Dirichlet Boundary Value Problem for the Second Order Asymptotically Linear System

Abstract

We consider the second order system [superscript]x[variant prime][variant prime][/superscript] =f(x) with the Dirichlet boundary conditions x(0)=0=x(1), where the vector field f∈[superscript]C1[/superscript] ([superscript]Rn[/superscript] ,[superscript]Rn[/superscript] ) is asymptotically linear and f(0)=0. We provide the existence and multiplicity results using the vector field rotation theory.

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A. Gritsans 1 and F. Sadyrbaev 1,2 and I. Yermachenko 1

Academic Editor:Qingkai Kong

1; a, Institute of Life Sciences and Technologies, Daugavpils University, Parades iela 1^a , Daugavpils LV-5400, Latvia

2, Institute of Mathematics and Computer Science, University of Latvia, Raina bulv. 29, Riga LV-1459, Latvia

Received 27 July 2016; Accepted 5 October 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 form [figure omitted; refer to PDF] and the boundary conditions are [figure omitted; refer to PDF] 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 f in (1) is continuous and one is looking for classical (x∈^C2 ([a,b])) solution of the problem. If f is bounded, then problem (1) and (2) is solvable. This is true for scalar and vectorial cases. If f 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 ^{x[variant prime][variant prime]} +^k2 x=0,x(0)=A,x(1)=B has at most one solution for any A,B∈R if k is not multiple of π. The condition k≠0 (mod π) is also sufficient for solvability of the problem for any A,B.

This is not the case for nonlinear problems. The solvability and multiplicity of solutions may be observed simultaneously. The problem ^{x[variant prime][variant prime]} =-^x3 , x(0)=0,x(1)=0 is solvable and has a countable number of solutions. Another phenomenon was observed. Consider the problem ^{x[variant prime][variant prime]} +f(x)=0 together with Sturm-Liouville boundary conditions _a1 x(0)+_a2^{x[variant prime]} (0)=0,_b1 x(1)-_b2^{x[variant prime]} (1)=0. It is convenient to look at this problem in a phase plane (x,^{x[variant prime]} ). Suppose that f(x)[approximate]^k2 x at zero and f(x)[approximate]^l2 x at infinity, where k and l 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 n second-order differential equations. The approach is the same. However, there is need for employing the respective results concerning rotation of n-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 [varphi] Associated with the Dirichlet Boundary Value Problem

Consider the system [figure omitted; refer to PDF] given with the boundary conditions [figure omitted; refer to PDF] and the initial conditions [figure omitted; refer to PDF] where 0=(0,0,...,0[...]n^)T ∈^Rn .

We suppose that the following conditions are fulfilled.

(A1) : f∈^C1 (^Rn ,^Rn ).

(A2) : f(0)=0, and hence system (3) has the trivial solution x=0.

(A3) : The vector field f is asymptotically linear ; that is, there exists n×n matrix ^{f[variant prime]} (∞) with real entries such that [figure omitted; refer to PDF]

The norms are standard everywhere. The matrix ^{f[variant prime]} (∞) is called the derivative of the vector field f at infinity [18].

It follows from the above conditions that [figure omitted; refer to PDF] where h∈^C1 (^Rn ,^Rn ), h(0)=0, and [figure omitted; refer to PDF] It follows from (7) and (8) that the vector field f is asymptotically linear if and only if for any [straight epsilon]>0 there exists M([straight epsilon])>0 such that [figure omitted; refer to PDF] The asymptotically linear vector field f is linearly bounded . Indeed, fix _{[straight epsilon]0} >0 and consider the corresponding _M0 =M(_{[straight epsilon]0} )>0. Then, it follows from (7) and (9) that [figure omitted; refer to PDF] where ((^{f[variant prime]} (∞)))=_max(β)=1 (^{f[variant prime]} (∞)β)≥0, _a1 =_M0 >0, and _b1 =((^{f[variant prime]} (∞)))+_{[straight epsilon]0} >0.

Rewrite system (3) in the equivalent form [figure omitted; refer to PDF] where F(z)=(y,f(x)^)T , z=(x,y^)T ∈^RN , y=^{x[variant prime]} , and N=2n.

Proposition 1.

Suppose that conditions (A1) , (A2), and (A3) are fulfilled. Then, the vector field F has the following properties.

