Abstract
We consider the second order system with the Dirichlet boundary conditions , where the vector field is asymptotically linear and . We provide the existence and multiplicity results using the vector field rotation theory.
1. Introduction
The theory of nonlinear boundary value problems (BVPs in short) is intensively developed since the first works on calculus of variations where BVPs naturally appear in a classical problem of minimizing the integral functional considered on curves with fixed end points. The Euler equation for the problems of the calculus of variations can be written in the formand the boundary conditions areif the problem of fixed end points is considered. The methods for investigation of this problem are diverse. For the existence of a solution, a lot of papers use topological approaches. The main scheme is the following. Imagine in (1) is continuous and one is looking for classical () solution of the problem. If is bounded, then problem (1) and (2) is solvable. This is true for scalar and vectorial cases. If is not bounded, then a priori estimates for a possible solution should be proved first in order to reduce given problem to that with bounded nonlinearity. The interested reader may consult books [1, Ch. 12] and [2–4] for details. We would like to mention also articles [5–8]. The diverse approaches to the subject were used in relatively recent contributions to the theory [9–16].
In all the above-mentioned references, the main question is about the existence of a solution. The problem of the uniqueness of a solution is the next important one, especially for purposes of numerical investigation. It is to be mentioned that both problems (existence and uniqueness) are closely related for linear problems. Indeed, the linear problem has at most one solution for any if is not multiple of The condition is also sufficient for solvability of the problem for any
This is not the case for nonlinear problems. The solvability and multiplicity of solutions may be observed simultaneously. The problem is solvable and has a countable number of solutions. Another phenomenon was observed. Consider the problem together with Sturm-Liouville boundary conditions It is convenient to look at this problem in a phase plane Suppose that at zero and at infinity, where and are essentially different constants. Then, the problem generally has multiple solutions due to the fact that trajectories of solutions of the equation have essentially different rotation speed near the origin and at infinity. This is evident geometrically and one of the first works employing this type of arguments is in the book [17, Ch. 15].
When passing to systems of the second-order differential equations, the analogous approach can be applied. The geometrical interpretation fails however. One should think of a substitute for the rotation (angular) speed. It appears that apparatus of vector fields is good enough. It is possible to construct special vector fields (based on the form of boundary conditions and on the behaviour of nonlinearities of a system) in the vicinity of the origin and “at infinity.” This approach was applied to study BVPs for a system of the two second-order nonlinear differential equations in the work [16]. The considered system was supposed to be asymptotically linear (of one kind) at zero and quasi-linear (linear plus bounded nonlinearity) of another kind at infinity. Special vector fields were considered and the appropriate rotation numbers were invented.
The current article considers the case of second-order differential equations. The approach is the same. However, there is need for employing the respective results concerning rotation of -dimensional vector fields. The main object is a system of the second-order ordinary differential equations given together with the Dirichlet type boundary conditions. The main difference compared with paper [16] is that the computation of rotation numbers at zero and “at infinity” is more complicated and uses an advanced technique.
The structure of the work is the following. In Section 2, the general idea is discussed and useful references and needed definitions are given. In Section 3, the analysis of the vector field at zero (i.e., for solutions with small initial values) is carried out. The similar work is done in Section 4 for the infinity. Section 5 contains the main result. The example and the conclusions complete the article.
2. The Vector Field Associated with the Dirichlet Boundary Value Problem
Consider the systemgiven with the boundary conditionsand the initial conditionswhere .
We suppose that the following conditions are fulfilled.(A1)(A2), and hence system (3) has the trivial solution .(A3)The vector field is asymptotically linear; that is, there exists matrix with real entries such thatThe norms are standard everywhere. The matrix is called the derivative of the vector field at infinity [18].
It follows from the above conditions thatwhere , , andIt follows from (7) and (8) that the vector field is asymptotically linear if and only if for any there exists such thatThe asymptotically linear vector field is linearly bounded. Indeed, fix and consider the corresponding . Then, it follows from (7) and (9) thatwhere , , and .
Rewrite system (3) in the equivalent formwhere , , , and .
Proposition 1. Suppose that conditions (A1), (A2), and (A3) are fulfilled. Then, the vector field has the following properties. (1).(2), where .(3)The vector field is asymptotically linear since there exists matrix where and are unity and zero matrices, respectively, such that (4)The vector field is linearly bounded.
