Existence of Solutions for Nonlinear Four-Point -Laplacian Boundary Value Problems on Time Scales

Abstract

We are concerned with proving the existence of positive solutions of a nonlinear second-order four-point boundary value problem with a -Laplacian operator on time scales. The proofs are based on the fixed point theorems concerning cones in a Banach space. Existence result for -Laplacian boundary value problem is also given by the monotone method.

1. Introduction

Let be any time scale such that be subset of . The concept of dynamic equations on time scales can build bridges between differential and difference equations. This concept not only gives us unified approach to study the boundary value problems on discrete intervals with uniform step size and real intervals but also gives an extended approach to study on discrete case with non uniform step size or combination of real and discrete intervals. Some basic definitions and theorems on time scales can be found in [1, 2].

In this paper, we study the existence of positive solutions for the following nonlinear four-point boundary value problem with a -Laplacian operator:

(1.1)

(1.2)

where is an operator, that is, for , where , , , with :

• (H1) the function ,

• (H2) the function and does not vanish identically on any closed subinterval of and ,

• (H3) is continuous and satisfies that there exist such that for .

In recent years, the existence of positive solutions for nonlinear boundary value problems with -Laplacians has received wide attention, since it has led to several important mathematical and physical applications [3, 4]. In particular, for or is linear, the existence of positive solutions for nonlinear singular boundary value problems has been obtained [5, 6]. -Laplacian problems with two-, three-, and m-point boundary conditions for ordinary differential equations and difference equations have been studied in [7–9] and the references therein. Recently, there is much attention paid to question of positive solutions of boundary value problems for second-order dynamic equations on time scales, see [10–13]. In particular, we would like to mention some results of Agarwal and O'Regan [14], Chyan and Henderson [5], Song and Weng [15], Sun and Li [16], and Liu [17], which motivate us to consider the -Laplacian boundary value problem on time scales.

The aim of this paper is to establish some simple criterions for the existence of positive solutions of the -Laplacian BVP (1.1)-(1.2). This paper is organized as follows. In Section 2 we first present the solution and some properties of the solution of the linear -Laplacian BVP corresponding to (1.1)-(1.2). Consequently we define the Banach space, cone and the integral operator to prove the existence of the solution of (1.1)-(1.2). In Section 3, we state the fixed point theorems in order to prove the main results and we get the existence of at least one and two positive solutions for nonlinear -Laplacian BVP (1.1)-(1.2). Finally, using the monotone method, we prove the existence of solutions for -Laplacian BVP in Section 4.

2. Preliminaries and Lemmas

In this section, we will give several fixed point theorems to prove existence of positive solutions of nonlinear -Laplacian BVP (1.1)-(1.2). Also, to state the main results in this paper, we employ the following lemmas. These lemmas are based on the linear dynamic equation:

(2.1)

Lemma 2.1.

Suppose condition (H2) holds, then there exists a constant that satisfies

(2.2)

Furthermore, the function

(2.3)

is a positive continuous function, therefore, has a minimum on , hence one supposes that there exists such that for .

Proof.

It is easily seen that is continuous on .

Let

(2.4)

Then, from condition (H2), we have that the function is strictly monoton nondecreasing on and , the function is strictly monoton nonincreasing on and , which implies

Throughout this paper, let , then is a Banach space with the norm . Let

(2.5)

Lemma 2.2.

Let and be as in Lemma 2.1, then

(2.6)

Proof.

Suppose We have three different cases.

• (i) . It follows from the concavity of that each point on the chard between and is below the graph of , thus

  

  (2.7)

  then

  

  (2.8)

  this means for

• (ii) . If , similarly, we have

  

  (2.9)

  If , similarly, we have

  

  (2.10)

  this means for .

• (iii) . Similarly we have

  

  (2.11)

  then

  

  (2.12)

  this means for From the above, we know

  

  (2.13)

Lemma 2.3.

Suppose that condition (H3) holds. Let and . Then -Laplacian BVP (2.1)-(1.2) has a solution

(2.14)

where is a solution of the following equation

(2.15)

where

(2.16)

Proof.

