Eigenvalue Problems for -Laplacian Functional Dynamic Equations on Time Scales

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

(1.1)

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

(1.3)

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

(1.4)

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

(1.5)

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

(2.1)

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

(2.2)

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

Define as

(2.3)

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

(2.4)

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

(2.5)

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

(2.6)

We know that Let where

(2.7)

From above, we have

(2.8)

Let and Then

(2.9)

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

(2.10)

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

(2.11)

We choose such that

(2.12)

such that

(2.13)

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

(2.14)

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

(2.15)

where Set

(2.16)

and Then

(2.17)

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

(2.18)

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

(2.19)

leads to

(2.20)

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

(2.21)

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

(2.22)

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

(2.23)

Hence,

(2.24)

From above, we have

(2.25)

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

(2.26)

Set

(2.27)

Similar to Lemma 2.3, it is easy to obtain that

(2.28)

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

(2.29)

So, has at least two fixed points in . The proof is complete.