistry, and so forth. Integral conditions come up when values of the function on the boundary are connected to values inside the domain or when direct measurements on the boundary are not possible. The presence of an integral term in the boundary condition leads to great difficulties. Our aim, in this work, is the study of existence, uniqueness, and positivity of solution for the following second-order boundary value problem: where α, β ∈ R , to prove the uniqueness of solution, we apply Banach contraction principle, by using Guo-Krasnosel'skii fixed point theorem in cone we study the existence of positive solution. As applications, some examples to illustrate our results are given. Various types of boundary value problems with integral boundary conditions were studied by many authors using different methods see 1-9 . In 2 Benchohra et al. have studied 1.1 with the integral condition u 0 0, u 1 1 0 g s u s ds, the authors assumed that the function f depends only on t and u and the condition 1.3 holds for α ∈ 0, 1 , so our work is new and more general than 2 . Similar boundary value problems for thirdorder differential equations with one of the following conditions u 0 0, u 0 0, u 1 1 0 g s u s ds, or u 0 1 0 g s u s ds, u 0 θ, u 1 0, were investigated by Zhao et al. in 6 , they established the existence and nonexistence and the multiplicity of positive solutions in ordered Banach spaces basing on fixed point theory in cone. For more knowledge about the nonlocal boundary value problem, we refer to the references 10-17 .
This paper is organized as follows. In Section 2, we give some notations, recall some concepts and preparation results. In the third Section, we give two main results, the first result based on Banach contraction principle and the second based nonlinear alternative of Leray-Schauder type. In Section 4, we treat the positivity of solutions with the help of Guo-Krasnosel'skii fixed point theorem in cone. Some examples are given to demonstrate the application of our main results, ending this paper.

Preliminaries Lemmas and Materials
In this section, we introduce notations, definitions, and preliminary facts that will be used in the sequel. ii the map u, v → f t, u, v is continuous on R 2 for almost each t ∈ 0, 1 ; iii for each r > 0, there exists an ϕ r ∈ L 1 0, 1 such that |f t, u, v | ≤ ϕ r t for almost each t ∈ 0, 1 and |u| |v| ≤ r.
We recall the definition of positive solution.
Definition 2.6. A function u is called positive solution of 1.1 if u t ≥ 0, for all t ∈ 0, 1 .
We expose the well-known Guo-Krasnosel'skii fixed point Theorem on cone 19 .
be a completely continuous operator such that Throughout this paper, let E C 1 0, 1 , R , with the norm u 1 u u , where · denotes the norm in C 0, 1 , R defined by u max t∈ 0,1 |u t |. One can obtain the following result. Lemma 2.8. Let δ ∈ E, then the solution of the following boundary value problem:

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Proof. Integrating two times the equation u t −δ t from 0 to t, one can obtain The condition u 0 0 gives C 2 0. The second condition u 1 1 0 tu t dt implies it is easy to get where G t, s is given by 2.4 .
We have the following result which is useful in what follows.
Remark 2.9. The function G t, s is continuous, nonnegative and satisfies for any t, s ∈ 0, 1 , max G t, s 3/2.
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Existence and Uniqueness Theorems
This section deals with the existence and uniqueness of solutions for the problem 1.1 -1.2 .
Theorem 3.1. Suppose that the following hypotheses hold.
H1 f is an L 1 -Carathéodory function.
H2 There exist two nonnegative functions g 1 , applying hypothesis H3 to the right-hand side of the above inequality, we obtain On the other hand we have for any t ∈ 0, 1 : Applying hypothesis H3 again gives Combining inequalities 3.5 and 3.8 we obtain thus, T is a contraction mapping on E. By applying the well-known Banach's contraction mapping principle we know that the operator T has a unique fixed point on E. Therefore, the problem 1.1 -1.2 has an unique solution.

