- Open Access
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.
- Anderson DR: Discrete third-order three-point right-focal boundary value problems. Comput. Math. Appl. 2003, 45: 861-871. 10.1016/S0898-1221(03)80157-8MathSciNetView ArticleGoogle Scholar
- Anderson D, Avery RI: Multiple positive solutions to a third-order discrete focal boundary value problem. Comput. Math. Appl. 2001, 42: 333-340. 10.1016/S0898-1221(01)00158-4MathSciNetView ArticleGoogle Scholar
- Avery RI, Peterson AC: Three positive fixed points of nonlinear operators on ordered Banach spaces. Comput. Math. Appl. 2001, 42: 313-322. 10.1016/S0898-1221(01)00156-0MathSciNetView ArticleGoogle Scholar
- Aykut N: Existence of positive solutions for boundary value problems of second-order functional difference equations. Comput. Math. Appl. 2004, 48: 517-527. 10.1016/j.camwa.2003.10.007MathSciNetView ArticleGoogle Scholar
- Cai X, Yu J: Existence theorems for second-order discrete boundary value problems. J. Math. Anal. Appl. 2006, 320: 649-661. 10.1016/j.jmaa.2005.07.029MathSciNetView ArticleGoogle Scholar
- Cheung W-S, Ren J, Wong PJY, Zhao D: Multiple positive solutions for discrete nonlocal boundary value problems. J. Math. Anal. Appl. 2007, 330: 900-915. 10.1016/j.jmaa.2006.08.034MathSciNetView ArticleGoogle Scholar
- Graef JR, Henderson J: Double solutions of boundary value problems for 2 m th-order differential equations and difference equations. Comput. Math. Appl. 2003, 45: 873-885. 10.1016/S0898-1221(03)00063-4MathSciNetView ArticleGoogle Scholar
- He Z: On the existence of positive solutions of p -Laplacian difference equations. J. Comput. Appl. Math. 2003, 161: 193-201. 10.1016/j.cam.2003.08.004MathSciNetView ArticleGoogle Scholar
- Yaslan Karaca I: Discrete third-order three-point boundary value problem. J. Comput. Appl. Math. 2007, 205: 458-468. 10.1016/j.cam.2006.05.030MathSciNetView ArticleGoogle Scholar
- Liu Y: Existence results for positive solutions of non-homogeneous BVPs for second order difference equations with one-dimensional p -Laplacian. J. Korean Math. Soc. 2010, 47: 135-163. 10.4134/JKMS.2010.47.1.135MathSciNetView ArticleGoogle Scholar
- Xia J, Debnath L, Jiang H, Liu Y: Three positive solutions of Sturm-Liouville type multi-point BVPs for second order p-Laplacian difference equations. Bull. Pure Appl. Math. 2010, 4: 266-287.MathSciNetGoogle Scholar
- Liu Y: Studies on nonhomogeneous multi-point BVPs of difference equations with one-dimensional p -Laplacian. Mediterr. J. Math. 2011, 8: 577-602. 10.1007/s00009-010-0089-1MathSciNetView ArticleGoogle Scholar
- Xia J, Liu Y: Positive solutions of BVPs for infinite difference equations with one-dimensional p -Laplacian. Miskolc Math. Notes 2012, 13: 149-176.MathSciNetGoogle Scholar
- Bian L-H, He X-P, Sun H-R: Multiple positive solutions of m -point BVPs for third-order p -Laplacian dynamic equations on time scales. Adv. Differ. Equ. 2009., 2009: Article ID 262857Google Scholar
- Xu F, Meng Z: The existence of positive solutions for third-order p -Laplacian m -point boundary value problems with sign changing nonlinearity on time scales. Adv. Differ. Equ. 2009., 2009: Article ID 169321Google Scholar
- Han W, Liu M: Existence and uniqueness of a nontrivial solution for a class of third-order nonlinear p -Laplacian m -point eigenvalue problems on time scales. Nonlinear Anal. 2009, 70(5):1877-1889. 10.1016/j.na.2008.02.088MathSciNetView ArticleGoogle Scholar
- Wu LM, Yang J: Positive solutions to third-order nonlinear p -Laplacian functional dynamic equations on time scales. J. Lanzhou Univ. Nat. Sci. 2010, 46(6):100-104.MathSciNetGoogle Scholar
- Song C, Gao X: Positive solutions for third-order p -Laplacian functional dynamic equations on time scales. Bound. Value Probl. 2011., 2011: Article ID 585378Google Scholar
- Bai Z, Ge W: Existence of three positive solutions for some second-order boundary value problems. Comput. Math. Appl. 2004, 48: 699-707. 10.1016/j.camwa.2004.03.002MathSciNetView ArticleGoogle Scholar
- Avery RI: A generalization of the Leggett-Williams fixed point theorem. Math. Sci. Res. Hot-Line 1999, 3: 9-14.MathSciNetGoogle Scholar
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