Obviously and , beside these and . So, there must be an intersection point between and for and , which is a solution , since and are continuous. It is easy to verify that is a solution of (2.1)-(1.2). If (2.1) has a solution, denoted by , then . There exists a constant such that . If it does not hold, without loss of generality, one supposes that for . From the boundary conditions, we have

(2.17)

which is a contradiction.

Integrating (2.1) on we get

(2.18)

Then, we have

(2.19)

Using the second boundary condition and the formula (2.18) for , we have

(2.20)

Also, using the formula (2.18), we have

(2.21)

Similarly, integrating (2.1) on we get

(2.22)

Throughout this paper, we assume that .

Lemma 2.4.

Suppose that the conditions in Lemma 2.3 hold. Then there exists a constant such that the solution of -Laplacian BVP (2.1)-(1.2) satisfies

(2.23)

Proof.

It is clear that satisfies

(2.24)

Similarly,

(2.25)

If we define , we get

(2.26)

Now, we define a mapping given by

(2.27)

Because of

(2.28)

we get , for and , for , thus the operator is monotone increasing on and monotone decreasing on and also is the maximum point of the operator . So the operator is concave on and . Therefore, .

Lemma 2.5.

Suppose that the conditions (H1)–(H3) hold. is completely continuous.

Proof.

Suppose is a bounded set. Let be such that , . For any , we have

(2.29)

Then, is bounded.

By the Arzela-Ascoli theorem, we can easily see that is completely continuous operator.

For convenience, we set

(2.30)

In order to follow the main results of this paper easily, now we state the fixed point theorems which we applied to prove Theorems 3.1–3.4.

Theorem 2.6 (see [18] (Krasnoselskii fixed point theorem)).

Let be a Banach space, and let be a cone. Assume and are open, bounded subsets of with , and let

(2.31)

be a completely continuous operator such that either

• (i) for for

• (ii) for for

hold. Then has a fixed point in .

Theorem 2.7 (see [19] (Schauder fixed point theorem)).

Let be a Banach space, and let be a completely continuous operator. Assume is a bounded, closed, and convex set. If , then has a fixed point in .

Theorem 2.8 (see [20] (Avery-Henderson fixed point theorem)).

Let be a cone in a real Banach space . Set

(2.32)

If and are increasing, nonnegative, continuous functionals on , let be a nonnegative continuous functional on with such that for some positive constants and ,

(2.33)

for all . Suppose that there exist positive numbers such that for all and

If is a completely continuous operator satisfying

• (i) for all

• (ii) for all

• (iii) and for all ,

then has at least two fixed points and such that

(2.34)

3. Main Results

In this section, we will prove the existence of at least one and two positive solution of -Laplacian BVP (1.1)-(1.2). In the following theorems we will make use of Krasnoselskii, Schauder, and Avery-Henderson fixed point theorems, respectively.

Theorem 3.1.

Assume that (H1)–(H3) are satisfied. In addition, suppose that satisfies

• (A1) for

• (A2) for

where and . Then the -Laplacian BVP (1.1)-(1.2) has a positive solution such that .

Proof.

Without loss of generality, we suppose . For any , by Lemma 2.2, we have

(3.1)

We define two open subsets and of such that and .

For , by (3.1), we have

(3.2)

For , if holds, we will discuss it from three perspectives.

• (i) If , thus for , by and Lemma 2.1, we have

  

  (3.3)

• (ii) If , thus for , by and Lemma 2.1, we have

  

  (3.4)

• (iii) If , thus for , by and Lemma 2.1, we have

  

  (3.5)

Therefore, we have ,

On the other hand, as , we have , by , we know

(3.6)

Then, has a fixed point . Obviously, is a positive solution of the -Laplacian BVP (1.1)-(1.2) and .

Existence of at least one positive solution is also proved using Schauder fixed point theorem (Theorem 2.7). Then we have the following result.

Theorem 3.2.

Assume that (H1)–(H3) are satisfied. If satisfies

(3.7)

where satisfies

(3.8)

then the -Laplacian BVP (1.1)-(1.2) has at least one positive solution.

Proof.

Let . Note that is closed, bounded, and convex subset of to which the Schauder fixed point theorem is applicable. Define as in (2.27) for . It can be shown that is continuous. Claim that . Let . By using the similar methods used in the proof of Theorem 3.1, we have

(3.9)

which implies . The compactness of the operator follows from the Arzela-Ascoli theorem. Hence has a fixed point in .

