Positive Solutions of a Nonlinear Three-Point Integral Boundary Value Problem

The study of the existence of solutions of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il’in and Moiseev 1 . Then Gupta 2 studied three-point boundary value problems for nonlinear second-order ordinary differential equations. Since then, nonlinear second-order three-point boundary value problems have also been studied by several authors. We refer the reader to 3–19 and the references therein. However, all these papers are concerned with problems with three-point boundary condition restrictions on the slope of the solutions and the solutions themselves, for example,


Introduction
The study of the existence of solutions of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il'in and Moiseev 1 . Then Gupta 2 studied three-point boundary value problems for nonlinear second-order ordinary differential equations. Since then, nonlinear second-order three-point boundary value problems have also been studied by several authors. We refer the reader to 3-19 and the references therein. However, all these papers are concerned with problems with three-point boundary condition restrictions on the slope of the solutions and the solutions themselves, for example,  In this paper, we consider the existence of positive solutions to the equation with the three-point integral boundary condition where 0 < η < 1. We note that the new three-point boundary conditions are related to the area under the curve of solutions u t from t 0 to t η. The aim of this paper is to give some results for existence of positive solutions to 1.2 -1.3 , assuming that 0 < α < 2/η 2 and f is either superlinear or sublinear. Set Then f 0 0 and f ∞ ∞ correspond to the superlinear case, and f 0 ∞ and f ∞ 0 correspond to the sublinear case. By the positive solution of 1.2 -1.3 we mean that a function u t is positive on 0 < t < 1 and satisfies the problem 1.2 -1.3 .
The proof of the main theorem is based upon an application of the following Krasnoselskii's fixed point theorem in a cone.

Preliminaries
We now state and prove several lemmas before stating our main results.
has a unique solution For t ∈ 0, 1 , integration from 0 to t, gives For t ∈ 0, 1 , integration from 0 to t yields that So, Boundary Value Problems

2.9
From 2.2 , we obtain that Thus, Therefore, 2.1 -2.2 has a unique solution t − s y s ds.
Proof. If u 1 0, then, by the concavity of u and the fact that u 0 0, we have u t 0 for t ∈ 0, 1 .
Moreover, we know that the graph of u t is concave down on 0, 1 , we get Proof. Assume 2.1 -2.2 has a positive solution u.
If u 1 > 0, then η 0 u s ds > 0, it implies that u η > 0 and which contradicts the concavity of u.
In the rest of the paper, we assume that 0 < αη 2 < 2. Moreover, we will work in the Banach space C 0, 1 , and only the sup norm is used. where γ : min η, αη 2 Proof. Set u τ u . We divide the proof into three cases.

2.23
This implies that This completes the proof.

Main Results
Now we are in the position to establish the main result.

3.1
Denote that where γ is defined in 2.18 . It is obvious that K is a cone in C 0, 1 . Moreover, by Lemmas 2.2 and 2.4, AK ⊂ K. It is also easy to check that A : K → K is completely continuous.

3.8
Hence, Au u , u ∈ K ∩ ∂Ω 2 . By the first past of Theorem 1.1, A has a fixed point in Sublinear Case (f 0 ∞ and f ∞ 0). Let

3.15
Thus Au u , u ∈ K ∩ ∂Ω 4 . By the second part of Theorem 1.1, A has a fixed point u in K ∩ Ω 4 \ Ω 3 , such that H 3 u H 4 . This completes the sublinear part of the theorem. Therefore, the problem 1.2 -1.3 has at least one positive solution.