Existence Results of Three-Point Boundary Value Problems for Second-order Ordinary Differential Equations
© S.-P.Wang and L.-Y. Tsai. 2011
Received: 19 May 2010
Accepted: 24 September 2010
Published: 28 September 2010
We establish existence results of the following three-point boundary value problems: , , and , where and . The approach applied in this paper is upper and lower solution method associated with basic degree theory or Schauder's fixed point theorem. We deal with this problem with the function which is Carathéodory or singular on its domain.
In the mathematical literature, a number of works have appeared on nonlocal boundary value problems, and one of the first of these was . Il'in and Moiseev initiated the research of multipoint boundary value problems for second-order linear ordinary differential equations, see [2, 3], motivated by the study [4–6] of Bitsadze and Samarskii.
Recently, nonlinear multipoint boundary value problems have been receiving considerable attention, and have been studied extensively by using iteration scheme (e.g., ), fixed point theorems in cones (e.g., ), and the Leray-Schauder continuation theorem (e.g., ). We refer more detailed treatment to more interesting research [10, 11] and the references therein.
The theory of upper and lower solutions is also a powerful tool in studying boundary value problems. For the existence results of two-point boundary value problem, there already are lots of interesting works by applying this essential technique (see [12, 13]). Recently, it is shown that this method plays an important role in proving the existence of solutions for three-point boundary value problems (see [14–16]).
Last but not least, as the singular source term appearing in two-point problems, singular three-point boundary value problems also attract more attention (e.g., ).
In this paper, we will discuss the existence of solutions of some general types on three-point boundary value problems by using upper and lower solution method associated with basic degree theory or Schauder's fixed point theorem.
By direct computations, we get the following results.
3. Carathéodory Case
In this section we first introduce the Carathéodory function as follows.
And we get a contradiction.
which is impossible.
It is clear that is a closed, bounded and convex set in and one can show that is a completely continuous mapping by Arzelà-Ascoli theorem and Lebesgue dominated convergence theorem. By applying Schauder's fixed point theorem, we obtain that has a fixed point in which is a solution of problem (3.3) and (1.2). From Proposition 3.4, this fixed point is also a solution of problem (1.1) and (1.2). Hence, we complete the proof.
with the boundary condition (1.2).
Then, problem (3.18) and (1.2) has at least one solution.
3.2. Nontangency Solution
In this subsection, we afford another stronger lower and upper solutions to get a strict inequality of the solution between them.
Now we are going to show that the solution curve of problem (1.1) and (1.2) cannot be tangent to upper or lower solutions from below or above.
And we get a contradiction.
This is a consequence of Theorem 3.5 and Proposition 3.10 and hence, we omits this proof.
4. Singular Case
In this section we give a more general existence result than Theorem 3.11 by assuming the existence of -lower and upper solutions. This makes us to deal with problem (1.1) and (1.2), where the function is singular at the end point and .
Notice that in Theorem 4.1, one can only deal with the case that is singular at end points , . However, when is singular at , there is no hope to obtain the solutions directly from Theorem 4.1. We will establish the following theorem to deal with this case by constructing upper and lower solutions to solve this problem.
Remark 4.5 (see [12, Remark ]).
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