Abstract

A discrete logistic steady-state equation with both positive and negative birth rate of population will be considered. By using sub- and upper-solution method, the existence of bounded positive solutions and the existence and uniqueness of positive solitons will be established. To this end, the Dirichlet eigenvalue problem with positive and negative coefficients is considered, and a general sub- and upper-solution theorem is also obtained.

1. Introduction

In this paper, we consider the second-order difference equation where is a set of all integers, and are constants, , , and will be specified later.

This equation is associated with the famous discrete logistic equation with diffusion: for and , where the parameter corresponds to the rate at which the population diffuses, and the unknown function corresponds to the density of a population. The term in the equation corresponds to the fact that the population is self-limiting and the corresponds to the birth rate of population if the self-limitation is ignored. At points where (<0) the population, ignoring self-limitation, has positive (negative) birth rate. Hence, we assume throughout this paper that takes on both positive and negative values on ; we further assume that the sequence is bounded and there exists an integer such that .

If we write , we see that the steady-state solutions of (2) must satisfy (1). Since the sequence represents a population density, it is nonnegative. Such solutions correspond to possible steady-state distributions of population. So our purpose in this paper is to establish the existence of positive solutions or the existence and uniqueness of positive solitons for (1). A discrete soliton is a spatially localized solution of (1) and decays to at infinity; that is,

Recently, the existence of nontrivial discrete solitons for the general equation has been extensively established by a number of authors [1–12]. Among the methods used are the principle of anticontinuity [1–3], variational methods [4–10], center manifold reduction [11], Nehari manifold approach [12], and so forth. On the other hand, a soliton of (4) is also its homoclinic orbit or homoclinic solution. By using the variational methods, the existence of homoclinic orbits or homoclinic solutions has also been extensively discussed by a number of authors; see [4–10, 13–16]. However, as far as we know, there are few papers concerned with the existence of positive solitons.

Equation (2) is a discrete analogue of the well-known logistic equation of the form Indeed, by means of standard finite difference methods, we set up a grid in the , plane with grid spacings and and then replace the second derivative with a central difference and with a forward difference. By writing , , , and , a finite difference scheme for (5) is obtained: or which has the steady-state equation: We note that the steady-state equation of (5) is or where . When , (10) is reduced to In [17], the authors considered the existence and uniqueness of positive solitons of (11) by using sub- and upper-solution method. The present work is motivated by [17].

To obtain a positive subsolution of (1), we need a positive eigenvalue and its corresponding positive eigenfunction of the eigenvalue problem Note that takes on both positive and negative values; thus, it is indefinite in a very strong sense. The corresponding problem for ordinary differential equation had been considered in [18] by using variational method. However, such problem is new for the above discrete problem; see [19–21]. In Section 2, we will consider the above discrete eigenvalue problem by using the matrix and vector method.

As far as we know, for the discrete problem (1) there is no general sub- and upper-solution theorem on the set . Thus, a general sub- and upper-solution theorem will be firstly obtained in Section 3 and such theorem will be used for (1). On the other hand, our solutions are classical; however, [17] cannot insure this fact because some points of their subsolution are not derivative.

Our results will give some theory groundwork for the numerical calculation of (11). Note that when . This will lead to different results between (1) and (11).

2. Preliminaries

For any with , we consider the eigenvalue problem of the form where , is a real sequence, and there exists such that .

Denote Then problem (13) can be rewritten by matrix and vector in the form

Let be a set of all real sequences with . For any , the inner product is defined as and the norm is defined as .

For any , we define Consider the Rayleigh quotient and let

For such defined , we have the following important result.

Lemma 1. is a positive eigenvalue of the problem (13) or (16). Moreover is simple and the corresponding eigenfunction can be chosen such that for all .

