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Existence of positive solutions for boundary value problems of p-Laplacian difference equations
Advances in Difference Equations volume 2014, Article number: 263 (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.
The aim of this paper is to study the existence of positive solutions for a third order p-Laplacian difference equation
, 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.
Liu  studied the following second order p-Laplacian difference equation with multi-point boundary conditions:
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 .
Liu  studied the following boundary value problem:
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.
Let γ and θ be nonnegative continuous convex functionals on , α be a nonnegative continuous concave functional on and ψ be a nonnegative continuous functional on . Then for positive real numbers t, v, w, and z, we define the following convex sets of :
and a closed set
The following fixed point theorem is fundamental and important to the proof of our main result.
Lemma 2.1 ()
Let be a real Banach space and be a cone in . Let γ, θ be nonnegative continuous convex functionals on , let α be a nonnegative continuous concave functional on , and let ψ be a nonnegative continuous functional on satisfying for all , such that for some positive numbers z and M,
Suppose that is completely continuous and there exist positive numbers t, v, and w with such that
and for ;
for with ;
and for with .
Then T has at least three fixed points such that
Let () be a positive sequence. Consider the following BVP:
Lemma 2.2 If y is a solution of BVP (2.1), then there exists unique such that and .
Proof Suppose y satisfies (2.1). It follows that
The BCs in (2.1) imply that
It follows that
Similarly, we get
The BCs in (2.1) imply that
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 .
Proof We get from Lemma 2.2 the result that there exists unique such that and . It follows from (2.1) that
It follows from being positive, , and that
So for all . From BCs in BVP (2.1), we get
Hence for all . The proof is complete. □
Lemma 2.4 If y is a solution of BVP (2.1), then
Proof It follows from Lemma 2.2 and Lemma 2.3 that for . Suppose that . Since and , we get . For , it is easy to see that
Since for all , we get for all . Then . It follows that . Then
Similarly, if , we get
Lemma 2.5 If y is a solution of BVP (2.1), then
where satisfies the equation
Proof The proof follows from Lemma 2.2 and is omitted. □
Lemma 2.6 If y is a solution of BVP (2.1), then there exists an such that
Proof It follows from Lemma 2.2 that there is such that and , for all and for all . Then
there exists such that
It is easy to see from (2.1) that
Here . So (2.6)-(2.8) imply that
Lemma 2.3 implies that . Furthermore, one has from (2.6)
On the other hand, by a discussion similar to Lemma 2.2 and Lemma 2.3, we have , with
It follows that
One has from Lemma 2.3 . Therefore
Let . Then satisfies the following equation:
Let . We call for if for all .
Define the norm
It is easy to see that is a semi-ordered real Banach space.
where . Then is a cone in .
Define the operator by
for , . Then
Lemma 2.7 Suppose that (C1) holds. Then
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
It follows from (2.9) in the proof of Lemma 2.6 that
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.
In fact, for each bounded subset and . Suppose
and Step 1 implies that there exists a constant such that . Then
where . Similarly, one has
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 .
Define the functionals on by
For and we have
Hence, we obtain
Theorem 3.1 Suppose that (C1) holds. If there are positive numbers
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.
Step 1. We will show that (C2) implies that
For , we have . From (3.2) we get
This implies that (C2) holds. Then one has from (C2) and (2.9) in the proof of Lemma 2.6
Therefore, . Hence, by Lemma 2.7, we know that is completely continuous.
Step 2. We show that condition (i) in Lemma 2.1 holds.
Choose for all . It is easy to see that
since . Hence .
For , we have and for .
It follows from (C3) that
Similarly to Lemma 2.6 there exists such that and and
we get from (2.4), (C1), (C3), and Lemma 2.6
We conclude that condition (i) of Lemma 2.1 holds.
Step 3. We prove that condition (ii) of Lemma 2.1 holds. If and , then we have
Then condition (ii) of Lemma 2.1 is satisfied.
Step 4. Finally, we verify that (iii) of Lemma 2.1 also holds. Clearly, . Suppose that with . Then by condition (C4) and (3.1), we obtain
Thus, condition (iii) of Lemma 2.1 is satisfied.
From Steps 1-4 together with Lemma 2.1 we find that the operator T has at least three fixed points which are positive solutions , , and belonging to of (1.1) such that
4 An example
Example 4.1 Consider the following BVP:
where is continuous and positive for all . Corresponding to BVP (1.1), we have , , , , , , .
It is easy to see that (C1) holds.
Choose the constant , then , , . Taking , , and , it is easy to check that
, for all ;
for all ;
, for all ,
then Theorem 3.1 implies that BVP (4.1) has at least three positive solutions such that
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The authors would like to thank the referees for their valuable suggestions and comments.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.
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Fen, F.T., Karaca, I.Y. Existence of positive solutions for boundary value problems of p-Laplacian difference equations. Adv Differ Equ 2014, 263 (2014). https://doi.org/10.1186/1687-1847-2014-263
- positive solutions
- fixed point theorem
- p-Laplacian difference equation
- boundary value problem