Existence and Multiplicity of Positive Solutions of a Boundary-Value Problem for Sixth-Order ODE with Three Parameters

Abstract

We study the existence and multiplicity of positive solutions of the following boundary-value problem: , , , where R⁺ → R⁺ is continuous, , , and satisfy some suitable assumptions.

1. Introduction

The following boundary-value problem:

(1.1)

where are some given real constants and is a continuous function on , is motivated by the study for stationary solutions of the sixth-order parabolic differential equations

(1.2)

This equation arose in the formation of the spatial periodic patterns in bistable systems and is also a model for describing the behaviour of phase fronts in materials that are undergoing a transition between the liquid and solid state. When it was studied by Gardner and Jones [1] as well as by Caginalp and Fife [2].

If is an even periodic function with respect to and odd with respect to , in order to get the stationary spatial periodic solutions of (1.2), one turns to study the two points boundary-value problem (1.1). The periodic extension of the odd extension of the solution of problems (1.1) to the interval yields spatial periodic solutions of(1.2)

Gyulov et al. [3] have studied the existence and multiplicity of nontrivial solutions of BVP (1.1). They gained the following results.

Theorem 1.1.

Let be a continuous function and . Suppose the following assumptions are held:

as , uniformly with respect to in bounded intervals,

as , uniformly with respect to in bounded intervals,

then problem (1.1) has at least two nontrivial solutions provided that there exists a natural number such that , where is the symbol of the linear differential operator .

At the same time, in investigating such spatial patterns, some other high-order parabolic differential equations appear, such as the extended Fisher-Kolmogorov (EFK) equation

(1.3)

proposed by Coullet, Elphick, and Repaux in 1987 as well as by Dee and Van Saarlos in 1988 and Swift-Hohenberg (SH) equation

(1.4)

proposed in 1977.

In much the same way, the existence of spatial periodic solutions of both the EFK equation and the SH equation was studied by Peletier and Troy [4], Peletier and Rottschäfer [5], Tersian and Chaparova [6], and other authors. More precisely, in those papers, the authors studied the following fourth-order boundary-value problem:

(1.5)

The methods used in those papers are variational method and linking theorems.

On the other hand, The positive solutions of fourth-order boundary value problems (1.5) have been studied extensively by using the fixed point theorem of cone extension or compression. Here, we mention Li's paper [7], in which the author decomposes the fourth-order differential operator into the product of two second-order differential operators to obtain Green's function and then used the fixed point theorem of cone extension or compression to study the problem.

The purpose of this paper is using the idea of [7] to investigate BVP for sixth-order equations. We will discuss the existence and multiplicity of positive solutions of the boundary-value problem

(1.6)

(1.7)

and then we assume the following conditions throughout:

is continuous,

satisfy

(1.8)

Note.

The set of which satisfies is nonempty. For instance, if , then holds for .

To be convenient, we introduce the following notations:

(1.9)

2. Preliminaries

Lemma 2.1 (see [8]).

Set the cubic equation with one variable as follows:

(2.1)

Let

(2.2)

one has the following:

• (1) Equation (2.1) has a triple root if ,

• (2) Equation (2.1) has a real root and two mutually conjugate imaginary roots if ,

• (3) Equation (2.1) has three real roots, two of which are reroots if ,

• (4) Equation (2.1) has three unequal real roots if .

Lemma 2.2.

Let be the roots of the polynomial . Suppose that condition holds, then are real and greater than .

Proof.

There are in the equation . Since condition holds, we have

(2.3)

Therefore, the equation has three real roots in reply to Lemma 2.1.

By Vieta theorem, we have

(2.4)

Therefore , and hold if and only if

(2.5)

Then, we only prove that the system of inequalities (2.5) holds if and only if are all greater than .

In fact, the sufficiency is obvious, we just prove the necessity. Assume that are less than . By the first inequality of (2.5), there exist two roots which are less than and one which is greater than . Without loss of generality, we assume that , then we have . Multiplying the second inequality of (2.5) by , one gets

(2.6)

Compare with the third inequality of (2.5), we have

(2.7)

which is a contradiction. Hence, the assumption is false. The proof is completed.

Let be Green's function of the linear boundary-value problem

(2.8)

Lemma 2.3 (see [7]).

has the following properties:

• (i) ,

• (ii) , where is a constant,

• (iii) , where is a constant.

One denotes the following:

(2.9)

then . Let be the maximum norm of ,  and let   be the cone of all nonnegative functions in .

Let , then one considers linear boundary-value problem (LBVP) as follows:

(2.10)

with the boundary condition (1.7). Since

(2.11)

the solution of LBVP (2.10)–(1.7) can be expressed by

(2.12)

Lemma 2.4.

Let , then the solution of LBVP(2.10)–(1.7) satisfies

(2.13)

Proof.

From (2.12) and (ii) of Lemma 2.3, it is easy to see that

(2.14)

and, therefore,

(2.15)

that is,

(2.16)

Using (iii) of Lemma 2.3, we have

(2.17)

The proof is completed.

We now define a mapping by

(2.18)

It is clear that is completely continuous. By Lemma 2.4, the positive solution of BVP(1.6)-(1.7) is equivalent to nontrivial fixed point of . We will find the nonzero fixed point of by using the fixed point index theory in cones. For this, one chooses the subcone of by

(2.19)

where , we have the following.

Lemma 2.5.

Having , is completely continuous.

Proof.

For , let , then is the solution of LBVP(2.10)–(1.7). By Lemma 2.4, one has

(2.20)

namely . Therefore, . The complete continuity of is obvious.