Proof. () and () follow from (A1) and (A2).
() For every , one has that , where . Then, for any there exists such that for every () It follows that asymptotic linearity of the vector field implies its linear boundedness.
Since the vector field is linearly bounded, then [19, 20] its flow is complete and for any , where is the solution to the Cauchy problemLet . We consider for our purposes the restriction of time one flow , where is the solution to Cauchy problem (3) and (5). Denote the first component of by ; that is, Then, . The singular points of the vector field are such that and they are in one-to-one correspondence with the solutions to Dirichlet boundary value problem (3) and (4). It follows from condition (A2) that and hence the singular point of the vector field corresponds to the trivial solution to problem (3) and (4). Any singular point of the vector field generates a nontrivial solution to problem (3) and (4). In what follows, we investigate singular points of the vector field in terms of rotation numbers and provide the conditions which guarantee the existence of a solution (nontrivial) for the boundary value problem under consideration.
Consider a bounded open set . Suppose that the vector field is nonsingular on the boundary ; that is, Then [21, 22], there is an integer , which is associated with the vector field and called the rotation of the vector field on the boundary .
A singular point of the vector field is called isolated [21, 22], if there is neighbourhood containing no other singular points. In this case, the rotation is the same for any sufficiently small radius . This common value is called the index of the isolated singular point .
If the vector field is nonsingular for all of sufficiently large norm, then by definition the point is an isolated singular point of . In this case, the rotation is the same for sufficiently large radius . This common value is called the index of the isolated singular point [21, 22].
3. The Vector Field Near Zero
Suppose that conditions (A1) and (A2) hold. Then, there exists the derivative (the Jacobian matrix) of the vector field at zero and we can consider the linearized system at zerothe Dirichlet boundary conditionsand the initial conditions
If is a solution to Cauchy problem (18) and (20) and is the solution to the matrix Cauchy problemthen for every and . Let us define the linear vector field :Hence, for every .
Let us consider the following condition.(A4)The linearized system at zero (18) is nonresonant with respect to boundary conditions (19); that is, linear homogeneous problem (18) and (19) has only the trivial solution.
The spectrum of the scalar Dirichlet boundary value problemconsists of all such that boundary value problem (23) has a nontrivial solution.
Proposition 2. The following statements are equivalent. (1)Condition (A4) holds.(2).(3) is the unique singular point of the vector field .(4)No eigenvalue of matrix belongs to the spectrum of scalar Dirichlet boundary value problem (23).
Proof. The nonzero singular points of the vector field are in one-to-one correspondence with the nontrivial solutions to Dirichlet boundary value problem (18) and (19). Hence, the equivalence follows from (22).
Let us prove that .
If is the real Jordan form [23] of matrix , then there exists a real nonsingular matrix such that Cauchy problem (18) and (20) transforms to the Cauchy problemwhere and .
If is the solution to Cauchy problem (24) and is the solution to the matrix Cauchy problem then for every and . Let us consider the linear vector field such that Hence, for every .
The Jacobian matrices and are similar. Indeed, since and , one has that and hence and . Next we shall analyze .
The blocks of the real Jordan form of matrix are of two types [23]: a real eigenvalue of matrix generates blockswhere is the size of the block, but a pair and () of complex conjugate eigenvalues of matrix is associated with blockswhere is the size of the block andSuppose solves the matrix Cauchy problemLet be a real eigenvalue of matrix and , where . Then [15, 24],Sinceare upper triangular matrices, then matrix is upper triangular also with the diagonal elements It follows from (31) that function solves the Cauchy problem(a) If , then the solution to the Cauchy problemis . Hence, (b) If , then the solution to the Cauchy problemis Hence, (c) If , then the solution to the Cauchy problemis Hence, (d) Suppose and () are complex conjugate eigenvalues of matrix and , . Then [15, 24],The matrices are upper triangular block matrices of blocks, where and . Then, is upper triangular block matrix of blocks also with diagonal blockswhere It follows from (31) that the matrix solves the matrix Cauchy problem or Suppose that , where and . Then, functions and solve the Cauchy problem and henceTherefore,The determinant of is equal to the product of the determinants of the blocks corresponding to the eigenvalues of matrix . It follows from the above-mentioned considerations that if and only if the eigenvalues of matrix do not belong to the spectrum of scalar Dirichlet boundary value problem (23). Hence, .
