Dirichlet Boundary Value Problem for the Second Order Asymptotically Linear System

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.


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 form and the boundary conditions are if 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 ( ∈  2 ([, ])) 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][3][4] for details.We would like to mention also articles [5][6][7][8].The diverse approaches to the subject were used in relatively recent contributions to the theory [9][10][11][12][13][14][15][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   +  2  = 0, (0) = , (1) =  has at most one solution for any ,  ∈ R if  is not multiple of .The condition  ̸ = 0 (mod ) 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   = − 3 , (0) = 0, (1) = 0 is solvable and has a countable number of solutions.Another phenomenon was observed.Consider the problem   + () = 0 together with Sturm-Liouville boundary conditions  1 (0)+ 2   (0) = 0,  1 (1) −  2   (1) = 0.It is convenient to look at this problem in a phase plane (,   ).Suppose that () ≈  2  at zero and () ≈  2  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 2 International Journal of Differential Equations 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 secondorder 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.

The Vector Field 𝜙 Associated with the Dirichlet Boundary Value Problem
Consider the system given with the boundary conditions and the initial conditions where 0 = (0, 0, . . ., 0 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ We suppose that the following conditions are fulfilled. (A2) f(0) = 0, and hence system (3) has the trivial solution x = 0.
Proposition 1. Suppose that conditions (A1), (A2), and (A3) are fulfilled.Then, the vector field F has the following properties. ( ( (3) The vector field F is asymptotically linear since there exists  ×  matrix where   and   are  ×  unity and zero matrices, respectively, such that (4) The vector field F is linearly bounded.
Since the vector field F ∈  1 (R  , R  ) is linearly bounded, then [19,20] its flow Φ  () = z(; ) is complete and Φ  ∈  1 (R  , R  ) for any  ∈ R, where z(; ) is the solution to the Cauchy problem Let  = (, )  ∈ R  .We consider for our purposes the restriction of time one flow Φ Then,  ∈  1 (R  , R  ).The singular points of the vector field  are  ∈ R  such that () = 0 and they are in one-toone correspondence with the solutions to Dirichlet boundary value problem (3) and (4).It follows from condition (A2) that (0) = 0 and hence the singular point  = 0 of the vector field  corresponds to the trivial solution to problem (3) and (4).Any singular point  ̸ = 0 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 Ω ⊂ R  .Suppose that the vector field  is nonsingular on the boundary Ω; that is, Then [21,22], there is an integer (, Ω), which is associated with the vector field and called the rotation of the vector field  on the boundary Ω.
A singular point  0 ∈ R  of the vector field  is called isolated [21,22], if there is neighbourhood   ( 0 ) = {‖ −  0 ‖ < ,  ∈ R  } containing no other singular points.In this case, the rotation (,   ( 0 )) is the same for any sufficiently small radius .This common value ind ( 0 , ) is called the index of the isolated singular point  0 ∈ R  .
If the vector field  is nonsingular for all  ∈ R  of sufficiently large norm, then by definition the point ∞ is an isolated singular point of .In this case, the rotation (,   (0)) is the same for sufficiently large radius .This common value ind (∞, ) is called the index of the isolated singular point ∞ [21,22].

