Positive Solutions for Third-Order -Laplacian Functional Dynamic Equations on Time Scales

Abstract

The authors study the boundary value problems for a -Laplacian functional dynamic equation on a time scale, , , , , , . By using the twin fixed-point theorem, sufficient conditions are established for the existence of twin positive solutions.

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 (or measure chain). For notation, we shall use the convention that, for each interval of of , will denote time scales interval, that is, .

In this paper, let be a time scale such that , 0, . 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 ; and

the function is continuous,

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 ,

is continuous and satisfies that there are such that

(1.2)

-Laplacian problems with two-, three-, m-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 [5–8] and references therein. Especially, Kaufmann and Raffoul [8] considered a nonlinear functional dynamic equation on a time scale and obtained sufficient conditions for the existence of positive solutions. In this paper, we apply the twin fixed-point theorem to obtain at least two positive solutions of boundary value problem (BVP for short) (1.1) when growth conditions are imposed on . Finally, we present two corollaries, which show that under the assumptions that is superlinear or sublinear, BVP (1.1) has at least two positive solutions.

Given a nonnegative continuous functional on a cone of a real Banach space , we define for each the sets

(1.3)

The following twin fixed-point lemma due to [9] will play an important role in the proof of our results.

Lemma 1.1.

Let be a real Banach space, a cone of , and two nonnegative increasing continuous functionals, a nonnegative continuous functional, and . Suppose that there are two positive numbers and such that

(1.4)

is completely continuous. There are positive numbers such that

(1.5)

and

(i) for ,

(ii) for ,

(iii) and for .

Then, has at least two fixed points and satisfying

(1.6)

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 is concave and nonnegative valued on , and .

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

Define as

(2.2)

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

(2.3)

Then, denotes a positive solution of BVP (1.1).

It follows from (2.2) that

Lemma 2.1.

Let be defined by (2.2). If , then

(i).

(ii) is completely continuous.

(iii), .

(iv) is decreasing on .

The proof is similar to the proofs of Lemma  2.3 and Theorem  3.1 in [7], and is omitted.

Fix such that , and set

(2.4)

Throughout this paper, we assume and .

Now, we define the nonnegative, increasing, continuous functionals , , and on by

(2.5)

We have

(2.6)

Then,

(2.7)

We also see that

(2.8)

For the notational convenience, we denote , and , by

(2.9)

Theorem 2.2.

Suppose that there are positive numbers such that

(2.10)

Assume satisfies the following conditions:

for , uniformly in ,

for , uniformly in ,

(2.11)

for , uniformly in .

Then, BVP (1.1) has at least two positive solutions of the form

(2.12)

where , and , .

Proof.

By the definition of operator and its properties, it suffices to show that the conditions of Lemma 1.1 hold with respect to .

First, we verify that implies .

Since , one gets for . Recalling that (2.7), we know for . Then, we get

(2.13)

Secondly, we prove that implies .

Since implies , it holds that for , and for all implies

(2.14)

Then,

(2.15)

So, we have

(2.16)

Finally, we show that

(2.17)

It is obvious that . On the other hand, and (2.7) imply

(2.18)

Thus,

(2.19)

By Lemma 1.1, has at least two different fixed points and satisfying

(2.20)

Let

(2.21)

which are twin positive solutions of BVP (1.1). The proof is complete.

In analogy to Theorem 2.2, we have the following result.

Theorem 2.3.

Suppose that there are positive numbers such that

(2.22)

Assume satisfies the following conditions:

for , uniformly in ,

(2.23)

for , uniformly in ,

for , uniformly in ,

(2.24)

Then, BVP (1.1) has at least two positive solutions of the form

(2.25)

Now, we give theorems, which may be considered as the corollaries of Theorems 2.2 and 2.3.

Let

(2.26)

and choose , , such that

(2.27)

From above, we deduce that .

Theorem 2.4.

If the following conditions are satisfied:

, , uniformly in ,

there exists a such that for all , one has

(2.28)

Then, BVP (1.1) has at least two positive solutions of the form

(2.29)

Proof.

First, choose , one gets

(2.30)

Secondly, since , there is sufficiently small such that

(2.31)

Without loss of generality, suppose . Choose so that . For , we have and . Thus,

(2.32)

Thirdly, since , there is sufficiently large such that

(2.33)

Without loss of generality, suppose . Choose . Then,

(2.34)

We get now , and then the conditions in Theorem 2.2 are all satisfied. By Theorem 2.2, BVP (1.1) has at least two positive solutions. The proof is complete.

Theorem 2.5.

If the following conditions are satisfied:

, uniformly in ; ,

there exists a such that for all , one has

(2.35)

Then, BVP (1.1) has at least two positive solutions of the form

(2.36)

The proof is similar to that of Theorem 2.4 and we omitted it.

The following Corollaries are obvious.

Corollary 2.6.

If the following conditions are satisfied:

, , uniformly in ,

there exists a such that for all , one has

(2.37)

Then, BVP (1.1) has at least two positive solutions of the form

(2.38)

Corollary 2.7.

If the following conditions are satisfied:

, uniformly in , ;

there exists a such that for all , one has

(2.39)

Then, BVP (1.1) has at least two positive solutions of the form

(2.40)

3. Example

Example 3.1.

Let , , , , , .

We consider the following boundary value problem:

(3.1)

where and ; , .

Choosing , , , , direct calculation shows that

(3.2)

Consequently, and satisfies

for , uniformly in ,

(3.3)

for , uniformly in ,

for , uniformly in ,

(3.4)

Then all conditions of Theorem 2.3 hold. Thus, with Theorem 2.3, the BVP (3.1) has at least two positive solutions.