Research Article | Open Access
Bo Zheng, "Existence and Multiplicity of Solutions to Discrete Conjugate Boundary Value Problems", Discrete Dynamics in Nature and Society, vol. 2010, Article ID 364079, 26 pages, 2010. https://doi.org/10.1155/2010/364079
Existence and Multiplicity of Solutions to Discrete Conjugate Boundary Value Problems
We consider the existence and multiplicity of solutions to discrete conjugate boundary value problems. A generalized asymptotically linear condition on the nonlinearity is proposed, which includes the asymptotically linear as a special case. By classifying the linear systems, we define index functions and obtain some properties and the concrete computation formulae of index functions. Then, some new conditions on the existence and multiplicity of solutions are obtained by combining some nonlinear analysis methods, such as Leray-Schauder principle and Morse theory. Our results are new even for the case of asymptotically linear.
Let , and be the sets of all natural numbers, integers, and real numbers, respectively. For , define when . is the forward difference operator defined by , and . Let be an matrix. or denotes the transpose of matrix or vector . The set of eigenvalues of matrix will be denoted by , and the determinant of matrix will be denoted by .
Discrete boundary value problems (BVPs for short) arise in the study of solid state physics, combinatorial analysis, chemical reactions, population dynamics, and so forth. Besides, they are also natural consequences of the discretization of continuous BVPs. Thus, these problems have been studied by many scholars.
Discrete two-point BVPs often appear in electrical circuit analysis, mathematical physics, finite elasticity, and so forth as the mathematical models, where with , is a given integer, and are given constants.
We may think of (1.1) as being a discrete analogue of the continuous BVPs: which have been studied by many scholars because of its numerous applications in science and technology. In particular, Hale, Walter, Mawhin, and so forth have obtained some significant results on the existence, uniqueness, and multiplicity of solutions of (1.2). We refer the readers to [1–3] and references therein for further details.
Let Then (1.1) reduces to Hence, in the following, we can only consider the discrete conjugate BVPs, that is,
As being remarked in , the nature of the solution of a continuous problem is not identical with that of the solution of its discrete analogue. And since discrete analogs of continuous problems yield interesting dynamical systems in their own right, many scholars have investigated BVPs (1.5) independently. There are fundamental questions that arise for BVPs (1.5). Does a solution exist, is it unique, and how many solutions can be found if BVPs (1.5) have multiple solutions? How to find the lower bound or the upper bound of the number of solutions of BVPs (1.5)? Furthermore, how to obtain the precise number of solutions of BVPs (1.5)?
In recent years, the existence, uniqueness, and multiplicity of solutions of discrete BVPs have been studied by many authors. In fact, early in 1968, Lasota  studied the discretizations of (1.2) with replaced by and proved that the discrete problem had one and only one solution with satisfying a Lipschitz condition. Note that under certain conditions the solution of a nonhomogeneous BVPs can be expressed in terms of Green's functions. For example, suppose that is a solution of (1.1). Then where is Green's function for Let is a real-valued function defined on , and define by for in . Then there is a one-to-one correspondence between the fixed points of and the solutions of BVPs (1.1). When the nonlinearity satisfies growth conditions known as Lipschitz conditions, a unique solution of BVPs (1.1) can be obtained by using Contraction Mapping Theorem see [6, 7] for more details.
Note that discrete BVPs model numerous physical phenomena in nature hence it is of fundamental importance to know the criteria that ensure the existence of at least one meaningful solution. And since discrete BVPs often have multiple solutions, it is useful to have a collection of results that yield existence of solutions without the implication that the solutions must be unique. To this end, many scholars have obtained some significant results on the existence and multiplicity of solutions of discrete BVPs by using various analytic techniques and various fixed-point theorems, for example, the upper and lower solution method [8–10], the conical shell fixed point theorems [11, 12], the Brouwer and Schauder fixed point theorems [9, 13, 14], and topological degree theory [15, 16]. As we know, critical-point theory (which includes the minimax method and Morse theory, etc.) has played an important role in dealing with the existence and multiplicity of solutions to continuous systems [2, 17]. It is natural for us to think that critical-point theory may be applied to study the existence and multiplicity of solutions to discrete systems. In fact, in recent papers [18–25], the authors have applied critical-point theory to study the existence and multiplicity of periodic solutions to discrete systems. We also refer to [26–31] for the discrete BVPs. In , Agarwal et al. employed the Mountain Pass Lemma to study (1.5) and obtained the existence of multiple solutions. Very recently, B. Zheng and Q. Zhang  studied discrete BVPs (1.5) with and obtained the existence of exactly three solutions by using both Morse theory and degree theory, and so forth. To the best of our knowledge,  is among a few works dealing with discrete BVPs by using Morse theory. Hence, further studies on application of Morse theory to discrete BVPs are still perspective.