3.11
Then the BVP 1.1 -1.2 has at least one nontrivial solution.
Proof. First we show that T is a completely continuous mapping that we will prove in some steps: 1 T is continuous. In fact, let {u m } ∞ 1 be a convergent sequence in E such that u m → u, then u m → u and for each t ∈ 0, 1 we have

3.13
From the above discussion one can write Due to P1 f is Cathéodory, then Tu m − Tu 1 → 0 as m → ∞.
2 T maps bounded sets into bounded sets in E, to establish this step we use Arzela-Ascoli Theorem. Let B r {u ∈ E : u 1 ≤ r}, then from P2 , we have for any u ∈ B r and t ∈ 0, 1

3.15
consequently T u 1 ≤ 4 h L 1 r α g L 1 r β k L 1 3.16 that implies T maps bounded sets into bounded sets. 3 T maps bounded sets into equicontinuous sets of E. Let u ∈ B r and t 1 , t 2 ∈ 0, 1 , t 1 < t 2 and |t 1 − t 2 | < δ, then using P2 it yields

3.17
In addition, we have

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Since G t, s is continuous, then |Tu t 1 − Tu t 2 | tend to 0 when t 1 → t 2 , and we have immediately that |T u t 1 − T u t 2 | → 0, this yields that T is equicontinuous. Then T is completely continuous.
Secondly, we apply the nonlinear alternative of Leray-Schauder to prove the existence of solution. Let us make the following notations: Since f t, 0, 0 / 0, then there exists an interval η, τ ⊂ 0, 1 such that min η≤t≤r |f t, 0, 0 | > 0 hence N > 0. From hypothesis P3 , we know that M < 1/8. Putting m M/N. Setting Ω {u ∈ E : u 1 < 1} and let u ∈ ∂Ω, λ > 1, such that Tu t λu t . Using the same argument that to get 3.16 , it yields as u 1 1 then λ ≤ 4 M N . First, if m ≤ 1 then λ ≤ 8N < 1, hence λ < 1, this contradicts the fact that λ > 1. By Lemma 2.4 we conclude that T has a fixed point u * ∈ Ω and then problem 1.1 -1.2 has a nontrivial solution u * ∈ E. Second, if m ≥ 1 then λ ≤ 8M < 1. By arguing as above we complete the proof.

Existence of Positive Solutions
In this section the existence results for positive solutions for problem 1.1 -1.2 are presented. We make the following hypotheses: Q1 f t, u, v a t f 1 u, v where a ∈ C 0, 1 , R and f 1 ∈ C R × R, R ; Q2 There exists 0 < τ < 1 such that

4.5
The proof is complete.  Define the quantities A 0 and A ∞ by The case A 0 0 and A ∞ ∞ is called superlinear case and the case A 0 ∞ and A ∞ 0 is called sublinear case. The main result of this section is the following. To prove this theorem we apply the well-known Guo-Krasnosel'skii fixed point Theorem in cone.

Proof. Denote E
{u ∈ E, u t ≥ 0, for all t ∈ 0, 1 } and define the cone K by where γ is given in Lemma 4.2.
It is easy to check that K is a nonempty closed and convex subset of E. Using Lemma 4.2 we see that TK ⊂ K. Applying Arzela-Ascoli Theorem we know that T : K → E is completely continuous for u ∈ K. On the basis of hypothesis Q1 , one can write 4.14 Let us consider the superlinear case. Since A 0 0, then for any ε > 0, there exists The first statement of Theorem 2.7 implies that T has a fixed point in K ∩ Ω 2 \ Ω 1 such that R 2 ≤ u 1 ≤ R. Applying similar techniques as above, we prove the sublinear case. The proof of Theorem 4.3 is complete.
To illustrate the main results, we consider the following examples.

4.26
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Choosing g 1 t 1/2 t 3 ln t 3 , g 2 t e −t /8 1 t 4 , it is easy to see that g 1 L 1 g 2 L 1 ≈ 0.185 < 1/4 and hypotheses H1 -H3 of Theorem 3.1 are satisfied, then, problem P1 has a unique solution in E. has at least one solution in E. In fact, we have |f t, u, u | ≤ h t |u t | 1/5 g t |u t | 4/3 k t , where h t k t 1/10 t 2 ln t 2 , g t 1/2 3t 1 4 , by calculation we obtain h L 1 g L 1 ≈ 0, 09 < 1/8 and k L 1 1 0 1/10 t 2 ln t 2 dt 0.0363 < 1/8. From Theorem 3.2, we deduce the existence of at least one solution in E.