(1) F∈^C1 (^RN ,^RN ).

(2) F(o)=o∈^RN , where o=(0,0^)T .

(3) The vector field F is asymptotically linear since there exists N×N matrix [figure omitted; refer to PDF]

: where _In and _On are n×n unity and zero matrices, respectively, such that [figure omitted; refer to PDF]

(4) The vector field F is linearly bounded.

Proof.

(1) and (2) follow from (A1) and (A2) .

(3) For every z=(x,y^)T ∈^RN , one has that F(z)=^{F[variant prime]} (∞)z+H(z), where H(z)=(0,h(x)^)T . Then, for any [straight epsilon]>0 there exists M([straight epsilon])>0 such that for every z=(x,y^)T ∈^RN [figure omitted; refer to PDF] (4) It follows that asymptotic linearity of the vector field F implies its linear boundedness.

Since the vector field F∈^C1 (^RN ,^RN ) is linearly bounded, then [19, 20] its flow ^Φt (γ)=z(t;γ) is complete and ^Φt ∈^C1 (^RN ,^RN ) for any t∈R, where z(t;γ) is the solution to the Cauchy problem [figure omitted; refer to PDF] Let γ=(α,β^)T ∈^RN . We consider for our purposes the restriction of time one flow _{(^Φ1 )α=0} (γ)=(x(1;β),^{x[variant prime]} (1;β)), where x(1;β) is the solution to Cauchy problem (3) and (5). Denote the first component of _{(^Φ1 )α=0} by [varphi]; that is, [figure omitted; refer to PDF] Then, [varphi]∈^C1 (^Rn ,^Rn ). The singular points of the vector field [varphi] are β∈^Rn such that [varphi](β)=0 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 [varphi](0)=0 and hence the singular point β=0 of the vector field [varphi] corresponds to the trivial solution to problem (3) and (4). Any singular point β≠0 of the vector field [varphi] generates a nontrivial solution to problem (3) and (4). In what follows, we investigate singular points of the vector field [varphi] 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⊂^Rn . Suppose that the vector field [varphi] is nonsingular on the boundary ∂Ω; that is, [figure omitted; refer to PDF] Then [21, 22], there is an integer γ([varphi],Ω), which is associated with the vector field and called the rotation of the vector field [varphi] on the boundary ∂Ω.

A singular point _β0 ∈^Rn of the vector field [varphi] is called isolated [21, 22], if there is neighbourhood _Br (_β0 )={(β-_β0 )<r, β∈^Rn } containing no other singular points. In this case, the rotation γ([varphi],_Br (_β0 )) is the same for any sufficiently small radius r. This common value ind (_β0 ,[varphi]) is called the index of the isolated singular point _β0 ∈^Rn .

If the vector field [varphi] is nonsingular for all β∈^Rn of sufficiently large norm, then by definition the point ∞ is an isolated singular point of [varphi]. In this case, the rotation γ([varphi],_BR (0)) is the same for sufficiently large radius R. This common value ind (∞,[varphi]) is called the index of the isolated singular point ∞ [21, 22].

3. The Vector Field [varphi] Near Zero

Suppose that conditions (A1) and (A2 ) hold. Then, there exists the derivative ^{f[variant prime]} (0) (the Jacobian matrix) of the vector field f at zero x=0 and we can consider the linearized system at zero [figure omitted; refer to PDF] the Dirichlet boundary conditions [figure omitted; refer to PDF] and the initial conditions [figure omitted; refer to PDF]

If u(t;β) is a solution to Cauchy problem (18) and (20) and P(t) is the solution to the n×n matrix Cauchy problem [figure omitted; refer to PDF] then u(t;β)=P(t)β for every t∈R and β∈^Rn . Let us define the linear vector field _[varphi]0 :^Rn [arrow right]^Rn : [figure omitted; refer to PDF] Hence, ^{[varphi]0[variant prime]} (β)=^{[varphi]0[variant prime]} (0)=P(1) for every β∈^Rn .

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 _σD ={-(jπ⁾² :j∈N} of the scalar Dirichlet boundary value problem [figure omitted; refer to PDF] consists of all λ such that boundary value problem (23) has a nontrivial solution.

Proposition 2.

The following statements are equivalent.

(1) Condition (A4) holds.