Here, we consider the case that is, we consider the following discrete conjugate BVPs: where for every , denotes the gradient of with respect to , and , are given integers.
Assume as , where , and for every , and , , denotes the group of real nonsingular symmetric matrices, and denotes the Euclidean norm of in . Throughout this paper, for any , we denote if is semi-positive definite, and we denote if is positive definite. For any , we denote if for every , and we denote if for every and .
If in (1.10), then (1.10) is usually called an asymptotically linear condition. So here we call (1.10) and (1.11) generalized asymptotically linear conditions. Our results are new even for the case of asymptotically linear case.
The rest of this paper is organized as follows. In Section 2, firstly, we classify the linear systems for every . This classification gives a pair of integers . We call and the index and nullity of , respectively. Secondly, we give some properties of the index and nullity together with the concrete computation formulae. And finally, we introduce the definition of relative Morse index and give its precise description. By using both results in Section 2 and Leray-Schauder principle, we obtain some solvable conditions of (1.9) in Section 3. However, we cannot exclude the possibility that the solution we found is trivial. To this end, we make use of Morse theory to obtain the existence and multiplicity of nontrivial solutions to (1.9). Examples are also included to illustrate the results obtained.
2. Index Theory for Linear Systems
To establish the index theory for (1.12), we introduce the following finite dimensional sequence space: where for every . Define the inner product on as follows: by which the norm on can be induced by where is the usual inner product on , and is the usual norm on .
Define a linear map by It is easy to see that the map defined in (2.4) is a linear homeomorphism, and is a Hilbert space, which can be identified with .
Define For any , if , we say that and are orthogonal. For any two subspaces and of , if for any and , we say that and are orthogonal.
For any subspace of , we say that is positive definite (or negative definite) on if (or ) for all . And if for all , then is called a null subspace of .
Proposition 2.1. For any , the following results hold.(1)There are with such that has a nontrivial solution. If denotes the subspace of solutions with respect to , then and .(2)The space has a orthogonal decomposition such that is positive definite, negative definite, and null on , and , respectively.
To prove Proposition 2.1, we need the following lemma.
Lemma 2.2. For any , the following inequalities hold.
Proof. Note that where Assume that is an eigenvalue of and that is an eigenvector associated to . Define the sequence as Then satisfies Equation (2.12) has a nontrivial solution if and only if see . So with and Noticing that for any real symmetric matrix , we have Since , the inequalities (2.8) now follow from (2.9) and (2.15).
Remark 2.3. In the following, we rewrite (2.8) as for simplicity.
Proof of Proposition 2.1. (1) We claim that the norm induced by the inner product
is equivalent to , where is a positive number satisfying . In fact, it is easy to see that there exists such that
Define a bilinear function
Hence, there exists a unique continuous linear operator satisfying
It is easy to see that is self-adjoint, and hence all the eigenvalues of are real. Therefore, there exist , and , such that
where is the multiplicity of with . By (2.22) and (2.23) we have
In particular, . Without loss of generality we assume that is strictly monotonously decreasing, that is, Denote and , then . We claim that for every , is a nontrivial solution of (2.6). In fact, by (2.24), for any , we have
and since , the above equality means
Therefore satisfies (2.6). Now, we have proved the first result of Proposition 2.1 except .
Set ; then (2.6) is equivalent to which is also equivalent to where Since , is a nonsingular matrix for every . So, we can assume that is the fundamental matrix of equation satisfying . The general solution of can be given by , where and . Set then By and , we have and Hence, .
(2) For any with , by (2.23) and (2.24), we have Hence, if we denote then the results hold.
Definition 2.4. For any , define the index of as , and define the nullity of as .
In the following we shall discuss the properties of .
Proof. (1) By Proposition 2.5, if for any , then (1.12) has only a trivial solution, and hence , . If for some with multiplicity , then by the proof of Proposition 2.1, is the solution subspace of (1.12) and .
(2) By the proof of Proposition 2.1, , and and are orthogonal if . Hence the result holds.
Proposition 2.7. For any , the following results hold.(1)If , then .(2)If , then .(3)If , then .
To prove Proposition 2.7, we firstly prove the following lemma.
