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Eigenvalue Problems for
-Laplacian Functional Dynamic Equations on Time Scales
Advances in Difference Equations volume 2008, Article number: 879140 (2008)
Abstract
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.
1. Introduction
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,
In this paper, let be a time scale such that
We are concerned with the existence of positive solutions of the
-Laplacian dynamic equation on a time scale

where is the
-Laplacian operator, that is,
, where
.
-
(H1)
The function
is continuous and nondecreasing about each element;
-
(H2)
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
.
-
(H3)
is continuous and
.
-
(H4)
is continuous,
for all
.
-
(H5)
is continuous and nondecreasing;
and satisfies that there exist
such that
(1.2) -
(H6)
uniformly in
-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 [7] considered a nonlinear functional dynamic equation on a time scale and obtained sufficient conditions for the existence of positive solutions, Li and Liu [10] 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.
For convenience, we list the following well-known definitions which can be found in [11–13] and the references therein.
Definition 1.1.
For and
, define the forward jump operator
and the backward jump operator
, respectively, as

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
Definition 1.2.
For and
define the deltaderivative of
to be the number (when it exists), with the property that, for any
, there is a neighborhood
of
such that

For and
define the nabla derivative of
to be the number (when it exists), with the property that, for any
, there is a neighborhood
of
such that

If , then
If
, then
is forward difference operator while
is the backward difference operator.
Definition 1.3.
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 ([14]).
Let be a Banach space and let
be a cone in
. For
, define
Assume that
is completely continuous such that
for
-
(i)
If
for
then
-
(ii)
If
for
then
2. Positive Solutions
We note that is a solution of (1.1) if and only if

Let be endowed with the norm
and define the cone of
by

Clearly, is a Banach space with the norm
. For each
, extend
to
with
for
.
Define as

We seek a fixed point, , of
in the cone
. Define

Then denotes a positive solution of BVP (1.1).
It follows from (2.3) that the following lemma holds.
Lemma 2.1.
Let be defined by (2.3). If
, then
-
(i)
-
(ii)
is completely continuous.
The proof of Lemma 2.1 can be found in [15].
We need to define further subsets of with respect to the delay
. Set

Throughout this paper, we assume and
Lemma 2.2.
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
.
Proof.
Set

We know that Let
where

From above, we have

Let and
Then

By the Lebesgue dominated convergence theorem [16] together with (H3), it follows that decreases to a fixed point
of the operator
The proof is complete.
Lemma 2.3.
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
Proof.
Set

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
Then

We choose such that

such that

In view of (H6) there exists an sufficiently large such that
For
we have

which is a contradiction. The proof is complete.
Lemma 2.4.
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
Proof.
Let be the fixed point of the operator
at
. Then

where Set

and
Then

where is also defined by (2.6), which implies that
decreases to a fixed point
of the operator
, and
The proof is complete.
Lemma 2.5.
Suppose that (H1)–(H6) hold. Let have at least one fixed point at
in
. Then
is bounded above.
Proof.
Suppose to the contrary that there exists a fixed point sequence of
at
such that
Then we need to consider two cases:
-
(i)
there exists a constant
such that
-
(ii)
there exists a subsequence
such that
which is impossible by Lemma 2.3.
Only (i) is considered. We can choose such that
, and further
. For
, we have

Now we consider (2.18). Assume that the case (i) holds. Then

leads to

which is a contradiction. The proof is complete.
Lemma 2.6.
Let Then
where
is defined just as in Lemma 2.5.
Proof.
In view of Lemma 2.4, it follows that We only need to prove
In fact, by the definition of
, we may choose a distinct nondecreasing sequence
such that
Let
be the positive fixed point of
at
By Lemma 2.3,
is uniformly bounded, so it has a subsequence denoted by
converging to
Note that

Taking the limitation to both sides of (2.21), and using the Lebesgue dominated convergence theorem [16], we have

which shows that has a positive fixed point
at
The proof is complete.
Theorem 2.7.
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.
Proof.
Assume that (H1)–(H5) hold. Then there exists a such that
has a fixed point
at
In view of Lemma 2.4,
also has a fixed point
and
Note that
is continuous on
. For
there exists a
such that

Hence,

From above, we have

Set for
and
. We have
for
By Lemma 2.1,
In view of (H6), we can choose
such that

Set

Similar to Lemma 2.3, it is easy to obtain that

In view of Lemma 2.1, By the additivity of fixed point index,

So, has at least two fixed points in
. The proof is complete.
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Acknowledgments
This work was supported by Grant 10571064 from NNSF of China, and by a grant from NSF of Guangdong.
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Song, C. Eigenvalue Problems for -Laplacian Functional Dynamic Equations on Time Scales.
Adv Differ Equ 2008, 879140 (2008). https://doi.org/10.1155/2008/879140
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DOI: https://doi.org/10.1155/2008/879140