(2) det ^{[varphi]0[variant prime]} (0)=det P(1)≠0.

(3) β=0 is the unique singular point of the vector field _[varphi]0 .

(4) No eigenvalue of matrix ^{f[variant prime]} (0) belongs to the spectrum _σD of scalar Dirichlet boundary value problem (23).

Proof.

The nonzero singular points of the vector field _[varphi]0 are in one-to-one correspondence with the nontrivial solutions to Dirichlet boundary value problem (18) and (19). Hence, the equivalence (1)[...](2)[...](3) follows from (22).

Let us prove that (2)[...](4).

If J is the real Jordan form [23] of matrix ^{f[variant prime]} (0), then there exists a real nonsingular matrix M such that J=^{M-1f[variant prime]} (0)M. Cauchy problem (18) and (20) transforms to the Cauchy problem [figure omitted; refer to PDF] where v=^M-1 u and η=^M-1 β.

If v(t;η) is the solution to Cauchy problem (24) and Q(t) is the solution to the n×n matrix Cauchy problem [figure omitted; refer to PDF] then v(t;η)=Q(t)η for every t∈R and η∈^Rn . Let us consider the linear vector field _ψ0 :^Rn [arrow right]^Rn such that [figure omitted; refer to PDF] Hence, ^{ψ0[variant prime]} (η)=^{ψ0[variant prime]} (0)=Q(1) for every η∈^Rn .

The Jacobian matrices ^{[varphi]0[variant prime]} (β) and ^{ψ0[variant prime]} (η) are similar. Indeed, since v(t;η)=^M-1 u(t;β) and η=^M-1 β, one has that [figure omitted; refer to PDF] and hence P(1)=MQ(1)^M-1 and det ^{[varphi]0[variant prime]} (0)=det P(1)=det Q(1). Next we shall analyze det Q(1).

The blocks of the real Jordan form J of matrix ^{f[variant prime]} (0) are of two types [23]: a real eigenvalue λ of matrix ^{f[variant prime]} (0) generates blocks [figure omitted; refer to PDF] where k is the size of the block, but a pair λ=a+ib and λ¯=a-ib (b≠0) of complex conjugate eigenvalues of matrix ^{f[variant prime]} (0) is associated with blocks [figure omitted; refer to PDF] where k=2m is the size of the block and [figure omitted; refer to PDF]

Suppose _Qk (t)=(_qij (t)) solves the k×k matrix Cauchy problem [figure omitted; refer to PDF]

Let λ be a real eigenvalue of matrix ^{f[variant prime]} (0) and λ=^r2 sgn λ, where r=(λ). Then [15, 24], [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] are upper triangular matrices, then matrix _Qk (t) is upper triangular also with the diagonal elements [figure omitted; refer to PDF] It follows from (31) that function q(t)=_qkk (t) solves the Cauchy problem [figure omitted; refer to PDF]

(a) If λ=0, then the solution to the Cauchy problem [figure omitted; refer to PDF] is q(t)=t. Hence, [figure omitted; refer to PDF]

(b) If λ=^r2 >0 (r>0), then the solution to the Cauchy problem [figure omitted; refer to PDF] is q(t)=sinh[...](rt)/r. Hence, [figure omitted; refer to PDF]

(c) If λ=-^r2 <0 (r>0), then the solution to the Cauchy problem [figure omitted; refer to PDF] is q(t)=sin[...](rt)/r. Hence, [figure omitted; refer to PDF]

(d) Suppose λ=a+ib and λ¯=a-ib (b≠0) are complex conjugate eigenvalues of matrix ^{f[variant prime]} (0) and _Jk (λ)=_C2m (λ), k=2m. Then [15, 24], [figure omitted; refer to PDF] The matrices [figure omitted; refer to PDF] are k×k upper triangular block matrices of 2×2 blocks, [figure omitted; refer to PDF] where a=ρcos[...][straight phi] and b=ρsin[...][straight phi]. Then, _Q2m (t) is k×k upper triangular block matrix of 2×2 blocks also with diagonal blocks [figure omitted; refer to PDF] where [figure omitted; refer to PDF] It follows from (31) that the matrix [figure omitted; refer to PDF] solves the matrix Cauchy problem [figure omitted; refer to PDF] or [figure omitted; refer to PDF] Suppose that λ=a+ib=^μ2 =(α+iβ⁾² , where a=^α2 -^β2 and b=2αβ≠0 (α≠0, β≠0). Then, functions u(t) and v(t) solve the Cauchy problem [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF]

