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Positive Solutions of a General Discrete Dirichlet Boundary Value Problem
A steady state equation of the discrete heat diffusion can be obtained by the heat diffusion of particles or the difference method of the elliptic equations. In this paper, the nonexistence, existence, and uniqueness of positive solutions for a general discrete Dirichlet boundary value problem are considered by using the maximum principle, eigenvalue method, sub- and supersolution technique, and monotone method. All obtained results are new and valid on any -dimension finite lattice point domain. To the best of our knowledge, they are better than the results of the corresponding partial differential equations. In particular, the methods of proof are different.
In this paper, we consider the discrete Dirichlet boundary value problemwhere is a finite domain and defined on is a continuous function about the second variable.
To understand the above problem, we need some terminology. Denote the set of integers by . A lattice point is defined as a point with integer coordinates. Two lattice points are said to be neighbors if their Euclidean distance is one. The lattice points are said to form a path with terminals and if is a neighbor of , is a neighbor of , and so forth. A set of lattice points is said to be connected if any two of its points are terminals of a path of points contained in the set. A nonempty connected set of lattice points is called a domain. Given a domain , a lattice point is an exterior boundary point of if it does not belong to but has at least one neighbor in . We will denote the set of exterior boundary points of by , and the set of lattice points will be denoted by . For a finite domain , the number of the lattice points is denoted by .
Given a sequence , the partial differences of are defined byThe discrete Laplace operator will be denoted by which is defined by
Problem (1) can be regarded as the discrete analogue of the elliptic boundary problem of the formwhere and satisfy some conditions and is the Laplacian operator defined by
There exist many differences between (1) and (4). As a simplest example, let us consider the continuous linear equation:It is well known that the operator has a sequence of eigenvalues: when the weight function is assumed to be in with and on a subset of positive measures of . In particular, from Krein-Rutman theorem, the principal eigenvalue is simple, in the sense that the eigenspace is one-dimensional, and the corresponding eigenfunction can be chosen to be positive. At the same time, we also have . Note that when is a finite domain, the discrete problemis only a finite dimension problem. This will induce some essential differences between (6) and (8); see [1, 2] and the listed references. First of all, we need to ask for eigenvalue problem (6); however, corresponding eigenvalue problem (8) does not need to add any conditions; secondly, problem (8) has at most eigenvalues; however, the number of eigenvalues of (6) is infinite; thirdly, in general case, the existence of solutions for (4) will be considered in Sobolev space (see [3–6]), but (1) is not necessary, and so forth. Thus, it is necessary to consider the existence of solutions for problem (1).
By using the difference method, we can obtain problem (1); see . In fact, problem (1) can also be obtained by the discrete heat diffusion; see . Recently, the existence of solutions for problem (1) has been extensively established in the rectangle domain; for example, see [8–20] (by using the eigenvalue method [8, 9], critical point theory [10–16], fixed point theorems and the degree theory [17, 18], and contraction method and monotone method ). For the -dimension case, we only see [21–25]. Pao [21–23] considered the existence of numerical solutions for nonlinear elliptic boundary value problems by using the matrix and vector method. However, [21–23] restricted that the coefficient matrix of system needs to be irreducible. In fact, it is difficult to prove that a matrix is irreducible. In [24, 25] the authors mostly established the existence of radial positive solutions by using the fixed point theorems and degree theory.
In this paper, we will consider the nonexistence, existence, and uniqueness of positive solutions of (1) by using the maximum principle, eigenvalue method, sub- and supersolution technique, and monotone method. To this end, the elementary theory will be introduced in the next section. The sub- and supersolution technique, the nonexistence, existence, and uniqueness will be considered in Section 3. In the final section, we will give some applications.
Our results are suitable for any -dimension space and the shape of domain is not restrained.
In this section, we hope to obtain some preliminaries results. They are important for establishing our main results. The obtained results seem to be similar to the corresponding continuous elliptic boundary problem; however, their conditions are different. In particular, the methods of proof are also different.
Consider first the eigenvalue problemwhere is real and is a finite domain.
It is well known that the maximum principle is important. Thus, we firstly give a maximum principle.
Lemma 1. Assume that is nonnegative and satisfies the difference inequalityThen cannot achieve a nonnegative maximum (nonpositive minimum) in the interior of unless it is constant.
Proof. Suppose that there exists such that . Then we have . In view of (10), we must have . That is, which implies thatIn the following, we will assert that for any point , . Let , be a chain of points contained in . By using (12), we have . If , the proof is complete. Otherwise, we may repeat the above argument to deduce . By using finite times induction, we can prove that . The result for follows by replacement of by .
In the following, we will consider eigenvalue problem (9).
Lemma 2. Eigenvalue problem (9) has real eigenvalues which satisfy where is simple and the corresponding eigenfunction can be chosen to be positive. Moreover, the other eigenfunctions must change sign in .
Proof. Denote the points in by . Let be matrix, where if and are neighbors and otherwise. Then (9) can be written as where is the identity matrix, , and ; see . Clearly, the matrix is positive definite and symmetric. The rest of the proof is the same as Lemma 1 in .
A comparison result will be obtained in the following.
Lemma 3. Assume that for . Then . Moreover, if there exists such that , then .
Proof. By the representation of the first eigenvalue we see that for . In view of Lemma 2, let with being the eigenfunction corresponding to ; then we have for and .
Now, we consider the principal eigenvalue and the corresponding principal function or the corresponding principal vector.
Lemma 4. Let . Then, for any sequence , the discrete Dirichlet boundary value problemhas a unique solution. Moreover, if is nonnegative and is nonnegative and not identically zero, the solution is positive.
