# Multiple Solutions for Generalized Asymptotical Linear Hamiltonian Systems Satisfying Bolza Boundary Conditions

Authors: Yuan Shan; Baoqing Liu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper is devoted to multiple solutions of generalized asymptotical linear Hamiltonian systems satisfying Bolza boundary conditions. We classify the linear Hamiltonian systems by the index theory and obtain the existence and multiplicity of solutions for the Hamiltonian systems, based on an application of the classical symmetric mountain pass lemma.


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## Body

## 1. Introduction

This paper is concerned with a classical problem on the existence of solutions for Hamiltonian systems when a more general form of twist condition between the origin and infinity holds for the Hamiltonian function. More precisely consider the system(1)x˙=JH′(t,x),Px(0)=0=Px(1),
where H∈C(R×R2n,R), and H′∈C(R×R2n,R2n)and H′  satisfies the following:(H1)
H′(t,x)=B0(t,x)x+o(|x|), as  |x|→0 uniformly in t∈[0,1],(H2)
H′(t,x)=B∞(t,x)x+o(|x|), as  |x|→∞ uniformly in t∈[0,1],for B0(t,x),  B∞∈L∞((0,1)×R2n;GLs(R2n)). Here and below, we use H′ to denote the first derivative of H with respect to x∈R2n.A quantitative way to measure the twisting is given by the Maslov-type index. As in [1, 2], an index for the second-order and first-order linear Hamiltonian systems was defined and developed in [3] for the study of linear operator equation with infinite Morse index. In [4, 5], by Conley and Zehnder and Long, an index theory for symplectic path was defined. We refer to two excellent books [6, 7] for systematical treatment.In [2], Dong discussed the classification of the linear Hamiltonian systems with Bolza boundary conditions as follows:
(2)x˙=JB(t)x,Px(0)=0=Px(1),
where B∈L∞((0,1);GLs(R2n)) and Px=(x1,…xn)∈Rn, for any x=(x1,…,x2n). That is, for any B∈L∞((0,1);GLs(R2n)), he associated it with a pair of numbers (i(B),ν(B))∈Z×{0,1,…,n}. This pair of integers is called index and nullity of B, respectively. And he defined the nullity ν(B) as the dimension of the solution space of (1). Let E[α] be the integer a∈Z such that a<α≤a+1. So that E[a]=a-1 for all integer a defined as in [6]. In order to process the definition of i(B), he defined i(λI2n)=nE[λ/π]. In particular, when λ∈(kπ,(k+1)π), i(λI2n)=nk. We will introduce the definitions and properties of i(B) in detail in Section 2. After the discussion of the index theory, we will prove our main result in Section 3.Throughout this paper,∥x∥C denotes the usual norm in C[0,1]. For any A1, A2∈GLs(R2n), we write A1≤A2 if A2-A1 is positively semidefinite, and we write A1<A2 if A2-A1 is positive definite. For any A1, A2∈L∞((0,1);GLs(R2n)), we write A1≤A2 if A1(t)≤A2(t) for a.e. t∈(0,1), and we write A1<A2 if A1≤A2 and A1(t)<A2(t) on a subset of (0,1) with a nonzero measure.Remark 1.
LetC1, C2∈(kπ,(k+1)π), C3, C4∈((k+1)π,(k+2)π) with C1<C2, C3<C4. Assume that Ai=CiI2n, i=1,2, and Bi=CiI2n, i=3,4. Then, we have i(A1)=i(A2)=nk, i(B1)=i(B2)=n(k+1), and ν(Ai)=ν(Bi)=0, for i=1,2; (1) has at least n pairs of solutions.We make use of the critical point theory [8, 9] to prove Theorem 8. The novelty of our result is that it suffices to assume that H(t,x)+(1/2)μ|x|2 is convex. The twisting between origin and infinity is reflected in (H1*) and (H2*). Thus, our results complement with Theorem 3.8 in [2] and Theorem 1.1 in [10]. For other results, we refer to [11–15].This paper is organized as follows. In Section2, we introduce some preliminaries including index theory, and establish the μ-index theory which is needed in the proofs. In Section 3, we present the proofs of the results.