The determinant of Q(1) is equal to the product of the determinants of the blocks _Qk (1) corresponding to the eigenvalues λ of matrix ^{f[variant prime]} (0). It follows from the above-mentioned considerations that det ^{[varphi]0[variant prime]} (0)=det P(1)=det Q(1)≠0 if and only if the eigenvalues of matrix ^{f[variant prime]} (0) do not belong to the spectrum _σD ={-(jπ⁾² :j∈N} of scalar Dirichlet boundary value problem (23). Hence, (2)[...](4).

Proposition 3.

Suppose that condition (A4) holds. If matrix ^{f[variant prime]} (0) does not have negative eigenvalues with odd algebraic multiplicities, then ind (0,_[varphi]0 )=1. If matrix ^{f[variant prime]} (0) has k (1<=k<=n) different negative eigenvalues _λj (1<=j<=k) with odd algebraic multiplicities, then [figure omitted; refer to PDF]

Proof.

Suppose that condition (A4) holds. It follows from Proposition 2 that det[...]^{[varphi]0[variant prime]} (0)=det[...]P(1)≠0 and β=0 is the unique singular point of the vector field _[varphi]0 . Hence [21, 22], [figure omitted; refer to PDF] The sign of det[...]Q(1) is equal to the product of the signs of det _Qk (1) for the blocks _Qk (1) corresponding to the eigenvalues λ of matrix ^{f[variant prime]} (0). It follows from the proof of Proposition 2 that sgn det _Qk (1)=1 for the blocks _Qk (1) corresponding to nonnegative and complex eigenvalues λ of matrix ^{f[variant prime]} (0). Let λ=-^r2 <0 (r=|λ|) be a negative eigenvalue of matrix ^{f[variant prime]} (0) with algebraic multiplicity μ and geometric multiplicity γ, 1<=γ<=μ<=n. Then, matrix Q(1) has γ blocks _Qk1 (1),...,_Qkγ (1) corresponding to the eigenvalue λ and [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] If matrix ^{f[variant prime]} (0) does not have negative eigenvalues with odd algebraic multiplicities, then ind (0,_[varphi]0 )=1. If matrix ^{f[variant prime]} (0) has k (1<=k<=n) different negative eigenvalues _λj (1<=j<=k) with odd algebraic multiplicities, then formula (53) is valid.

Theorem 4.

Suppose that conditions (A1) , (A2), and (A4) hold. Then, β=0 is an isolated singular point of the vector field [varphi] and ind (0,[varphi])=ind (0,_[varphi]0 ).

Proof.

We already mentioned that the flow ^Φt (γ)=z(t;γ) of the vector field F is of class ^C1 for every t∈R, where z(t;γ) is the solution to Cauchy problem (15). Then, there exist continuous partial derivatives (∂_zi /∂_γj )(t;γ) (i,j=1,2,...,N) for every t∈R and γ∈^RN . Matrix Z(t;γ)=((∂_zi /∂_γj )(t;γ)) solves [4, 19] the N×N matrix Cauchy problem [figure omitted; refer to PDF] where ^{F[variant prime]} (z(t;γ)) is the Jacobian matrix of the vector field F along the solution z(t;γ). One has for γ=(0,β), taking into account that z=(x,^{x[variant prime]} ), that matrix X(t;β)=((∂_xi /∂_βj )(t;β)) solves the n×n matrix Cauchy problem [figure omitted; refer to PDF] where ^{f[variant prime]} (x(t;β)) is the Jacobian matrix of the vector field f along the solution x(t;β) to Cauchy problem (3) and (5). If β=0, then it follows from condition (A2) that x(t;0)=0 and the matrix X(t;0)=((∂_xi /∂_βj )(t;0)) solves the n×n matrix Cauchy problem [figure omitted; refer to PDF] Uniqueness of solutions to n×n matrix Cauchy problems (21) and (59) implies that X(t;0)=P(t) for every t∈R. Hence, X(1;0)=P(1). Notice that X(1;0)=((∂_xi /∂_βj )(1;0))=^{[varphi][variant prime]} (0). Therefore, ^{[varphi][variant prime]} (0)=P(1). Since P(1)=^{[varphi]0[variant prime]} (0), one has that ^{[varphi][variant prime]} (0)=^{[varphi]0[variant prime]} (0). It follows from Proposition 2 that det ^{[varphi][variant prime]} (0)=det ^{[varphi]0[variant prime]} (0)≠0. Hence, [21, 22] β=0 is an isolated singular point of the vector field [varphi] and [figure omitted; refer to PDF]