Proof. Note that problem (17) can be written in matrix and vector form where and , and are as in (14). Since is not an eigenvalue of (17), is reversible. Hence, problem (17) has a unique solution. The positivity of the solution is a direct result of Lemma 1.
Consider now the eigenvalue problemwhere is nonnegative and not identically zero in and is a finite domain.
For eigenvalue problem (19), we have the following result.
Lemma 5. The first eigenvalue of problem (19) is simple and the corresponding eigenfunction can be chosen to be positive. In particular, if is a positive constant, then
Proof. Consider the eigenvalue problemIt is easy to see that is the first eigenvalue for (19) with corresponding eigenfunction if and only if is the first eigenvalue of (21) with corresponding eigenfunction . In view of Lemma 2, we complete the proof.
Lemma 6. Let satisfy the condition of (19) and . Then for any sequence , the discrete Dirichlet boundary value problemhas a unique solution. Moreover, if is nonnegative and not identically zero, then the solution is positive.
Proof. Problem (22) can be written in matrix and vector formwhere , , and and are as in (14). As Lemma 4, problem (22) has a unique solution for any . If , the positivity of the solution is a direct result of Lemma 1. In the following, we assume that . Assume without loss of generality that for . Write (23) in the formFor any , let denote the first eigenvalue of We claim that implies that . Indeed, suppose by contradiction that . Then Hence, for any there exists such that or equivalently By the representation of the first eigenvalue we get Let , we have That is contradiction. Thus . Write (24) in the form Denote . Since , the largest eigenvalue of is less than . Hence and the positivity of follows from the positivity of and .
3. Nonexistence, Existence, and Uniqueness of Positive Solutions
First of all, we give a nonexistence result.
Theorem 7. Assume that orholds. Then problem (1) has no nonzero solution.
Definition 8. A sequence is said to be a supersolution (subsolution) of (1) if
Theorem 9. Suppose that problem (1) has a subsolution and a supersolution , with . Assume that is continuous with and there exists a constant such thatfor all and with (where and ). Then problem (1) has solutions and such that . Moreover any solution of (1) with satisfies .
Proof. In view of Lemma 4, for any sequence , the problemhas a unique solution. This defines a mapping which is monotone in ; that is, if , then . Indeed, writing (41) with and and subtracting one from the other, we obtain By using (40) and the maximum principle, we can obtain .
Let , ; , . In the following, we claim that First of all, by using (41) and the definition of subsolution, we see that This implies by the maximum principle that in . A similar argument gives . The monotonicity of gives the rest. So there exist and such that The proof is complete.
Remark 10. For the corresponding partial differential equations, need to be () function; see .
In the following, we will give an existence result. To this end, we assume that is continuous with and satisfies the following three conditions:()there exists a nonnegative and nontrivial sequence and a positive number such that ()there exist nonnegative sequences and such that ()for any real number , there exists a constant such that is nondecreasing for .
Theorem 11. Assume that conditions (), (), and () hold. Suppose that and . Then Dirichlet problem (1) has a positive solution.
Proof. Note that . In view of Lemma 6, the Dirichlet problemhas a unique positive solution. It is a supersolution of (1) and any nonnegative solution of (1) satisfies . On the other hand, the eigenvalue problem has the first eigenvalue and the corresponding positive eigenfunction . Choose an enough small positive number such that is a subsolution of (1). In view of Theorem 9 and condition (), problem (1) has a positive solution.
Remark 12. For the partial differential equations, the corresponding result has been obtained in  when .
Proof. Assume that and are two distinct positive solutions. If neither nor , by using Theorem 9 we can obtain a positive solution of (1) such that and . Thus we may assume without loss of generality that and . Then, for we haveNote that the function is strictly monotone for , which implies that or . Thus, by Lemma 3, we have orHowever, from (50) and the fact that in we arrive at the contradictory statement that . The proof is complete.
Remark 14. For the corresponding partial differential equations, the conditions of Theorem 13 need to add that the function is uniformly continuous for for some or that there exist constants , , and such that for . See Theorems 2.4 and 2.5 in .
Theorem 15. Let . Suppose that there exists a constant such that for and and that is smooth and strictly decreasing in . Then problem (1) admits at most one positive solution.
Proof. By the maximum principle and the hypothesis, any solution of problem (1) satisfies in . Assume that and are two distinct positive solutions. Under the hypothesis, and are sub- and suppersolution of (1). If neither nor , by Theorem 9, there exists a maximal solution by using . Hence, and . Thus we may assume without loss of generality that and . Then, we haveHence, . On the other hand, by the assumption on , we have That is a contradiction.
Remark 16. The result of corresponding partial differential equations has been obtained in .
Consider first the discrete Dirichlet boundary value problemwhere and are as in (19); moreover, is bounded.
By the maximum principle, any nonnegative and nontrivial solution of (55) must satisfy . For any with , we have Let , then condition (40) holds. Assume that is the first eigenvalue of (19) and is the corresponding positive eigenfunction with . For , we can choose an enough small positive number such that is a subsolution of (55). On the other hand, for is a supersolution of (55). In view of Theorem 9, problem (55) has a positive solution. Since the function satisfies all conditions of Theorem 15, the positive solution is unique. Besides, in view of Theorem 7, we can immediately obtain the nonexistence result. Thus, we obtain the following theorem.
Now consider the discrete Dirichlet boundary value problemswhere . For any , we have the following:(a), ;(b) for , where is an arbitrarily small positive constant and is a constant large enough;(c) and are increasing for ;(d) and are strictly decreasing for .
In view of Theorem 13, we can immediately obtain the following result.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Natural Science Foundation of China (no. 11371277).
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