## 2. Index Theory for Linear Hamiltonian Systems Satisfying Bolza Boundary Value Conditions

First, we recall some definitions and propositions in [2]. We consider the following system:
(3)x˙=JB(t)x,(4)Px(0)=0=Px(1).Let H={x:[0,1]→R2n|x is continuous on [0,1], satisfies (4), and x˙∈L2} with the norm ∥x∥H=(∫01‍(|x′(t)|2+|x(t)|2)dt)1/2, (Λ1x)(t)=Jx′(t), (B-x)(t)=B(t)x(t), where L2:=L2((0,1);R2n), and let |x|=(∑i=12n‍|xi|)1/2 for any x=(x1,…,x2n)∈R2n.    Then, Λ1 and B- are self-adjoint operators, and B- is bounded.Definition 2 (see [2, Definitions 2.1, 2.3, 2.4, and 2.7]).
For anyB1,B2,B∈L∞((0,1),GLs(R2n)), one defines that(1)
ν(B)=dimker(Λ1+B-),(2)
I(B1,B2)=∑λ∈[0,1)‍ν((1-λ)B1+λB2) for B1<B2,(3)
i(B)=i(λI2n)-I(B,λI2n) for λ∈R satisfies B<λI2n.Using spectral theory, a Morse-type indexiμ(B) was established in [2]. More precisely, for any B∈L∞((0,1),GLs(R2n)), let μ∈R∖πZ with B+μI2n≥I2n. Then, ν(-μI2n=0), Λx:=Jx˙(t)-μx(t) is invertible, and the inverse Λ-1:L2→L2 is self-adjoint and compact. He put a quadratic form:
(5)qμ,B(u,u)=12∫01‍(Λ-1u(t),u(t))+(Cμ(t)u(t),u(t))dt∀u∈L2,(6)(Cμ(t)u(t),u(t))∶=∫01‍(Cμ(t)u(t),u(t)),
where Cμ(t):=(μI2n+B(t))-1. Then, (C-μ(t)u(t),u(t)) defines a Hilbert space structure on L2. Cμ-1Λ-1 is a self-adjoint and compact operator under this interior product. By the spectral theory, there is a basis ej, j∈N of L2, and a sequence λj→0 in R such that
(7)(Cμ(t)ei(t),ej(t))=δij,(Λ-1ej,u)=(Cμλjej,u)∀u∈L2.
For any u∈L2 as u=∑j=1∞‍ξjej, (5) can be rewritten as follows:
(8)qμ,B(u,u)=12∫01‍(Λ-1u,u)+(Cμ(t)u,u)dt=12∑j=1∞‍(1+λj)ξj2.
Define that
(9)Eμ-(B)∶={∑j=1∞‍ξjej∣ξj=0if1+λj≥0},Eμ0(B)∶={∑j=1∞‍ξjej∣ξj=0if1+λj≠0},Eμ+(B)∶={∑j=1∞‍ξjej∣ξj=0if1+λj≤0}.Eμ-(B), Eμ0(B), and Eμ+(B) are qμ,B-orthogonal, and Eμ-(B)⊕Eμ0(B)⊕Eμ+(B)=L2. Since λj→0 as j→∞, Eμ-(B) and Eμ0(B) are two finite-dimensional subspaces.Definition 3 (see [2, Definition 2.9]).
For anyB∈L∞((0,1),GLs(R2n)), μ∈R with μI2n+B≥I2n, one defines that