4. The Vector Field [varphi] at Infinity

Suppose that conditions (A1) and (A3 ) hold. Then, there exists the derivative ^{f[variant prime]} (∞) of the vector field f at infinity and we can consider the linearized system at infinity [figure omitted; refer to PDF] the Dirichlet boundary conditions [figure omitted; refer to PDF] and the initial conditions [figure omitted; refer to PDF]

If w(t;β) is the solution to Cauchy problem (61) and (63) and S(t) is the solution to n×n matrix Cauchy problem [figure omitted; refer to PDF] then w(t;β)=S(t)β for every t∈R and β∈^Rn . Let us define the linear vector field _[varphi]∞ :^Rn [arrow right]^Rn , [figure omitted; refer to PDF] Hence, ^{[varphi]∞[variant prime]} (β)=^{[varphi]∞[variant prime]} (0)=S(1) for every β∈^Rn .

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) det[...]^{[varphi]∞[variant prime]} (0)=det S(1)≠0.

(3) β=0 is the unique singular point of the vector field _[varphi]∞ .

(4) No eigenvalue of the matrix ^{f[variant prime]} (∞) belongs to the spectrum _σD of scalar Dirichlet boundary value problem (23).

Proposition 6.

Suppose that condition (A5) holds. If the matrix ^{f[variant prime]} (∞) does not have negative eigenvalues with odd algebraic multiplicities, then ind (0,_[varphi]∞ )=1. If the matrix ^{f[variant prime]} (∞) has s (1<=s<=n) different negative eigenvalues _μi (1<=i<=s) with odd algebraic multiplicities, then [figure omitted; refer to PDF]

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 [varphi] and ind (∞,[varphi])=ind (0,_[varphi]∞ ).

Proof.

First of all, we shall prove that the vector field [varphi] is asymptotically linear with the derivative at infinity ^{[varphi][variant prime]} (∞)=^{[varphi]∞[variant prime]} (0)=S(1). We proceed in the following steps.

Step 1 (auxiliary linear nonhomogeneous initial value problem) . Let us consider the function p(t;β)=(1/||β||)x(t;β)-w(t;β/||β||) for every t∈R and β∈^Rn \{0}, where x(t;β) is the solution to Cauchy problem (3) and (5) and w(t;β) is the solution to Cauchy problem (61) and (63). The function p(t;β) solves the Cauchy problem [figure omitted; refer to PDF] where g(t;β)=(1/(β))h(x(t;β)) for every t∈R and β∈^Rn \{0}. One can find [24, 25] that [figure omitted; refer to PDF]

Step 2 (estimates for ( _{[straight phi] m} ( 1 ; β ) ) ( m = 0,1 , 2 , ... ) ) . Suppose β≠0 and consider [figure omitted; refer to PDF] Taking into account (9) for any [straight epsilon]>0, one concludes that there exists M([straight epsilon])>0 such that [figure omitted; refer to PDF] Since 0<=1-τ<=1, one has [figure omitted; refer to PDF] In accordance with Proposition 1, the vector field F is asymptotically linear, and hence there exist A,B≥0 such that (F(z))<=A+B(z) for every z=(x,y^)T ∈^RN , y=^{x[variant prime]} . Consider the integral equation z(τ;γ)=γ+^∫0τ F(z(s;γ))ds equivalent to the Cauchy problem ^{z[variant prime]} (τ;γ)=F(z(τ;γ)), z(0;γ)=γ=(0,β^)T . Then, [figure omitted; refer to PDF] Using Grönwall's inequality [20], one has that [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] Since (x(τ;β))<=(z(τ;γ)) and (γ)=(β), we obtain [figure omitted; refer to PDF] It follows from (72) and (76) that [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF]

Step 3. Let us prove that _{lim(β)[arrow right]∞} (p(1;β))=0.