Lemma 2.8. Let be a subspace of satisfying and then Moreover, if then
Proof. Without loss of generality we can assume that and . Let , where , . To prove , we only need to prove that is linear independent. If not, there exist not all zero constants such that . So , and hence . This is a contradiction to (2.34). This implies that is linear independent and . The first part is proved.
To prove the second part, let , where , . To prove , we only need to prove that is linear independent. If not, there exist not all zero constants such that . So, and . This is a contradiction to (2.36).
Proof of Proposition 2.7. For any , denote , where , .(1)From Lemma 2.8, we only need to show that In fact, for every , if , then (2)From Lemma 2.8, we only need to show that In fact, for every , one has (3)From Lemma 2.8, we only need to show that In fact, for every with , , if , then If , , then is a nontrivial solution of From we have Hence (2.42) holds.
Proposition 2.9. If , that is, and , then for any , . In particular, for any , we have where is given by (2.13), and is the set of eigenvalues of .
Proof. Firstly, we claim that In fact, since if and only if , we can choose . By (2.22) and (2.23), it is easy to see that , and hence (2.47) holds. Therefore, by Definition 2.4, Since we have Note that the scalar eigenvalue problem has a nontrivial solution if and only if , By Proposition 2.1 and Definition 2.4, we see that for any , Since is the set of eigenvalues of , there exists an orthogonal matrix such that From (2.48), (2.50), and (2.52) we have This completes the proof.
Proposition 2.10. For any with , one has And the equality holds if and only if .
Proof. For any with , we have Because , by definition, for any . So the inequality holds. And the equality holds if and only if as , that is, .
By now, we have proved the monotonicity and have offered the computation formulae of the indices. These will play an important role in discussing nonlinear Hamiltonian systems in the next section. In the end of this section, we shall introduce the relative Morse index, which is a precise expression of the number as .
Definition 2.11. For any with , define
If , , where are two real numbers, then by Proposition 2.9, we have
So This gives us a steer toward the following result.
Proposition 2.12. For any with , one has
Proof. Denote for , ; then to prove (2.60), we only need to prove that
hold. In fact, if (2.61) and (2.62) hold, then the function is integer-valued, left continuous, and nondecreasing. So, for any , must equal to the sum of the jumps incurred in . By (2.61) and (2.62), this is precisely the sum of , , that is,
Hence, if we choose , then (2.60) holds.
From (3) of Proposition 2.7, to prove (2.61), we only need to prove which is also equivalent to . For any , set , we only need to prove Similar to the proof of Lemma 2.8, it is easy to know that for sufficiently small, if then (2.64) holds, where , . While as is sufficiently small and , we have where . Hence .
On the other hand, from (1) of Proposition 2.7, to prove (2.62), we only need to prove . By Lemma 2.8, to prove , we only need to prove where is sufficiently small, . And as is sufficiently small, we have where . This completes the proof.
Proposition 2.13. For any , there exists such that for any , one has
Proof. From Proposition 2.12 we have . From Definition 2.11 and Proposition 2.12, we know that is finite, and then there must exist some such that for any , and
This proves (2.69) and (2.72).
To prove (2.70) and (2.71), note that and Since is finite, there exists such that for any , . Hence This proves (2.70) and (2.71).
3. Main Results
In this section, firstly, we shall obtain the existence of solutions to (1.9) by using both the index theory in Section 2 and Leray-Schauder principle. Then, we obtain the multiplicity of solutions to (1.9) by using Morse theory.
Theorem 3.1. Assume that (1)there exist and which are both continuous with respect to the second variable, where as for every and (2)there exist satisfying , , such that for every . Then (1.9) has at least one solution.
Lemma 3.2. Assume that is a Banach space and that is completely continuous. If the set is bounded, then must have a fixed point in a closed ball in , where
Proof of Theorem 3.1. Assume that (3.2) holds. Since , from (1) of Proposition 2.5, we know that the system has only a trivial solution. Define as then is an invertible operator. Define as then finding the solutions to (1.9) is equivalent to finding solutions to in , which is also equivalent to finding the fixed points of in since is invertible. By Lemma 3.2, we only need to prove that the possible solutions to are priori bounded with respect to the norm in , where . If not, there exist with such that Denote , , ; then (3.9) is equivalent to From (3.1), as . We may assume that , and , where . Denote ; let in (3.10); we have On the other hand, (3.2) implies that , and hence . By , , and Proposition 2.7, we have . This contradicts the fact that (3.11) has a nontrivial solution.
Example 3.3. Let