In this paper, we will consider the positive solutions to the following three-point boundary value problem with p-Laplacian
where , , is a constant and .
The study of positive solutions on second-order boundary value problems for ordinary differential equations has aroused extensive interest, one may see - and references therein.
Among the substantial number of works dealing with nonlinear differential equations we mention the boundary value problem (1) and (2). One thing to be mentioned is that nonlinear ter f is involved with the first-order derivative explicitly.
Firstly, we present here some necessary definitions and background material of the theory of cones in ordered Banach spaces.
Definition 2.1. Let be a real Banach space. A nonempty closed set is said to be a cone provided that
1) for all and all , , and
2) implies .
Definition 2.2. The map is said to be a nonnegative continuous concave functional on provided that is continuous and
for all and . Similarly, we say the map is a nonnegative continuous convex functional on provided that is continuous and
for all and .
Definition 2.3. Let , be constants, is a nonnegative continuous concave functional and are nonnegative continuous convex functionals on the cone . Define the following convex sets
The following assumptions as regards the nonnegative continuous convex functions are used
: there exists such that , for all ;
: , for any , .
Next, we present a fixed point theorem established in , in which Bai and Ge generalized the Leggett-Williams’ fixed point theorem. The generalization is achieved by introducing on cone of continuous functionals satisfying certain properties. The technique using functionals to replace norms has been proved very useful in generalizing some fixed point theorems.
Theorem 2.1.  Let be a Banach space, is a cone and , . Assume the are nonnegative continuous convex functionals satisfying and , is a nonnegative concave functional on , such that
and is a completely continuous operator. Suppose that
: , for all ,
: , , for all ,
: for all with .
Then has at least three fixed points with
3. Multiple Positive Solutions for (1) and (2)
Let the Banach space be endowed with the norm
and define the cone by
Choose a natural number . For notational convenience,
Then are nonnegative continuous convex functionals satisfying and , and is nonnegative continuous concave functional on , it is also clear that for all .
Lemma 3.1. For then the boundary value problem
has a unique solution
where is the solution of the equation
Proof. The proof can be obtained by regular calculation, so we omit it here.
We define an operator by
where is the solution of the equation
by Lemma 3.1, we know that boundary value problem (1) and (2) has a solution if and only if is a fixed point of .
Lemma 3.2. defined by (3) is completely continuous.
Proof. From the definition of , we deduce that for each , there is
is nonnegative and satisfies (2).
Moreover, is the maximum value of on [0,1], since
is continuous and nonincreasing in [0,1] and . As is nonincreasing on [0,1], we have .
Hence, we get that .
Then according to Arzela-Ascoli theorem, is completely continuous if and only if is continuous about and maps a bounded subset of into a relatively compact set.
Let as on .
For , according to (3) and (4), we have
Hence, we obtain that is continuous.
Now, let be a bounded set, i.e., there exists a positive constant such that
, for all ,
from the expression of and we can obtain that is uniformly bounded according to the properties of . And it is also easy to get that, for any , , we have
which shows that is equicontinuous.
Then the Arzela-Ascoli theorem guarantees that is relatively compact, which means is compact. Then, we obtain that is completely continuous.
Thus, from what has been discussed above, we can draw the conclusion that is completely continuous.
We are now ready to apply the fixed point theorem due to Avery and Peterson to the operator in order to get sufficient conditions for the existence of multiple positive solutions to the problems (1) and (2).
Our main result is as follows.
Theorem 3.1. Assume that there exist constants ,
, such that . If the following assumptions hold
Then the boundary value problem (1) and (2) has at least three positive solutions , and such that
Proof. From we discussed earlier, is well defined. Problem (1) and (2) has a solution if and only if is a fixed point of .
We have already showed that is completely continuous.
In what following, we will prove the results step by step according to the Theorem 2.1.
The proof is divided into some steps.
Firstly, we will show that condition implies that
In fact, for , we have
and assumption implies
And by Lemma 3.1 and (4), we have
Secondly, we show that condition in Theorem 2.1 holds.
In order to check condition in Theorem 2.1, we choose .
It is easy to see that and , consequently,
If , then for .
From assumption , we have
Then we have
Then, we can get that
Consequently, condition in Theorem 2.1 holds.
Thirdly, We now show in Theorem 2.1 is satisfied.
If , then assumption yields
In the same way as in the first step, we can obtain that . Hence, condition in Theorem 2.1 is aslo satisfied.
Finally, we show in Theorem 2.1 is also satisfied.
Suppose that with . Then, by the definition of and , we have
Thus, condition in Theorem 2.1 is also satisfied.
Then, Theorem 3.1 is proved by Theorem 2.1.
Consequently, Theorem 3.1 is proved by Theorem 2.1. We obtain that the boundary value problem (1) and (2) has at least three positive solutions , and such that
The author thanks the referees for their valuable comments and suggestions. This work was supported by the Discipline Construction Fund of Central University of Finance and Economics.
 Sun, B., Ge, W. and Zhao, D. (2007) Three Positive Solutions for Multipoint One-Dimensional p-Laplacian Boundary Value Problems with Dependence on the First Order Derivative. Math. Comput. Model., 45, 1170-1178. https://doi.org/10.1016/j.mcm.2006.10.002
 Wang, J. and Zheng, D. (1997) On the Existence of Positive Solutions to a Three-Point Boundary Value Problem for the One-Dimensional p-Laplacian. ZAMM., 77, 477-479. https://doi.org/10.1002/zamm.19970770618
 Jiang, D., Nieto, J.J. and Zuo, W. (2004) On Monotone Method for First Order and Second Order Periodic Boundary Value Problems and Periodic Solutions of Functional Differential Equations. J. Math. Anal. Appl., 289, 691-699. https://doi.org/10.1016/j.jmaa.2003.09.020
 Ahmad, B. and Sivasundaram, S. (2006) The Monotone Iterative Technique for Impulsive Hybrid Set Valued Integro-Differential Equations. Nonlinear Anal., 65, 2260-2276. https://doi.org/10.1016/j.na.2006.01.033