Volume 2014, Issue 1 503496

Research Article

Open Access

Existence and Uniqueness of Positive Solitons for a Second-Order Difference Equation

Xinfu Li,

Xinfu Li

School of Science, Tianjin University of Commerce, Tianjin 300134, China tjcu.edu.cn

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Guang Zhang,

Corresponding Author

Guang Zhang

[email protected]

orcid.org/0000-0001-5745-9005

School of Science, Tianjin University of Commerce, Tianjin 300134, China tjcu.edu.cn

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Ying Wang,

Ying Wang

School of Science, Tianjin University of Commerce, Tianjin 300134, China tjcu.edu.cn

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Xinfu Li,

Xinfu Li

School of Science, Tianjin University of Commerce, Tianjin 300134, China tjcu.edu.cn

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Guang Zhang,

Corresponding Author

Guang Zhang

[email protected]

orcid.org/0000-0001-5745-9005

School of Science, Tianjin University of Commerce, Tianjin 300134, China tjcu.edu.cn

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Ying Wang,

Ying Wang

School of Science, Tianjin University of Commerce, Tianjin 300134, China tjcu.edu.cn

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First published: 02 December 2014

https://doi.org/10.1155/2014/503496

Citations: 3

Academic Editor: Juan J. Nieto

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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

(1)

where

is a set of all integers, α > 0 and λ > 0 are constants, Δx_i = x_i+1 − x_i, Δ²x_i−1 = Δ(Δx_i−1), and

will be specified later.

This equation is associated with the famous discrete logistic equation with diffusion:

(2)

for

and

, where the parameter d > 0 corresponds to the rate at which the population diffuses, and the unknown function x corresponds to the density of a population. The term −x^1+α in the equation corresponds to the fact that the population is self-limiting and the p corresponds to the birth rate of population if the self-limitation is ignored. At points where p_i > 0 (<0) the population, ignoring self-limitation, has positive (negative) birth rate. Hence, we assume throughout this paper that p 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 λ = 1/d, we see that the steady-state solutions of (2) must satisfy (1). Since the sequence x 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 0 at infinity; that is,

(3)

Recently, the existence of nontrivial discrete solitons for the general equation

(4)

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

(5)

Indeed, by means of standard finite difference methods, we set up a grid in the x, t plane with grid spacings Δx and Δt and then replace the second derivative u_xx with a central difference and u_t with a forward difference. By writing x_i = iΔx, t_n = nΔt, p_i = p(x_i), and

, a finite difference scheme for (5) is obtained:

(6)

(7)

which has the steady-state equation:

(8)

We note that the steady-state equation of (5) is

(9)

(10)

where μ = 1/D. When α = 1, (10) is reduced to

(11)

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

(12)

Note that p 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 λ = (Δx) ²/D ≠ μ when (Δx) ² ≠ 1. This will lead to different results between (1) and (11).

2. Preliminaries

For any

with a < b, we consider the eigenvalue problem of the form

(13)

where [a, b] = {a, a + 1, …, b},

is a real sequence, and there exists i₀ ∈ [a, b] such that

Denote

(14)

(15)

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

(16)

Let H be a set of all real sequences with x_a−1 = 0 = x_b+1. For any u, v ∈ H, the inner product is defined as and the norm ‖·‖ is defined as .

For any v ∈ H, we define

(17)

Consider the Rayleigh quotient

(18)

and let

(19)

For such defined λ₁, we have the following important result.

Lemma 1. λ₁ is a positive eigenvalue of the problem (13) or (16). Moreover λ₁ is simple and the corresponding eigenfunction φ can be chosen such that φ_i > 0 for all i ∈ [a, b].

Proof. First, we show λ₁ > 0. Clearly

(20)

for all v ∈ H. Moreover, by the spectral theorem, (Av, v) ≥ γ₁(v, v), where

(21)

is the first eigenvalue of the eigenvalue problem

(22)

Hence, if v ∈ H and

, then

(23)

Hence λ₁ > 0.

Second, consider the linear eigenvalue problem

(24)

and define (Su)_i = −Δ²u_i−1 − λ₁p_iu_i. It is easy to see that λ₁ is an eigenvalue for (13) with corresponding eigenfunction φ if and only if 0 is an eigenvalue of S and so of (24) with corresponding eigenfunction φ. The minimal eigenvalue α₁ of S is given by

(25)

Note that

for all v ∈ H; thus, we have α₁ ≥ 0. Because of how we defined λ₁, there exists a sequence v⁽ⁿ⁾ ∈ H and

such that

(26)

which implies that

(27)

Thus, we have α₁ ≤ 0. In this case, we know that α₁ = 0 is the minimal eigenvalue of (24). By Lemma A.1 in the Appendix, α₁ is simple and the corresponding eigenfunction can be chosen to be positive on H. Thus, the statement in this lemma is right.

Let λ > λ₁; consider the eigenvalue problem of the form

(28)

We have the following result.