(10)νμ(B)∶=dimEμ0(B),iμ(B)∶=dimEμ-(B).One callsνμ(B) and iμ(B)  μ-nullity and μ-index of B, respectively.Proposition 4 (see [2, Propositions 2.10, 2.13]).
The index, relative Morse index, andμ-index have the following properties.(1)
For anyB1, B2, B∈L∞((0,1),GLs(R2n)), with B1<B2, one has
(11)νμ(B)=ν(B),I(B1,B2)=iμ(B2)-iμ(B1),iμ(B1)-iμ(B2)=i(B1)-i(B2).(2)
There existsϵ0>0, such that, for any ϵ∈(0,ϵ0], one gets
(12)ν(B+ϵI2n)=0=ν(B-ϵI2n),i(B-ϵI2n)=i(B),i(B+ϵI2n)=i(B)+ν(B).In particular, ifν(B)=0, one obtains i(B+ϵI2n)=i(B) for ϵ∈(0,ϵ0].(3)
iμ(B)-i(B) is a constant for B satisfying B+μI2n≥I2n, that is, iμ(B)-i(B)=iμ(B1)-i(B1) for any other B1∈L∞((0,1),GLs(R2n)) with B1+μI2n≥I2n. For any μ>1, one has
(13)iμ(0)=nE[μπ],iμ(B)=nE[μπ]+n+i(B),for anyB∈L∞((0,1),GLs(R2n)) with B+μI2n≥I2n.Remark 5.
From this proposition, there isϵ0>0, ϵ∈(0,ϵ0] such that νμ(B+ϵI2n)=0=νμ(B-ϵI2n) and iμ(B-ϵI2n)=iμ(B),iμ(B+ϵI2n)=iμ(B)+νμ(B).In order to prove Theorem8, we make use of minimax arguments for the multiplicity of critical points in the presence of symmetry. We state two results of this type from [8, 9].Lemma 6 (cf. Chang [9, Theorem 4.3.4]).
Assume thatf∈   C1(X,R1) satisfies the (PS) condition, f(θ)=0, f(-x)=f(x), and(1)
there are an m-dimensional subspaceX1 and a constant ρ>0 such that
(14)supx∈X1∩Sρf(x)<0,(2)
there is a j-dimensional subspaceX2 such that
(15)infx∈X2⊥f(x)>-∞,whereX2⊥ is a subset of X such that X2⊥⊕X2=X. Then, f has at least m-j pairs of critical points if m-j>0.Lemma 7 (see [8, Theorem 2.2.8]).
Suppose thatf∈   C′(H,R) satisfies (C) condition, f(θ)≥0, even, and there exist two closed subspaces H+H- of H, with codimH-<+∞, and two constants c∞>c0≥f(0) such that(a)
f(u)≥c0 for all u∈Sρ∩H+,(b)
f(u)≤c∞ for all u∈H-.
Then, ifdimH-≥codimH+, f possesses at least m=dimH--codimH+ distinct pairs of critical points whose corresponding critical values belong to [c0, c∞].