(1) Suppose that ^{f[variant prime]} (∞)=_On . It follows from (68) and (78) that [figure omitted; refer to PDF] Hence, [figure omitted; refer to PDF] Since [straight epsilon]>0 can be arbitrary, _{lim(β)[arrow right]∞} (p(1;β))=0.

(2) Suppose that B=^{f[variant prime]} (∞)≠_On . Then, [figure omitted; refer to PDF] Let us prove that the series ^∑k=0∞Bk_{[straight phi]2k+1} (1;β) is absolutely convergent; that is, the number series ^∑k=0∞ (^Bk_{[straight phi]2k+1} (1;β)) is convergent. It follows from (78) that [figure omitted; refer to PDF] The series ^∑k=0∞ (^((B))k /(2k+1)!)((M([straight epsilon])+[straight epsilon]A^eB )/(β)+[straight epsilon]^eB ) converges and the sum is [figure omitted; refer to PDF] One can conclude from (82) by using the comparison test that the number series ^∑k=0∞ (^Bk_{[straight phi]2k+1} (1;β)) is convergent also and the sum is [figure omitted; refer to PDF] Hence, the series ^∑k=0∞Bk_{[straight phi]2k+1} (1;β) is absolutely convergent and [figure omitted; refer to PDF] Therefore, _{lim(β)[arrow right]∞} (p(1;β))<=[straight epsilon]^eB (sinh[...](((B)))/((B))). Since [straight epsilon]>0 can be chosen arbitrary, one has that _{lim(β)[arrow right]∞} (p(1;β))=0.

Step 4 (asymptotic linearity of the vector field [varphi] ) . If β≠0, then [figure omitted; refer to PDF] Hence, [figure omitted; refer to PDF] Since _{lim(β)[arrow right]∞} (p(1;β))=0, [figure omitted; refer to PDF] that is, the vector field [varphi] is asymptotically linear with the derivative at infinity ^{[varphi][variant prime]} (∞)=^{[varphi]∞[variant prime]} (0). It follows from Proposition 5 that det ^{[varphi][variant prime]} (∞)=det ^{[varphi]∞[variant prime]} (0)≠0. Hence [21, 22], the point ∞ is an isolated singular point of the vector field [varphi] and [figure omitted; refer to PDF]

5. The Main Theorem

Let us recall that the singular points of the vector field [varphi] are in one-to-one correspondence with solutions to Dirichlet boundary value problem (3) and (4). A solution x(t;β) of problem (3) and (4) is called nondegenerate , if the singular point β of the vector field [varphi] is nondegenerate; that is, det ^{[varphi][variant prime]} (β)≠0.

Theorem 8.

Suppose that conditions (A1) to (A5) hold. Then, the points β=0 and ∞ are isolated singular points of the vector field [varphi].

(a) If ind (0,[varphi])≠ind (∞,[varphi]), then boundary value problem (3) and (4) has a nontrivial solution.

(b) If ind (0,[varphi])≠ind (∞,[varphi]) 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 β=0 and ∞ are isolated singular points of the vector field [varphi]. Hence, one can find positive r, R such that r<R and the sets [figure omitted; refer to PDF] contain no singular points of the vector field [varphi]. The vector field [varphi] is nonsingular on the spheres _Sr (0)=∂_Br (0) and _SR (0)=∂_BR (0) and the rotations on these spheres are different: [figure omitted; refer to PDF] Using [22, Theorem 2], one can conclude that the n-dimensional annulus [figure omitted; refer to PDF] contains a singular point _β0 ≠0 of the vector field [varphi], which generates a nontrivial solution to Dirichlet boundary value problem (3) and (4).

