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Triple Positive Solutions of a Nonlocal Boundary Value Problem for Singular Differential Equations with p-Laplacian
We establish the existence of triple positive solutions of an m-point boundary value problem for the nonlinear singular second-order differential equations of mixed type with a p-Laplacian operator by Leggett-William fixed point theorem. At last, we give an example to demonstrate the use of the main result of this paper. The conclusions in this paper essentially extend and improve the known results.
The existence and multiplicity of positive solutions for differential equations boundary value problems (BVPs) with the p-Laplacian operator subject to Dirichlet, Sturm-Liouville, or nonlinear boundary value conditions have been extensively investigated in recent years; see [1–10] and the references therein. Particularly, the following differential equations with one-dimensional p-Laplacian have been studied subject to different kinds of boundary conditions; see [1–4] and the references therein. The methods mainly depend on Kransnosel’skii fixed point theorem, upper and lower solution technique, Leggett-Williams fixed point theorem, and some new fixed point theorems in cones, and so forth.
Recently, in , Kong et al. have studied the existence of triple positive solutions for the following BVP:
Firstly, we confirm that the mistakes which have been pointed out in  exist. At the same time, we think that the value of designed in Theorem 3.1 in  is not suitable, since the proof needs the condition , but in fact this condition does not always hold.
Motivated by the work above, in this paper, we will study the following more extensive second-order m-point BVP: where is an increasing function, , ; , , , and ; and are two linear operators defined by in which , , , , , , , and .
Throughout this paper, we always suppose the following conditions hold:; may be singular at and , so it is easy to see that there exists a constant such that ; is nonincreasing and continuous, and for .
2. Preliminary Results
In this section, we firstly present some definitions, theorems, and lemmas, which will be needed in the proof of the main result.
Definition 1. Let be a real Banach space. A nonempty closed convex set is called a cone if it satisfies the following two conditions:(i), implies ;(ii), implies .
Definition 2. Given a cone in a real Banach space , a continuous map is called a concave (resp., convex) functional on if and only if, for all and , it holds: , (resp., .
We consider the Banach space equipped with norm , where .
We denote, for any fixed constants , , = is concave and nonincreasing on , , . It’s easy to see that is a cone in .
Theorem 3 (Leggett-William). Let be a completely continuous map and let be a nonnegative continuous concave functional on with for any . Suppose there exist constants , and with such that(i) and for all ;(ii) for all ;(iii) for all with .
Then has at least three fixed points , , and satisfying
Lemma 4. Suppose with satisfies Then, is concave and nonincreasing on , that is, .
Proof. Since , we know that is nonincreasing, that is, is nonincreasing, which means is concave. At the same time, we have , so , namely . Then, ; that is to say, is nonincreasing. So . Above all, . This completes the proof.
Lemma 5. Let and , then, BVP
has a unique solution
Define the operator by Obviously, is well defined and is a solution of BVP (4) if and only if is a fixed point of .
Lemma 6. is completely continuous.
Proof. It is similar to the proof of Lemma 2.2 in .
Lemma 7. For any , one has , .
Proof. From (10), we obtain
Since , so we have , which completes the proof.
3. Main Results
For any , we define a nonnegative continuous concave function by . Obviously, the following two conclusions hold:
The main result of this paper is following.
Theorem 8. Let and . Suppose , , and hold. Suppose further that there exist numbers and such that , and, for × × ;, for × × ;, for × × , where ; ≥ , where × .
Then, BVP (4) has at least three positive solutions , and such that
Proof. We divide the proof into three steps.
Step 1. We prove , ; that is, (ii) of Theorem 3. By Lemma 6, we have , so , we get , , , . For and by Hence, and . Similarly, we obtain .
Step 2. We show that is, (i) of Theorem 3.
Let , then . Hence, (15) holds. For any , we have , so by , we have Hence (16) holds.
Step 3. We show that for all with , that is, (iii) of Theorem 3.
If with , we obtain , , , , for any , and so by , we have Furthermore, we have Therefore, by Lemma 7, we have Hence, by Theorem 3, the results of Theorem 8 hold. This completes the proof of Theorem 8.
Consider the following BVP: where
Proof. Since and , , , , , , , and , then we can obtain , and
Next, we show that – are satisfied.
If , then So is satisfied.
If , then So is satisfied.
If , then So is satisfied.
For any , we have Hence, it’s easy to know that is satisfied.
So by Theorem 8, we conclude that the BVP (21) has three positive solutions , , and satisfying
This paper is supported by the Natural Science Foundation of China (10901045) and (11201112), the Natural Science Foundation of Hebei Province (A2009000664) and (A2011208012) and the Foundation of Hebei University of Science and Technology (XL200757).
D. Kong, L. Liu, and Y. Wu, “Triple positive solutions of a boundary value problem for nonlinear singular second-order differential equations of mixed type with -Laplacian,” Computers & Mathematics with Applications, vol. 58, no. 7, pp. 1425–1432, 2009.View at: Publisher Site | Google Scholar | MathSciNet