Existence of positive solutions for boundary value problems of p-Laplacian difference equations
© Fen and Karaca; licensee Springer. 2014
Received: 17 July 2014
Accepted: 25 September 2014
Published: 13 October 2014
In this paper, by using the Avery-Peterson fixed point theorem, we investigate the existence of at least three positive solutions for a third order p-Laplacian difference equation. An example is given to illustrate our main results.
MSC:34B10, 34B15, 34J10.
, and with ;
f and q are continuous and positive;
is called p-Laplacian, with , its inverse function is denoted by with with ;
if and , where ℤ is the integer set, denote for with .
Difference equations, the discrete analog of differential equations, have been widely used in many fields such as computer science, economics, neural network, ecology, cybernetics, etc. . In the past decade, the existence of positive solutions for the boundary value problems (BVPs) of the difference equations has been extensively studied; to mention a few references, see [1–13] and the references therein. Also there has been much interest shown in obtaining the existence of positive solutions for the third order p-Laplacian dynamic equations on time scales. To mention a few papers along these lines, see [14–18].
We now discuss briefly several of the appropriate papers on the topic.
The sufficient conditions to guarantee the existence of at least three positive solutions of the above multi-point boundary value problem were established by using a new fixed point theorem obtained in .
By using the five functionals fixed point theorem , Liu obtained the existence criteria of at least three positive solutions.
Therefore, in this paper, we will consider the existence of at least three positive solutions for the third order p-Laplacian difference equation (1.1) by using the Avery-Peterson fixed point theorem .
Throughout this paper we assume that the following condition holds:
(C1) and are continuous.
This paper is organized as follows. In Section 2, we give some preliminary lemmas which are key tools for our proof. The main result is given in Section 3. Finally, in Section 4, we give an example to demonstrate our result.
In this section we present some lemmas, which will be needed in the proof of the main result.
The following fixed point theorem is fundamental and important to the proof of our main result.
Lemma 2.1 ()
and for ;
for with ;
and for with .
Lemma 2.2 If y is a solution of BVP (2.1), then there exists unique such that and .
Since , and with and a positive sequence, one can easily see that and . It follows from , , and the fact that is decreasing on that there exists unique such that and . The proof is complete. □
Lemma 2.3 If y is a solution of BVP (2.1), then , , and for all .
Hence for all . The proof is complete. □
Proof The proof follows from Lemma 2.2 and is omitted. □
Let . We call for if for all .
It is easy to see that is a semi-ordered real Banach space.
where . Then is a cone in .
- (i)Ty satisfies the following:(2.11)
for each .
y is a solution of BVP (1.1) if and only if y is a solution of the operator equation .
is completely continuous.
By the definition of Ty, we get (2.11).
Note the definition of . Since (C1) holds, for , (2.11), Lemma 2.2, Lemma 2.3 and Lemma 2.4 imply that is decreasing on and for all . Together with (2.11), it follows that .
It is easy to see from (2.11) that y is a solution of BVP (1.1) if and only if y is a solution of the operator equation .
It suffices to prove that T is continuous on and T is relative compact.
We divide the proof into three steps:
Step 1. For each bounded subset , prove that is bounded in ℝ for
Hence is bounded in ℝ.
Step 2. For each bounded subset , and each , it is easy to prove that T is continuous at .
Step 3. For each bounded subset , prove that T is relative compact on D.
It follows that T Ω is bounded. Since , one knows that T Ω is relative compact. Steps 1, 2, and 3 imply that T is completely continuous. □
3 Main result
In this section, our objective is to establish the existence of at least three positive solutions for BVP (1.1) by using the Avery-Peterson fixed point theorem .
Choose , where denotes the largest integer not greater than x, and denote .
such that the following conditions are satisfied:
(C2) for all ;
(C3) for all ;
(C4) for all ,
then BVP (1.1) has at least three positive solutions.
Proof We choose positive numbers t, v, , z with , . Next we show that all the conditions of Lemma 2.1 are satisfied.
It is clear that for and , there are , . From (3.2), we have . Furthermore, and therefore .
Now the proof is divided into four steps.
Therefore, . Hence, by Lemma 2.7, we know that is completely continuous.
Step 2. We show that condition (i) in Lemma 2.1 holds.
since . Hence .
For , we have and for .
We conclude that condition (i) of Lemma 2.1 holds.
Then condition (ii) of Lemma 2.1 is satisfied.
Thus, condition (iii) of Lemma 2.1 is satisfied.
4 An example
where is continuous and positive for all . Corresponding to BVP (1.1), we have , , , , , , .
It is easy to see that (C1) holds.
, for all ;
for all ;
, for all ,
The authors would like to thank the referees for their valuable suggestions and comments.
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