(b) Let ind (0,[varphi])≠ind (∞,[varphi]) and suppose x(t;_β0 ) is a nontrivial nondegenerate solution to boundary value problem (3) and (4), or equivalently _β0 ∈^Rn is a nonzero nondegenerate singular point of the vector field [varphi]. Then [21, 22], ind (_β0 ,[varphi])=sgn det ^{[varphi][variant prime]} (_β0 )∈{-1,1}. Suppose the contrary that x(t;_β0 ) is the unique nontrivial solution to boundary value problem (3) and (4) or equivalently _β0 is the unique singular point of the vector field [varphi] in the set ^Rn \{0}. Hence [21, 22], [figure omitted; refer to PDF] If ind (∞,[varphi])=1 and ind (0,[varphi])=-1, then ind (_β0 ,[varphi])=ind (∞,[varphi])-ind (0,[varphi])=1-(-1)=2. If ind (∞,[varphi])=-1 and ind (0,[varphi])=1, then ind (_β0 ,[varphi])=ind (∞,[varphi])-ind (0,[varphi])=-1-1=-2. The contradiction proves that there exists a singular point _β1 ∈^Rn \{0} of the vector field [varphi] such that _β1 ≠_β0 or equivalently that there exists a solution x(t;_β1 ) to boundary value problem (3) and (4), which is different from x(t;_β0 ).

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[variant prime]} (0) and ^{f[variant prime]} (∞) must be calculated. If the eigenvalues do not belong to spectrum _σD ={-(jπ⁾² :j∈N} of scalar Dirichlet boundary value problem (23), then the indices ind (0,[varphi]) and ind (∞,[varphi]) 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 ind (0,[varphi])≠ind (∞,[varphi]). If boundary value problem (3) and (4) has an odd number of nontrivial nondegenerate solutions x(t;_βi ) (i=0,1,...,2k), where _βi ≠0 and det ^{[varphi][variant prime]} (_βi )≠0 (i=0,1,...,2k), then there exists yet another nontrivial solution to problem (3) and (4). Suppose the contrary that the set ^Rn \{0} contains only an odd number of singular points _βi (i=0,1,...,2k) of the vector field [varphi] and these points are nondegenerate. Then [21, 22], ind (∞,[varphi])-ind (0,[varphi])=^∑i=02k ind (_βi ,[varphi]), where the left hand side is equal to ±2, but the right hand side is odd.

6. Example

Consider the system [figure omitted; refer to PDF] where k and m are nonzero integers, together with the boundary conditions [figure omitted; refer to PDF] Consider the vector field f:^R2 [arrow right]^R2 : [figure omitted; refer to PDF] Obviously conditions (A1) and (A2) are fulfilled. Due to the boundedness of arctan function, the vector field f is asymptotically linear and ^{f[variant prime]} (∞)=_O2 , and hence condition (A3) is fulfilled also.

Matrix ^{f[variant prime]} (∞) has the only eigenvalue μ=0∉_σD , and hence matrix ^{f[variant prime]} (∞) does not have negative eigenvalues with odd algebraic multiplicities. It follows from Proposition 6 and Theorem 7 that ind (∞,[varphi])=1.

The matrix ^{f[variant prime]} (0)=(-^k2 -^k2 -^m2m2 ) has the characteristic equation [figure omitted; refer to PDF] and the eigenvalues are [figure omitted; refer to PDF] Since (^k2 +^m2)2 +4^k2m2 >^k2 +^m2 >^m2 -^k2 , one has that _λ1 >0 and _λ2 <0. Obviously _λ1 ∉_σD ={-(jπ⁾² :j∈N}. Note that _λ2 ∉_σD also since _λ2 is an algebraic number (_λ2 is the root of the characteristic polynomial p(λ) with rational coefficients), but the spectrum _σD consists of the transcendental numbers (the product -(jπ⁾² of the algebraic number -^j2 with the transcendental number ^π2 is transcendental number). Theorem 8 is applicable, taking into account Proposition 3 and Theorem 4, if ind (0,[varphi])=sgn[...]sin[...]|_λ2 |=-1. Hence, Theorem 8 guarantees the existence of a nontrivial solution to boundary value problem (94) and (95) for all nonzero integers k and m such that [figure omitted; refer to PDF] The pairs (k,m) (1<=k,m<=10) with integer coordinates which satisfy condition (99) are depicted in Figure 1.

Figure 1: The points (k,m) (1<=k,m<=10) with integer coordinates such that k and m satisfy condition (99).

[figure omitted; refer to PDF]

7. Conclusions

For an asymptotically linear system of n 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 n-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.

References

[1] P. Hartman Ordinary Differential Equations , John Wiley & Sons, New York, NY, USA, 1964.

[2] S. R. Bernfeld, V. Lakshmikantham An Introduction to Nonlinear Boundary Value Problems , Academic Press, New York, NY, USA, 1974.

