The Existence of Positive Solutions for Third-Order -Laplacian -Point Boundary Value Problems with Sign Changing Nonlinearity on Time Scales
© F. Xu and Z. Meng. 2009
Received: 25 February 2009
Accepted: 2 June 2009
Published: 5 July 2009
We study the following third-order -Laplacian -point boundary value problems on time scales , , , , , where is -Laplacian operator, that is, , , , . We obtain the existence of positive solutions by using fixed-point theorem in cones. In particular, the nonlinear term is allowed to change sign. The conclusions in this paper essentially extend and improve the known results.
The theory of time scales was initiated by Hilger  as a mean of unifying and extending theories from differential and difference equations. The study of time scales has lead to several important applications in the study of insect population models, neural networks, heat transfer, and epidemic models, see, for example [2–6]. Recently, the boundary value problems with -Laplacian operator have also been discussed extensively in literature; for example, see [7–18]. However, to the best of our knowledge, there are not many results concerning the higher-order -Laplacian mutilpoint boundary value problem on time scales.
A time scale is a nonempty closed subset of . We make the blanket assumption that are points in . By an interval , we always mean the intersection of the real interval with the given time scale; that is .
author studied the existence of solutions for the nonlinear boundary value problem by using Krasnoselskii's fixed point theorem and Leggett and Williams fixed point theorem, respectively.
where . He obtained the existence of at least double and triple positive solutions of the problems by using a new double fixed point theorem and triple fixed point theorem, respectively.
They established a corresponding iterative scheme for the problem by using the monotone iterative technique.
All the above works were done under the assumption that the nonlinear term is nonnegative. The key conditions used in the above papers ensure that positive solution is concave down. If the nonlinearity is negative somewhere, then the solution needs no longer to be concave down. As a result, it is difficult to find positive solutions of the -Laplacian equation when the nonlinearity changes sign. In particular, little work has been done on the existence of positive solutions for higher order -Laplacian -point boundary value problems with nonlinearity being nonnegative on time scales. Therefore, it is a natural problem to consider the existence of positive solution for higher order -Laplacian equations with sign changing nonlinearity on time scales. This paper attempts to fill this gap in literature.
where is -Laplacian operator, that is, , , and , , , satisfy
, , , ;
is continuous, , and there exists such that .
2. Preliminaries and Lemmas
for all . If , is said to be right scattered, if , is said to be left scattered; if , is said to be right dense, and if , is said to be left dense. If has a right scattered minimum , define ; otherwise set . If has a left scattered maximum , define ; otherwise set .
for all .
for all .
A function is left-dense continuous (i.e., -continuous), if is continuous at each left-dense point in and its right-sided limit exists at each right-dense point in .
By caculating, we can easily get (2.7). So we omit it.
where . The proof is complete.
The proof is completed.
where , .
This completes the proof.
Let be endowed with the ordering if for all and is defined as usual by maximum norm. Clearly, it follows that is a Banach space.
where , , and . Obviously, is a solution of the BVP(1.6) if and only if is a fixed point of operator .
is completely continuous.
It is easy to see that by and Lemma 2.8. By Arzela-Ascoli theorem and Lebesgue dominated convergence theorem, we can easily prove that operator is completely continuous.
Let be a cone in a Banach space . Let be an open bounded subset of with and . Assume that is a compact map such that for . Then the following results hold.
(1)If , , then .
(2)If there exists such that for all and all , then .
(3)Let be open in such that . If and , then has a fixed point in . The same result holds if and , where denotes fixed point index.
Lemma 2.11 (see ).
defined above has the following properties:
(b) is open relative to K;
(c) if and only if
(d)if , then for .
By we can know that ,
This implies that for . Hence by Lemma 2.10(1) it follows that .
a contradiction. Hence by Lemma 2.10(2) it follows that .
3. Main Results
We now give our results on the existence of positive solutions of BVP (1.6).
Suppose that conditions and hold, and assume that one of the following conditions holds.
There exist with such that
(i) , ;
(ii) , , moreover , .
There exist with such that
(i) , ;
(ii) , .
Then, the BVP (1.6) has at least one positive solution.
This means that is a fixed point of operator .
This means that is a fixed point of operator . Therefore, the BVP (1.6) has at least one positive solution.
Assume that conditions and hold, and suppose that one of the following conditions holds.
There exist , and with , and such that
(i) , ;
(ii) , , moreover , , ;
(iii) , .
There exist , and with such that
(i) , ;
(ii) , , , ;
(iii) , , moreover, , .
Then, the BVP (1.6) has at least two positive solutions.
This means that is a fixed point of operator . Then, the BVP (1.6) has at least two positive solutions.
When condition holds, the proof is similar to the above, and so we omit it here.
4. An Example
In the section, we present some simple examples to explain our results.
where , , , , .
By computing, we can know , , , . Obviously, , .
Then, by the definition of we have
(i) , ;
(ii) , , moreover , .
So condition holds, and by Theorem 3.1, BVP (4.1) has at least one positive solution.
This project was supported by the National Natural Science Foundation of China (10471075, 10771117).
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