Nodal Solutions of a Nonlinear Weighted Eigenvalue Problem on time scales
DONG ZHANG1, SHU-HONG BU2
1Department of Mathematics, Cangzhou Normal College, Hebei 061000, China
2Department of Information Engineering, Baoding Science and Technology Normal School, Hebei 071000, China
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Abstract: This paper investigates the existence of nodal solutions of a nonlinear weighted boundary value problem
, The main tool which the authors establish the existence theorems used is global bifurcation theory.
Keywords: Weighted eigenvalue problem; Bifurcation; Nodal solutions
MR(2010) Subject Classification : O175.15
1. Introduction
In this paper, we consider the existence of nodal solutions of the eqution
(1.1)
with the boundary condition
(1.2)
where is nonnegative and has nonzero zero points. When the function and are nonnegative, the boundary problem (1.1)-(1.2) has been studied by several authors[1-5] . The existence of solutions for a class of nonlinear problem on timescales is discussed[6-10]. The approaches used in the literature are usually topological degree theory and fixed-point theorems in cones, and the assumptions on the nonlinearities are related to the principle eigenvalue of corresponding linear eigenvalue problems, at the same time the results on the nonlinear equation can be improved in general. Recently, Luo[1] showed the following result
Theorem A. Suppose that
(A1) and when ,
(A2) and when ,
(A3) and when ,
(A4) there exist satisfying .
Ifsatisfies one of the following conditions:
（1）,
（2）.
Then the BVP (1.1)-(1.2) has at least one positive solutionand one negative solution satisfying and when .
Seeing Theorem A, We cannot but ask “whether or not nodal solutions can be drown when satisfies suitable conditions”. The purpose of this study is to establish a simple criterion for the existence of nodal solutions of (1.1)-(1.2), which extend the result in [1] . The key conditions we used are closely related to the principle eigenvalue of corresponding weighted linear eigenvalue problems. The proof of the above work was based on an application of global bifurcation theory on time scales and spectral theory of weighted linear eigenvalue problem.
2. Preliminaries
If T represents time scales, denote all the continuous operator on T. Define with normand with norm+. A linear mappingrepresents where . It is very easy to confirm that is completely continuous.
Lemma 2.1. [2] Let satisfy condition (A1)-(A2), the nonlinear weighted eigenvalue problem
(2.1)
(2.2)
has only simple positive eigenvalue which are real numbers satisfy .
3. Main Results
In what follows we shall make use of the following conditions:
() and when ,
() and when ,
() satisfies with and in case , and in case ,
() there exist satisfying .
For any , let be the set of the functionwhere
(i) u has only extended simple zeros in (0,1), and has exactlysuch zeros;
(ii) and .
Denote , they are open subsets and mutually disjoint in E. Define . Let be connected component which bifurcates from , and be connected component which bifurcates from .
Lemma 3.1. Let ()-() hold, furthermore f satisfies Lipschitsz condition on , then
(i) when , in case ,
(ii) and when , in case .
Proof. For any , just need to prove that
and . To prove this conclusion, we make use of proof by contradiction. Suppose there exist such that either
(3.1)
or (3.2)
As result of (), there exists two constants m such that
is strictly increasing on . (3.3)
Note to be zero points of u on T. From (3.1) , we conclude that there is such that
(3.4)
and
(3.5)
Consider two point boundary value problem
(3.6)
(3.7)
Since (3.3),
when, (3.8)
and , we conclude that
(3.9)
and
and (3.10)
Using maximum value theory, we get with , which is contradict (3.4). From (3.2) , we conclude that there is such that
(3.11)
and
(3.12)
Consider two point boundary value problem
(3.13)
(3.14)
Since (3.3),
when, (3.15)
and , we conclude that
(3.16)
and
and (3.17)
By use of minimum value theory, we get with , which is contradict (3.11). So the conclusion (i) of Lemma 3.1 hold..
Similarly, we can prove the conclusion (ii) of Lemma 3.1.
Remark. Suppose u is the nontrivial solution of problem (1.1)-(1.2) under the conditions ()-(), then there exist and such that .
Theorem 3.1. Let ()-() hold, assume , then
(i)when , the problem (1.1)-(1.2) has at least two solutions and which have exactly extended simple zeros on such that and .
(ii)when , the problem (1.1)-(1.2) has at least four solutions ,and which have exactly extended simple zeros onsuch that ,,
and .
Proof. Assume , such that and , in which apparently and . Investigate bifurcation of the problem
(3.18)
coming from general solution , and bifurcation of the problem
(3.19)
coming from infinity. It is easy to check that the problem (1.1)-(1.2) can be converted to (3.18) and (3.19). Let
,
then the problem (1.1)-(1.2) what we investigate is equivalent to the problem
(3.20)
with the boundary condition
. (3.21)
Due to the uniqueness of solution for the initial value problem (3.20)-(3.21) that u has only simple zeros. Make use of Rabinowitz global bifurcation, for any integer and there exist unbounded closed connected component for solutions of the problem (3.18), which bifurcates from. In the same way, there exist unbounded closed connected component for solutions of the problem (3.19), which bifurcates from . Since Remark , any nontrivial solution of problem (1.1)-(1.2) has the properties of nodal solution.
Consequently, k of and is used in accordance with . Since the problem (1.1)-(1.2) has only trivial solution when , so that crosses the hyperplane. Combining Remark, and are unbounded. Then . Let be a interval satisfying . Denote domain of , such that the projection of M on R fall into and the projection of M on E keep away from 0 and unbounded.
Case 1. keeps bounded on , in this case intersect with trivial solution .
Case 2. keeps unbounded.
Moreover, if case 2 holds and the projection of on R is bounded, then joint where . From Lemma 3.1., the case 1 is impossible.The Remark guarantee is composed with the solutions of (3.19) and joins . We will demonstrate that be a connected component on . Otherwise if there exist and where and , then it has such that has repeated zero points on , which contradict with Remark. From above we have no and such that joins . Then is unbounded such that
. (3.22)
From Lemma 3.1, for any ,
. (3.23)
Combine (3.23) with (3.18), it follows
. (3.24)
This illustrate for any fixed constant , the set is bounded. Moreover joins to , it follows
. (3.2)
Theorem 3.2. Let ()-() hold, assume , then
(i) when , the problem (1.1)-(1.2) has at least two solutions and which have exactly extended simple zeros on such that and .
(ii)when , the problem (1.1)-(1.2) has at least four solutions , and which have exactly extended simple zeros onsuch that , , and .
Applying a similar argument to that used in Theorem 3.1., the result follows.
4. Conclusions
In this sample paper, we have presented the existence of nodal solution for nonlinear weighed equation
(4.1)
with the boundary condition
(4.2)
Construct conditions that the function satisfies with
andin case and in case , and there exist constants satisfying. Using global bifurcation theory and eigenvalue theory on time scales, what follows the existence theorems in relation to the k-th eigenvalue for the solutions of problem (4.1)-(4.2).
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