The Vector Field 𝜙 Near Zero
Suppose that conditions (A1) and (A2) hold.Then, there exists the derivative f  (0) (the Jacobian matrix) of the vector field f at zero x = 0 and we can consider the linearized system at zero the Dirichlet boundary conditions and the initial conditions If u(; ) is a solution to Cauchy problem ( 18) and ( 20) and () is the solution to the  ×  matrix Cauchy problem then u(; ) = () for every  ∈ R and  ∈ R  .Let us define the linear vector field  0 : R  → R  : Hence,   0 () =   0 (0) = (1) for every  ∈ R  .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   = {−() 2 :  ∈ N} of the scalar Dirichlet boundary value problem consists of all  such that boundary value problem (23) has a nontrivial solution.
Proposition 2. The following statements are equivalent.
(3)  = 0 is the unique singular point of the vector field  0 .
(4) No eigenvalue of matrix f  (0) belongs to the spectrum   of scalar Dirichlet boundary value problem (23).
If  is the real Jordan form [23] of matrix f  (0), then there exists a real nonsingular matrix  such that  =  −1 f  (0).Cauchy problem (18) and ( 20) transforms to the Cauchy problem where k =  −1 u and  =  −1 .
The blocks of the real Jordan form  of matrix f  (0) are of two types [23]: a real eigenvalue  of matrix f  (0) generates blocks where  is the size of the block, but a pair  =  +  and  =  −  ( ̸ = 0) of complex conjugate eigenvalues of matrix f  (0) is associated with blocks where  = 2 is the size of the block and Suppose   () = (  ()) solves the  ×  matrix Cauchy problem Let  be a real eigenvalue of matrix f  (0) and  =  2 sgn , where  = √||.Then [15,24], Since are 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  = 0, then the solution to the Cauchy problem , then the solution to the Cauchy problem , then the solution to the Cauchy problem (41) (d) Suppose  =  +  and  =  −  ( ̸ = 0) are complex conjugate eigenvalues of matrix f  (0) and   () =  2 (),  = 2.Then [15,24], The matrices are  ×  upper triangular block matrices of 2 × 2 blocks, where  =  cos  and  =  sin .Then,  2 () is  × upper triangular block matrix of 2 × 2 blocks also with diagonal blocks where It follows from (31) that the matrix solves the matrix Cauchy problem or Suppose that  =  +  =  2 = ( + ) 2 , where  =  2 −  2 and  = 2 ̸ = 0 ( ̸ = 0,  ̸ = 0).Then, functions () and V() solve the Cauchy problem and hence The determinant of ( 1) is equal to the product of the determinants of the blocks   (1) corresponding to the eigenvalues  of matrix f  (0).It follows from the abovementioned considerations that det   0 (0) = det (1) = det (1) ̸ = 0 if and only if the eigenvalues of matrix f  (0) do not belong to the spectrum   = {−() 2 :  ∈ N} of scalar Dirichlet boundary value problem (23).Hence, (2) ⇔ (4).

The Vector Field 𝜙 at Infinity
Suppose that conditions (A1) and (A3) hold.Then, there exists the derivative f  (∞) of the vector field f at infinity and we can consider the linearized system at infinity the Dirichlet boundary conditions and the initial conditions (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.
(3)  = 0 is the unique singular point of the vector field  ∞ .
(4) No eigenvalue of the matrix f  (∞) belongs to the spectrum   of scalar Dirichlet boundary value problem (23).
Proposition 6. Suppose that condition (A5) holds.If the matrix f  (∞) does not have negative eigenvalues with odd algebraic multiplicities, then ind (0, The proofs of Propositions 5 and 6 are analogous to the proofs of Propositions 2 and 3, respectively.
Proof.First of all, we shall prove that the vector field  is asymptotically linear with the derivative at infinity   (∞) =   ∞ (0) =  (1).We proceed in the following steps.

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 x(; ) of problem ( 3) and ( 4) is called nondegenerate, if the singular point  of the vector field  is nondegenerate; that is, det   () ̸ = 0. Proof.(a) It follows from Theorems 4 and 7 that the points  = 0 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   (0) =   (0) and   (0) =   (0) and the rotations on these spheres are different:  4), or equivalently  0 ∈ R  is a nonzero nondegenerate singular point of the vector field .Then [21,22], ind ( 0 , ) = sgn det   ( 0 ) ∈ {−1, 1}.Suppose the contrary that x(;  0 ) is the unique nontrivial solution to boundary value problem ( 3) and ( 4) or equivalently  0 is the unique singular point of the vector field  in the set R  \ {0}.Hence [21,22], ind (∞, ) = ind (0, ) + ind ( 0 , ) . ( The contradiction proves that there exists a singular point  1 ∈ R  \ {0} of the vector field  such that  1 ̸ =  0 or equivalently that there exists a solution x(;  1 ) to boundary value problem ( 3) and ( 4), which is different from x(;  0 ).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 f  (0) and f  (∞) must be calculated.If the eigenvalues do not belong to spectrum   = {−() 2 :  ∈ N} of scalar Dirichlet boundary value problem (23), then the indices ind (0, ) and ind (∞, ) 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.

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 twopoint boundary conditions such as the Neumann problem and mixed problem.