Lemma 2. If λ > λ₁, then the minimal eigenvalue μ₁ of (28) is negative and the corresponding eigenfunction φ⁽¹⁾ can be chosen so that for all i ∈ [a, b].

Proof. Let φ be the eigenvector obtained in Lemma 1 and ∥φ∥ = 1; then we have

(29)

Consequently,

(30)

In view of Lemma A.1 in the Appendix, the eigenvector corresponding to μ₁ 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 a < b, consider the Dirichlet boundary value problem of the form

(31)

We will denote by (31)_n the problem (31) when a = −n and b = n.

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

(32)

A sequence

is said to be a subsolution of (31) if

(33)

Theorem 4. Suppose that are functions with (where is defined by for ) and , are, respectively, a subsolution and an upper-solution of (31)_n for all large n. Assume that f(i, z) is continuous with z ∈ [α, β] and there exists a constant k > 0 such that

(34)

for all

and s₂ > s₁ with s₁, s₂ ∈ [α, β] (where

and

). Then problem (4) has a solution x such that

Proof. Since and provide sub- and upper-solutions for (31)_n , there exists a solution φ of (31)_n such that for all |i | ≤ n.

In fact, for any sequence , clearly, the problem

(35)

has a unique solution w. This defines a mapping T : u → w. We claim that T is increasing on

. In fact, for any sequences

with

, we have

(36)

where Γ is an operator defined as (Γu) _i = −Δ²u_i−1 + ku_i for i ∈ [−n, n]. Since

, by the strong monotonicity of Γ, we obtain

; see [21].

Let u^(m) = Tu^(m−1), ; v^(m) = Tv^(m−1), . In the following, we claim that

(37)

First of all, by using (35) and the definition of subsolution, we see

(38)

and

. This implies that u⁽⁰⁾ − u⁽¹⁾ ≤ 0 by the strong monotonicity of Γ. A similar argument gives v⁽¹⁾ ≤ v⁽⁰⁾. The monotonicity of Γ and T gives the rest. So there exist u and v such that

(39)

By the definition of u and v, we see that u and v satisfy (31)_n and

. Let x⁽ⁿ⁾ denote such solution corresponding to (31)_n .

Then standard a priori estimates and a diagonalization argument show that there exists a subsequence of {x⁽ⁿ⁾} which converges to solution x of (4) on every bounded subset of . Moreover, since for all n, 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 n>|i₀|, consider the eigenvalue problem (13) when a = −n and b = n and denote the corresponding positive eigenvalue λ₁ defined in (19) by

. Then

(40)

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

Proof. For any , clearly, there exists a positive integer n₁≥|i₀| such that i₀ ∈ [−n₁, n₁] and . In view of Lemma 2, the eigenvalue problem

(41)

has a negative eigenvalue

and its corresponding eigenfunction

with

can be chosen so that

for all i ∈ [−n₁, n₁]. For any ɛ > 0, we define

(42)

At this time, we have

(43)

for i ∈ [−n₁, n₁] and sufficiently small ϵ. When i = n₁ + 1,

(44)

and

for i > n₁ + 1. Similarly, we can also prove that

for i ≤ −n₁ − 1. Thus, the sequence

defined in (42) is a subsolution of (1).

Note that α > 0 and the sequence {p_i} is bounded. Thus, the sufficiently large positive constant is a supersolution of (1).

For , we have

(45)

where ξ ∈ (s₁, s₂). Note that the sequences {p_i},

, and M are bounded; thus, (1) satisfies condition (34). By Theorem 4, problem (1) has a solution u such that

. We claim that u > 0. Suppose there exists

such that

; then we obtain

since

. Thus we obtain

. So u ≡ 0, which contradicts u≢0. The proof is complete.

Theorem 6. Assume that and there exists a constant C > 6/λ and n₀ > |i₀| such that

(46)

for |i | > n₀. Then (1) has a positive soliton and there exists a constant M > 0 and n₂ ≥ n₀ such that

(47)

Proof. Let be the subsolution of (1) obtained in Theorem 5 and

(48)

where M > 0 is a constant determined later. By simple calculation, we have

(49)

By the assumption on p, there exist n₂ ≥ n₀ and M₁ > 0 such that

(50)

for M ≥ M₁ and |i | ≥ n₂.

For any M ≥ M₁, define

(51)

We will show that ψ is upper-solution of (1) by appropriate choice of M. In fact,

(a)
for |i | > n₂, we have by the choice of M;
(b)
for |i | < n₂ − 1, we have
(52)
by choosing M > M₁ large enough since α > 0 and p is bounded;
(c)
similarly, for i = n₂, n₂ − 1, −n₂, −n₂ + 1, one can show that for M > M₁ large enough.

In view of (a)–(c), we construct upper-solution ψ of (1) with . Using the sub- and upper-solution Theorem 4, we complete the proof.