Proof. First, we show . Clearly for all . Moreover, by the spectral theorem, , where is the first eigenvalue of the eigenvalue problem Hence, if and , then Hence .
Second, consider the linear eigenvalue problem and define . It is easy to see that is an eigenvalue for (13) with corresponding eigenfunction if and only if is an eigenvalue of and so of (24) with corresponding eigenfunction . The minimal eigenvalue of is given by Note that for all ; thus, we have . Because of how we defined , there exists a sequence and such that which implies that Thus, we have . In this case, we know that is the minimal eigenvalue of (24). By Lemma A.1 in the Appendix, is simple and the corresponding eigenfunction can be chosen to be positive on . Thus, the statement in this lemma is right.

Let ; consider the eigenvalue problem of the form We have the following result.

Lemma 2. If , then the minimal eigenvalue of (28) is negative and the corresponding eigenfunction can be chosen so that for all .

Proof. Let be the eigenvector obtained in Lemma 1 and ; then we have Consequently, In view of Lemma A.1 in the Appendix, the eigenvector corresponding to can be chosen to be positive. This completes the proof.

3. Main Results

First of all, we introduce the definitions of the subsolution and upper-solution and give a general sub- and upper-solution theorem.

For any with , consider the Dirichlet boundary value problem of the form We will denote by (31) the problem (31) when and .

Definition 3. A sequence is said to be an upper-solution of (31) if A sequence is said to be a subsolution of (31) if

Theorem 4. Suppose that are functions with (where is defined by for ) and , are, respectively, a subsolution and an upper-solution of (31) for all large . Assume that is continuous with and there exists a constant such that for all and with (where and ). Then problem (4) has a solution such that .

Proof. Since and provide sub- and upper-solutions for (31), there exists a solution of (31) such that for all .
In fact, for any sequence , clearly, the problem has a unique solution . This defines a mapping . We claim that is increasing on . In fact, for any sequences , with , we have where is an operator defined as for . Since , , by the strong monotonicity of , we obtain ; see [21].
Let , ; , . In the following, we claim that First of all, by using (35) and the definition of subsolution, we see and , . This implies that by the strong monotonicity of . A similar argument gives . The monotonicity of and gives the rest. So there exist and such that By the definition of and , we see that and satisfy (31) and . Let denote such solution corresponding to (31).
Then standard a priori estimates and a diagonalization argument show that there exists a subsequence of which converges to solution of (4) on every bounded subset of . Moreover, since for all , it follows that on . The proof is complete.

In the following, we use the sub- and upper-solution theorem to establish the existence of positive solutions or the existence and uniqueness of positive solitons for (1).

For problem (1), we have assumed that there exists such that . For any , consider the eigenvalue problem (13) when and and denote the corresponding positive eigenvalue defined in (19) by . Then

Theorem 5. For any , (1) has a positive solution, where is defined in (40).

Proof. For any , clearly, there exists a positive integer such that and . In view of Lemma 2, the eigenvalue problem has a negative eigenvalue and its corresponding eigenfunction with can be chosen so that for all . For any , we define At this time, we have for and sufficiently small . When , and for . Similarly, we can also prove that for . Thus, the sequence defined in (42) is a subsolution of (1).
Note that and the sequence is bounded. Thus, the sufficiently large positive constant is a supersolution of (1).
For , we have where . Note that the sequences , , and are bounded; thus, (1) satisfies condition (34). By Theorem 4, problem (1) has a solution such that . We claim that . Suppose there exists such that ; then we obtain since . Thus we obtain . So , which contradicts . The proof is complete.

Theorem 6. Assume that and there exists a constant and such that for . Then (1) has a positive soliton and there exists a constant and such that

Proof. Let be the subsolution of (1) obtained in Theorem 5 and where is a constant determined later. By simple calculation, we have By the assumption on , there exist and such that for and .
For any , define We will show that is upper-solution of (1) by appropriate choice of . In fact,(a)for , we have by the choice of ;(b)for , we have  by choosing large enough since and is bounded;(c)similarly, for , one can show that for large enough.
In view of (a)–(c), we construct upper-solution of (1) with . Using the sub- and upper-solution Theorem 4, we complete the proof.