Corollary 3.3.

If is continuous and bounded on , then the -Laplacian BVP (1.1)-(1.2) has a positive solution.

Now we will give the sufficient conditions to have at least two positive solutions for -Laplacian BVP (1.1)-(1.2). Set

(3.10)

The function is positive and continuous on . Therefore, has a minimum on . Hence we suppose there exists such that

Also, we define the nonnegative, increasing continuous functions and by

(3.11)

We observe here that, for every , and from Lemma 2.2, . Also, for ,

Theorem 3.4.

Assume that (H1)–(H3) are satisfied. Suppose that there exist positive numbers such that the function f satisfies the following conditions:

• (i) for ,

• (ii) for ,

• (iii) for ,

for positive constants , and . Then the -Laplacian BVP (1.1)-(1.2) has at least two positive solutions such that

(3.12)

Proof.

Define the cone as in (2.5). From Lemmas 2.2 and 2.3 and the conditions (H1) and (H2), we can obtain . Also from Lemma 2.5, we see that is completely continuous.

We now show that the conditions of Theorem 2.8 are satisfied.

To fulfill property (i) of Theorem 2.8, we choose , thus . Recalling that , we have

(3.13)

Then assumption (iii) implies for . We have three different cases.

• (a) If , we have

  

  (3.14)

Thus we have .

• (b) If , we have

  

  (3.15)

Thus we have .

• (c) If , we have

  

  (3.16)

Thus we have and condition (i) of Theorem 2.8 holds. Next we will show condition (ii) of Theorem 2.8 is satisfied. If , then .

Noting that

(3.17)

we have , for .

Then (ii) yields for .

As so

(3.18)

So condition (ii) of Theorem 2.8 holds.

To fulfill property (iii) of Theorem 2.8, we note , is a member of and , so . Now choose , then and this implies that for . It follows from the assumption (i), we have for . As before we obtain the following cases.

• (a) If , we have

  

  (3.19)

Thus we have .

• (b) If , we have

  

  (3.20)

Thus we have .

• (c) If , we have

  

  (3.21)

Thus we have .

Therefore, condition (iii) of Theorem 2.8 holds. Since all conditions of Theorem 2.8 are satisfied, the -Laplacian BVP (1.1)-(1.2) has at least two positive solutions such that

(3.22)

4. Monotone Method

In this section, we will prove the existence of solution of -Laplacian BVP (1.1)-(1.2) by using upper and lower solution method. We define the set

(4.1)

Definition 4.1.

A real-valued function on is a lower solution for (1.1)-(1.2) if

(4.2)

Similarly, a real-valued function on is an upper solution for (1.1)-(1.2) if

(4.3)

We will prove when the lower and the upper solutions are given in the well order, that is, , the -Laplacian BVP (1.1)-(1.2) admits a solution lying between both functions.

Theorem 4.2.

Assume that (H1)–(H3) are satisfied and and are, respectively, lower and upper solutions for the -Laplacian BVP (1.1)-(1.2) such that on . Then the -Laplacian BVP (1.1)-(1.2) has a solution on .

Proof.

Consider the -Laplacian BVP:

(4.4)

where

(4.5)

for

Clearly, the function is bounded for and satisfies condition (H1). Thus by Theorem 3.2, there exists a solution of the -Laplacian BVP (4.4). We first show that on . Set . If on is not true, then there exists a such that has a positive maximum. Consequently, we know that and there exists such that on . On the other hand by the continuity of at we know there exists such that on Let , then we have on . Thus we get

(4.6)

Therefore,

(4.7)

which is a contradiction and thus cannot be an element of .

If , from the boundary conditions, we have

(4.8)

Thus we get

(4.9)

From this inequalities, we have

(4.10)

which is a contradiction.

If , from the boundary conditions, we have

(4.11)

Thus we get

(4.12)

From this inequalities, we have

(4.13)

which is a contradiction. Thus we have on .

Similarly, we can get on . Thus is a solution of -Laplacian BVP (1.1)-(1.2) which lies between and .