Eigenvalue Problems for -Laplacian Functional Dynamic Equations on Time Scales
© Changxiu Song. 2008
Received: 29 February 2008
Accepted: 25 June 2008
Published: 2 July 2008
This paper is concerned with the existence and nonexistence of positive solutions of the -Laplacian functional dynamic equation on a time scale, , , , , , . We show that there exists a such that the above boundary value problem has at least two, one, and no positive solutions for and , respectively.
Let be a closed nonempty subset of , and let have the subspace topology inherited from the Euclidean topology on . In some of the current literature, is called a time scale (please see [1, 2]). For notation, we will use the convention that, for each interval of will denote time-scale interval, that is,
where is the -Laplacian operator, that is, , where .
The function is continuous and nondecreasing about each element;
The function is left dense continuous (i.e., and does not vanish identically on any closed subinterval of . Here denotes the set of all left dense continuous functions from to .
is continuous and .
is continuous, for all .
- (H5)is continuous and nondecreasing; and satisfies that there exist such that(1.2)
-Laplacian problems with two-, three-, -point boundary conditions for ordinary differential equations and finite difference equations have been studied extensively, for example, see [1–4] and references therein. However, there are not many concerning the -Laplacian problems on time scales, especially for -Laplacian functional dynamic equations on time scales.
The motivations for the present work stems from many recent investigations in [5–10] and references therein. Especially, Kaufmann and Raffoul  considered a nonlinear functional dynamic equation on a time scale and obtained sufficient conditions for the existence of positive solutions, Li and Liu  studied the eigenvalue problem for second-order nonlinear dynamic equations on time scales. In this paper, our results show that the number of positive solutions of (1.1) is determined by the parameter . That is to say, we prove that there exists a such that (1.1) has at least two, one, and no positive solutions for and respectively.
If is said to be right scattered, and 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
If , then If , then is forward difference operator while is the backward difference operator.
If , then define the delta integral by If , then define the nabla integral by
The following lemma is crucial to prove our main results.
Lemma 1.4 ().
Let be a Banach space and let be a cone in . For , define Assume that is completely continuous such that for
If for then
If for then
2. Positive Solutions
Clearly, is a Banach space with the norm . For each , extend to with for .
Then denotes a positive solution of BVP (1.1).
It follows from (2.3) that the following lemma holds.
Let be defined by (2.3). If , then
is completely continuous.
The proof of Lemma 2.1 can be found in .
Throughout this paper, we assume and
Suppose that (H1)–(H5) hold. Then there exists a such that the operator has a fixed point at , where is the zero element of the Banach space .
By the Lebesgue dominated convergence theorem  together with (H3), it follows that decreases to a fixed point of the operator The proof is complete.
Suppose that (H1)–(H6) hold and that for some . Then there exists a constant such that for all and all possible fixed points of at , one has
We need to prove that there exists a constant such that for all If the number of elements of is finite, then the result is obvious. If not, without loss of generality, we assume that there exists a sequence such that , where is the fixed point of the operator defined by (2.3) at
which is a contradiction. The proof is complete.
Suppose that (H1)–(H5) hold and that the operator has a positive fixed point in at . Then for every the operator has a fixed point at , and
where is also defined by (2.6), which implies that decreases to a fixed point of the operator , and The proof is complete.
Suppose that (H1)–(H6) hold. Let have at least one fixed point at in . Then is bounded above.
there exists a constant such that
there exists a subsequence such that which is impossible by Lemma 2.3.
which is a contradiction. The proof is complete.
Let Then where is defined just as in Lemma 2.5.
which shows that has a positive fixed point at The proof is complete.
Suppose that (H1)–(H6) hold. Then there exists a such that (1.1) has at least two, one, and no positive solutions for and respectively.
So, has at least two fixed points in . The proof is complete.
This work was supported by Grant 10571064 from NNSF of China, and by a grant from NSF of Guangdong.
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