The main results of this paper are based on the theory of fixed point index in cones [9]. Let be a Banach space and be a closed convex cone in . Assume that is a bounded open subset of with boundary , and . Let be a completely continuous mapping. If for every , then the fixed point index is well defined. We have that if , then has a fixed point in .

Let and for every .

Lemma 2.6 (see [9]).

Let be a completely continuous mapping. If for every and , then .

Lemma 2.7 (see [9]).

Let be a completely continuous mapping. Suppose that the following two conditions are satisfied:

• (i) ,

• (ii) for every and ,

then .

Lemma 2.8 (see [9]).

Let be a Banach space, and let be a cone in . For , define . Assume that is a completely continuous mapping such that for every .

• (i) If for every , then .

• (ii) If for every , then .

3. Existence

We are now going to state our existence results.

Theorem 3.1.

Assume that (H1) and (H2) hold, then in each of the following case:

• (i) , ,

• (ii) , ,

the BVP(1.6)-(1.7) has at least one positive solution.

Proof.

To prove Theorem 3.1, we just show that the mapping defined by (2.18) has a nonzero fixed point in the cases, respectively.

Case(i): since , by the definition of , we may choose and so that

(3.1)

Let , we now prove that for every and . In fact, if there exist and such that , then, by definition of , satisfies differential equation the following:

(3.2)

and boundary condition (1.7). Multiplying (3.2) by and integrating on , then using integration by parts in the left side, we have

(3.3)

By Lemma 2.4, , and then . We see that , which is a contradiction. Hence, satisfies the hypotheses of Lemma 2.6, in . By Lemma 2.6 we have

(3.4)

On the other hand, since , there exist and such that

(3.5)

Let , then it is clear that

(3.6)

Choose . Let . Since , from (3.5) we see that

(3.7)

By Lemma 2.5, we have

(3.8)

Therefore,

(3.9)

from which we see that , namely the hypotheses (i) of Lemma 2.7 holds. Next, we show that if is large enough, then for any and . In fact, if there exist and such that , then satisfies (3.2) and boundary condition (1.7). Multiplying (3.2) by and integrating, from (3.6) we have

(3.10)

Consequently, we obtain that

(3.11)

By Lemma 2.4,

(3.12)

from which and from (3.11) we get that

(3.13)

Let , then for any and , . Hence, hypothesis (ii) of Lemma 2.7 also holds. By Lemma 2.7, we have

(3.14)

Now, by the additivity of fixed point index, combine (3.4) and (3.14) to conclude that

(3.15)

Therefore, has a fixed point in , which is the positive solution of BVP(1.6)-(1.7).

Case (ii): since , there exist and such that

(3.16)

Let , then for every , through the argument used in (3.9), we have

(3.17)

Hence, . Next, we show that for any and . In fact, if there exist and such that , then satisfies (3.2) and boundary (1.7). From (3.2) and (3.16), it follows that

(3.18)

Since , we see that , which is a contradiction. Hence, by Lemma 2.7, we have

(3.19)

On the other hand, since , there exist and such that

(3.20)

Set , we obviously have

(3.21)

If there exist and such that , then (3.2) is valid. From (3.2) and (3.21), it follows that

(3.22)

By the proof of (3.13), we see that . Let , then for any and , . Therefore, by Lemma 2.6, we have

(3.23)

From (3.19) and (3.23), it follows that

(3.24)

Therefore, has a fixed point in , which is the positive solution of BVP(1.6)-(1.7). The proof is completed.

From Theorem 3.1, we immediately obtain the following.

Corollary 3.2.

Assume that and hold, then in each of the following cases:

• (i) , ,

• (ii) , ,

the BVP(1.6)-(1.7) has at least one positive solution.

4. Multiplicity

Next, we study the multiplicity of positive solutions of BVP(1.6)-(1.7) and assume in this section that

(H3) there is a such that and imply , where .

(H4) there is a such that and imply , where .

Theorem 4.1.

If and and is satisfied, then BVP(1.6)-(1.7) has at least two positive solutions: and such that .

Proof.

According to the proof of Theorem 3.1, there exists , such that implies and implies .

We now prove that if is satisfied. In fact, for every , from the definition of we have

(4.1)

From (ii) of Lemma 2.8, we have

(4.2)

Combining (3.14) and (3.19), we have

(4.3)

Therefore, has fixed points and in and , respectively, which means that and are positive solutions of BVP(1.6)-(1.7) and . The proof is completed.

Theorem 4.2.

If and and is satisfied, then BVP(1.6)-(1.7) has at least two positive solutions: and such that .

Proof.

According to the proof of Theorem 3.1, there exists , such that implies and implies .

We now prove that if is satisfied. In fact, for every , from the proof of (i) of Theorem 3.1, we have

(4.4)

Therefore, , according to (i) of Lemma 2.8, .

Combining (3.4) and (3.23), we have

(4.5)

Therefore, has the fixed points and in and , respectively, which means that and are positive solutions of BVP(1.6)-(1.7) and . The proof is completed.

Theorem 4.3.

If and , and there exists that satisfies

• (i) if and ,

• (ii) if and

then BVP(1.6)-(1.7) has at least three positive solutions: , and such that .

Proof.

According to the proof of Theorem 3.1, there exists , such that implies and implies .

From the proof of Theorems 4.1 and 4.2, we have

(4.6)

Combining the four afore-mentioned equations, we have

(4.7)

Therefore, has the fixed points , and in , and , which means that , and are positive solutions of BVP(1.6)-(1.7) and . The proof is completed.