[3] N. I. Vasilyev, Y. A. Klokov 1978 (Russian) Foundations of the Theory of Boundary Value Problems for Ordinary Differential Equations , Apgads Zinatne, Riga, Latvia

[4] W. G. Kelley, A. C. Peterson The Theory of Differential Equations: Classical and Qualitative , Springer, 2010.

[5] R. Conti, "Equazioni differenziali ordinarie quasilineari con condizioni lineari," Annali di Matematica Pura ed Applicata , vol. 57, no. 1, pp. 49-61, 1962.

[6] P. Hartman, "On boundary value problems for systems of ordinary, nonlinear, second order differential equations," Transactions of the American Mathematical Society , vol. 96, no. 3, pp. 493-509, 1960.

[7] J. Mawhin, "Boundary value problems for nonlinear second-order vector differential equations," Journal of Differential Equations , vol. 16, no. 2, pp. 257-269, 1974.

[8] W. T. Reid, "Comparison theorems for non-linear vector differential equations," Journal of Differential Equations , vol. 5, no. 2, pp. 324-337, 1969.

[9] A. Abbondandolo Morse Theory for Hamiltonian Systems , Chapman & Hall/CRC, 2001.

[10] A. Capietto, F. Dalbono, A. Portaluri, "A multiplicity result for a class of strongly indefinite asymptotically linear second order systems," Nonlinear Analysis: Theory, Methods & Applications , vol. 72, no. 6, pp. 2874-2890, 2010.

[11] A. Capietto, W. Dambrosio, D. Papini, "Detecting multiplicity for systems of second-order equations: an alternative approach," Advances in Differential Equations , vol. 10, no. 5, pp. 553-578, 2005.

[12] Y. Dong, "Index theory, nontrivial solutions, and asymptotically linear second-order Hamiltonian systems," Journal of Differential Equations , vol. 214, no. 2, pp. 233-255, 2005.

[13] A. Margheri, C. Rebelo, "Multiplicity of solutions of asymptotically linear Dirichlet problems associated to second order equations in R²ⁿ⁺¹ ," Topological Methods in Nonlinear Analysis , vol. 46, no. 2, pp. 1107-1118, 2015.

[14] D. O'Regan, R. Precup Theorems of Leray-Schauder Type and Applications , CRC Press, New York, NY, USA, 2002.

[15] R. B. Taher, M. Rachidi, "Linear matrix differential equations of higher-order and applications," Electronic Journal of Differential Equations , vol. 2008, no. 95, pp. 1-12, 2008.

[16] I. Yermachenko, F. Sadyrbaev, "On a problem for a system of two second-order differential equations via the theory of vector fields," Nonlinear Analysis: Modelling and Control , vol. 20, no. 2, pp. 217-226, 2015.

[17] M. A. Krasnosel'ski[ibreve], A. I. Perov, A. I. Povolotskiy, P. P. Zabreiko Plane Vector Fields , Academic Press, New York, NY, USA, 1966.

[18] M. A. Krasnosel'skii The Operator of Translation Along the Trajectories of Differential Equations , American Mathematical Society, Providence, RI, USA, 1968.

[19] V. I. Arnold Ordinary Differential Equations , Springer, Berlin, Germany, 1992.

[20] E. Zehnder Lectures on Dynamical Systems: Hamiltonian Vector Fields and Symplectic Capacities , European Mathematical Society, Helsinki, Finland, 2010.

[21] M. A. Krasnosel'skii, P. P. Zabre[ibreve]ko Geometrical Methods of Nonlinear Analysis , Springer, Berlin, Germany, 1984.

[22] P. P. Zabrejko, M. Matzeu, A. Vignoli, "Rotation of vector fields: definition, basic properties, and calculation," Topological Nonlinear Analysis II: Degree, Singularity and Variations , vol. 27, of Progress in Nonlinear Differential Equations and Their Applications, pp. 445-601, Birkhäuser, Boston, Mass, USA, 1997.

[23] R. A. Horn, C. R. Johnson Matrix Analysis , Cambridge University Press, London, UK, 1990.

[24] F. R. Gantmacher The Theory of Matrices , AMS Chelsea Publishing, Providence, RI, USA, 1998.

[25] V. A. Zorich Mathematical Analysis I , Springer, Berlin, Germany, 2004.