Theorem 7. Assume that there exists a positive integer m > |i₀| such that p_i ≤ 0 for |i| ≥ m. Let x be a bounded positive solution of (1); then

(53)

Proof. At this time, we have

(54)

Hence Δx_i−1 is an increasing sequence for |i| ≥ m and so there exists a positive integer n₁ such that Δx_i is one sign for |i| > n₁. Thus x is eventually a monotone sequence for |i | > n₁. Note that x is bounded. Thus, lim⁡_n→+∞x_n and lim⁡_n→−∞x_n exist.

If lim⁡_n→+∞x_n ≠ 0, we have

(55)

which implies that

(56)

This is impossible since x is bounded and so we must have lim⁡_n→+∞x_n = 0. Similarly, we also have lim⁡_n→−∞x_n = 0. The proof is complete.

Theorem 8. For any λ ≠ 0, there exists at most one positive solution of (1) such that lim⁡_|i|→∞x_i = 0.

Proof. Suppose that u and v are such two distinct solutions. As before we can construct an arbitrarily small subsolution and so there must exist a solution w of (1) such that w ≤ u and w ≤ v. Multiplying the u-equation by w and the w-equation by u, then we have

(57)

(58)

Let n → +∞; we have

(59)

which implies that u = w. Similarly, we can also prove that v = w. The proof is complete.

Corollary 9. Assume that all conditions of Theorem 8 hold; then (1) has a unique positive soliton.

4. Conclusion

This paper studied a discrete logistic steady-state equation with both positive and negative birth rate of population. By using sub- and upper-solution method (Theorem 4), the existence of positive solution and positive soliton is obtained (Theorems 5 and 6). Uniqueness of homoclinic type solution is also obtained (Theorem 8).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the anonymous referees for their comments and suggestions on the paper.

Appendix

To obtain a positive eigenfunction of (13) or a positive eigenvector of (16), we need the following lemma. We give the proof by the method of [22].

Lemma A.1. For any (b − a + 1) × (b − a + 1) diagonal matrix

(A.1)

let μ₁ be the minimal eigenvalue of A + Q, where A is defined in (14). Then μ₁ is simple and the corresponding eigenvector φ⁽¹⁾ can be chosen to be positive.

Proof. Without loss of generality, we assume q_i > 0 for i ∈ [a, b]. If this is not the case, choose a constant C > 0 large enough such that q_i + C > 0. Consider the matrix A + Q + CE instead, where E is a (b − a + 1)×(b − a + 1) identity matrix. Denote ; then for any i ∈ [a, b].

Since A + Q is a real (b − a + 1)×(b − a + 1) symmetric positive definite matrix, A + Q has b − a + 1 real eigenvalues. We repeat each eigenvalue according to its multiplicity as follows:

(A.2)

and we choose eigenvectors φ⁽¹⁾, φ⁽²⁾, …, φ^(b−a+1) for A + Q such that

(A.3)

is an orthonormal basis for

. In the following we give the proof of Lemma A.1 in four steps.

Step 1. For any , define B[u, v] = v^T(A + Q)u. It is obvious that

(A.4)

Step 2. We claim that if and ∥u∥ = 1, then u is a solution of

(A.5)

if and only if

(A.6)

Obviously (A.5) implies (A.6). On the other hand, suppose (A.6) is valid. Then, writing d_k = (u, φ^(k)), we have

(A.7)

Hence

(A.8)

Consequently d_k = (u, φ^(k)) = 0, if μ_k > μ₁. It follows that

(A.9)

for some m, where (A + Q)φ^(k) = μ₁φ^(k) for k = 1,2, …, m. Therefore

(A.10)

This proves (A.5).

Step 3. We will show that if satisfies (A + Q)u = μ₁u, u ≠ 0, then either u > 0 or u < 0.

To see this, let us assume without loss of generality that ∥u∥ = 1 and note α + β = 1 for α = ∥u⁺∥², β = ∥u⁻∥², where and . In this case, we have

(A.11)

Thus

(A.12)

By Step 2, u⁺ and u⁻ satisfy (A + Q)u⁺ = μ₁u⁺, (A + Q)u⁻ = μ₁u⁻. Thus (A + Q)u⁺ ≥ 0, (A + Q)u⁻ ≥ 0. That is,

(A.13)

If there exists an i₀ ∈ [a, b] such that

, then u⁺ = 0. Similar arguments apply to u⁻, and so either u > 0 or u < 0. Thus, we can choose a positive eigenvector φ⁽¹⁾ corresponding to the minimal eigenvalue μ₁.

Step 4. Finally assume that u and are two eigenvectors corresponding to μ₁; in view of Step 3,

(A.14)

and so there exists a real constant k such that

(A.15)

(A.16)

But since

also satisfies

, by Step 3,

. In view of (A.16), we have

. Hence the eigenvalue μ₁ is simple. This completes the proof of Lemma A.1.

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Existence and Uniqueness of Positive Solitons for a Second-Order Difference Equation

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Conclusion

Conflict of Interests

Acknowledgment

Appendix

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Existence and Uniqueness of Positive Solitons for a Second-Order Difference Equation

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Conclusion

Conflict of Interests

Acknowledgment

Appendix

References

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