Proposition 3. Suppose that condition (A4) holds. If matrix does not have negative eigenvalues with odd algebraic multiplicities, then . If matrix has different negative eigenvalues with odd algebraic multiplicities, then
Proof. Suppose that condition (A4) holds. It follows from Proposition 2 that and is the unique singular point of the vector field . Hence [21, 22], The sign of is equal to the product of the signs of for the blocks corresponding to the eigenvalues of matrix . It follows from the proof of Proposition 2 that for the blocks corresponding to nonnegative and complex eigenvalues of matrix . Let be a negative eigenvalue of matrix with algebraic multiplicity and geometric multiplicity , . Then, matrix has blocks corresponding to the eigenvalue and Therefore, If matrix does not have negative eigenvalues with odd algebraic multiplicities, then . If matrix has different negative eigenvalues with odd algebraic multiplicities, then formula (53) is valid.
Theorem 4. Suppose that conditions (A1), (A2), and (A4) hold. Then, is an isolated singular point of the vector field and .
Proof. We already mentioned that the flow of the vector field is of class for every , where is the solution to Cauchy problem (15). Then, there exist continuous partial derivatives for every and . Matrix solves [4, 19] the matrix Cauchy problemwhere is the Jacobian matrix of the vector field along the solution . One has for , taking into account that , that matrix solves the matrix Cauchy problem where is the Jacobian matrix of the vector field along the solution to Cauchy problem (3) and (5). If , then it follows from condition (A2) that and the matrix solves the matrix Cauchy problemUniqueness of solutions to matrix Cauchy problems (21) and (59) implies that for every . Hence, . Notice that . Therefore, . Since , one has that . It follows from Proposition 2 that . Hence, [21, 22] is an isolated singular point of the vector field and
4. The Vector Field at Infinity
Suppose that conditions (A1) and (A3) hold. Then, there exists the derivative of the vector field at infinity and we can consider the linearized system at infinitythe Dirichlet boundary conditionsand the initial conditions
If is the solution to Cauchy problem (61) and (63) and is the solution to matrix Cauchy problemthen for every and . Let us define the linear vector field ,Hence, for every .
Let us consider the following condition.(A5)The linearized system at infinity (61) is nonresonant with respect to boundary conditions (62); that is, linear homogeneous problem (61) and (62) has only the trivial solution.
Proposition 5. The following statements are equivalent. (1)Condition (A5) holds.(2).(3) is the unique singular point of the vector field .(4)No eigenvalue of the matrix belongs to the spectrum of scalar Dirichlet boundary value problem (23).
Proposition 6. Suppose that condition (A5) holds. If the matrix does not have negative eigenvalues with odd algebraic multiplicities, then . If the matrix has different negative eigenvalues with odd algebraic multiplicities, then
The proofs of Propositions 5 and 6 are analogous to the proofs of Propositions 2 and 3, respectively.
Theorem 7. Suppose that conditions (A1), (A3), and (A5) hold. Then, the point is an isolated singular point of the vector field and .
Proof. First of all, we shall prove that the vector field is asymptotically linear with the derivative at infinity . We proceed in the following steps.
Step 1 (auxiliary linear nonhomogeneous initial value problem). Let us consider the function for every and , where is the solution to Cauchy problem (3) and (5) and is the solution to Cauchy problem (61) and (63). The function solves the Cauchy problemwhere for every and . One can find [24, 25] thatStep 2 (estimates for ). Suppose and consider Taking into account (9) for any , one concludes that there exists such that Since , one hasIn accordance with Proposition 1, the vector field is asymptotically linear, and hence there exist such that for every , . Consider the integral equation equivalent to the Cauchy problem , . Then, Using Grönwall’s inequality [20], one has that Therefore, Since and , we obtainIt follows from (72) and (76) that Therefore, Step 3. Let us prove that .
() Suppose that . It follows from (68) and (78) that Hence, Since can be arbitrary, .
() Suppose that . Then, Let us prove that the series is absolutely convergent; that is, the number series is convergent. It follows from (78) thatThe series converges and the sum isOne can conclude from (82) by using the comparison test that the number series is convergent also and the sum is Hence, the series is absolutely convergent andTherefore, Since can be chosen arbitrary, one has that .
Step 4 (asymptotic linearity of the vector field ). If , thenHence, Since , that is, the vector field is asymptotically linear with the derivative at infinity . It follows from Proposition 5 that . Hence [21, 22], the point is an isolated singular point of the vector field and
5. The Main Theorem
Let us recall that the singular points of the vector field are in one-to-one correspondence with solutions to Dirichlet boundary value problem (3) and (4). A solution of problem (3) and (4) is called nondegenerate, if the singular point of the vector field is nondegenerate; that is, .
Theorem 8. Suppose that conditions (A1) to (A5) hold. Then, the points and are isolated singular points of the vector field .(a)If , then boundary value problem (3) and (4) has a nontrivial solution.(b)If and boundary value problem (3) and (4) has a nontrivial nondegenerate solution, then there exists yet another nontrivial solution to problem (3) and (4).
Proof. (a) It follows from Theorems 4 and 7 that the points and are isolated singular points of the vector field . Hence, one can find positive such that and the sets contain no singular points of the vector field . The vector field is nonsingular on the spheres and and the rotations on these spheres are different: Using [22, Theorem ], one can conclude that the -dimensional annulus contains a singular point of the vector field , which generates a nontrivial solution to Dirichlet boundary value problem (3) and (4).
(b) Let and suppose is a nontrivial nondegenerate solution to boundary value problem (3) and (4), or equivalently is a nonzero nondegenerate singular point of the vector field . Then [21, 22], . Suppose the contrary that is the unique nontrivial solution to boundary value problem (3) and (4) or equivalently is the unique singular point of the vector field in the set . Hence [21, 22], If and , then . If and , then . The contradiction proves that there exists a singular point of the vector field such that or equivalently that there exists a solution to boundary value problem (3) and (4), which is different from .
Remark 9. The practical implementation of Theorem 8 is based on Propositions 3 and 6 and Theorems 4 and 7. Firstly the eigenvalues of the matrices and must be calculated. If the eigenvalues do not belong to spectrum of scalar Dirichlet boundary value problem (23), then the indices and must be calculated accordingly with Propositions 3 and 6. If these indices are different, then Theorem 8 is applicable and the existence of a nontrivial solution to boundary value problem (3) and (4) can be concluded.
Remark 10. Suppose that conditions (A1) to (A5) hold and . If boundary value problem (3) and (4) has an odd number of nontrivial nondegenerate solutions , where and , then there exists yet another nontrivial solution to problem (3) and (4). Suppose the contrary that the set contains only an odd number of singular points of the vector field and these points are nondegenerate. Then [21, 22], , where the left hand side is equal to , but the right hand side is odd.
6. Example
Consider the systemwhere and are nonzero integers, together with the boundary conditionsConsider the vector field : Obviously conditions (A1) and (A2) are fulfilled. Due to the boundedness of function, the vector field is asymptotically linear and , and hence condition (A3) is fulfilled also.
Matrix has the only eigenvalue , and hence matrix does not have negative eigenvalues with odd algebraic multiplicities. It follows from Proposition 6 and Theorem 7 that .
The matrix has the characteristic equation and the eigenvalues are Since , one has that and . Obviously . Note that also since is an algebraic number ( is the root of the characteristic polynomial with rational coefficients), but the spectrum consists of the transcendental numbers (the product of the algebraic number with the transcendental number is transcendental number). Theorem 8 is applicable, taking into account Proposition 3 and Theorem 4, if . Hence, Theorem 8 guarantees the existence of a nontrivial solution to boundary value problem (94) and (95) for all nonzero integers and such thatThe pairs with integer coordinates which satisfy condition (99) are depicted in Figure 1.
7. Conclusions
For an asymptotically linear system of the second-order ordinary differential equations that are assumed to have the trivial solution to the conditions for existence of nontrivial solutions of the Dirichlet boundary value problem are given. The technique and concepts of the theory of rotation of -dimensional vector fields are used. The existence conditions are formulated in terms of eigenvalues of coefficient matrices of linearized systems at zero (at the trivial solution) and at infinity. The proposed approach is applicable to other two-point boundary conditions such as the Neumann problem and mixed problem.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.