## 3. Proof of Main Results

We state the main result in this paper. We further make the following assumptions.(H1*)
There existsA1, A2∈L∞((0,1);GLs(R2n)) such that A1(t)≤B0(t,x)≤A2(t) with i(A1)=i(A2), ν(A2)=0.(H2*)
There existsB1, B2∈L∞((0,1);GLs(R2n)) such that B1(t)≤B∞(t,x)≤B2(t) with i(B1)=i(B2), ν(B2)=0.(H3)
H(t,θ)=0, H(t,x)=H(t,-x).(H4)
H(t,x)+(1/2)μ|x|2 is convex.Our main result reads as follows.Theorem 8.
Assume that (H1), (H1*), (H2), (H2*), (H3), (H4)    are satisfied, then (1) has at least |i(A1)-i(B1)| pairs of solutions.Theorem 9.
Ifi(A1)>i(B1), then (1) has at least i(A1)-i(B1) pairs of solutions.
Ifi(A1)>i(B1), let μ∈R∖πZ with A1+μI2n≥I2n, B1+μI2n≥I2n. Recalling that Λx=Jx˙-μI2n,  one denotes   Hμ(t,x)=H(t,x)+(1/2)μ|x|2. Then, one obtains
(16)Hμ*(t,x)=supy∈R2n{(x,y)-Hμ(t,y)}.
Thus, Hμ*(t,-x)=H*(t,x) and
(17)x=Hμ*′(t,x*)iffx*=Hμ′(t,x)
(by the Fenchel conjugate formula; see [6, Proposition II]). Hence, |x|→∞ if and only if |x*|→∞. Consider the functional defined by
(18)ψ(u)=∫01‍(12Λ-1u(t),u(t))+Hμ*(t,u(t))dt∀u∈L2.In order to prove Theorem9, we need the following lemmas.Lemma 10.
ψ satisfies the (PS) condition.Proof.
Assume that{uj} is a sequence in L2 such that ψ(uj) is bounded and ψ′(uj)→0 in L2. It suffices to prove that {uj} has convergent sequence. By (18), we have
(19)(ψ′(u),v)=∫01‍(Λ-1u(t),v(t))+(Hμ*′(t,u(t)),v(t))dt,
for all v∈L2. Hence, we get
(20)ψ′(uj)=Λ-1uj+Hμ*′(t,uj(t))⟶θ,inL2.
Let xj=Λ-1uj, yj=uj/∥xj∥C, and zj=ψ′(uj)-Λ-1(uj). If ∥xj∥C→∞, then ∥xj∥L2→∞. Joining (H2) with (17) and (20), we have
(21)uj(t)=Hμ′(t,zj(t))=(B∞(t,zj(t))+μI2n)zj(t)+o(|zj(t)|)=-(B∞(t,zj(t))+μI2n)Λ-1uj+(B∞(t,zj(t))+μI2n)ψ′(uj)+o(|zj(t)|).
Using the preceding notations, we have
(22)yj(t)=-(B∞(t,zj(t))+μI2n)Λ-1yj+(B∞(t,zj(t))+μI2n)ψ′(uj)∥xj∥C-1+o(|zj(t)|)∥xj∥C-1.
So yj(t) is bounded in L2. We assume that yj⇀y- in L2, and hence Λ-1yj→Λ-1y-. From (22), there exists B-∈L∞((0,1);GLs(R2n)), such that
(23)B∞(t,zj(t))v(t)⇀B-(t)v(t)foranyv∈L2.
And we have B1(t)≤B-(t)≤B2(t). Taking the limit as j→∞ in (22), we obtain y-(t)=-(B-(t)+μI2n)Λ-1y-. Let x-=Λ-1y-. We get
(24)Jx-+B-(t)x-=0,Px-(0)=0=Px-(1).
By assumption (H2*), we have i(B-)=0 and x-=0, which is impossible. Since ∥x-∥C=1, ∥Λ-1uj∥C is bounded. From (17) and (20), ∥uj∥L2 is bounded. Assume that uj⇀u0 in L2; then Λ-1uj→Λ-1u0. Let ξj:=Λ-1uj+Hμ*′(t,uj); then Hμ*′(t,uj)=ξj-Λ-1uj→-Λ-1u0. Fenchel conjugate formula gives uj=Hμ*′(t,ξj-Λ-1uj)→Hμ*′(t,-Λ-1u0) in L2  (by [6, Preposition II, Theorem 4]).Lemma 11.
The assumption (1) of Lemma6 is valid, where f is defined as in (18).Proof.
From (H1) and (H1*), we have that, for any ϵ>0, there exists δ>0 such that
(25)((A1(t)-ϵ)x,x)≤(H′(t,x),x)≤((A2(t)+ϵ)x,x),cccccccccccccccccccccccccccccccccccccforany|x|≤δ.
Thus,
(26)Hμ(t,y)=(∫01‍Hμ′(t,θy)dθ,y)≥12(((A1(t)-ϵI2n)+μI2n)y,y),forany|y|≤δ.
Letg(y)=(x,y)-Hμ(t,y). By (16) and (H4), g(y) is strictly concave. Then, |y|→∞ as g(y)→-∞. So, g achieves its maximum at a unique point. Because of g′(y)=x-Hμ′(t,y), we get that
(27)x=Hμ′(t,y)has  a  unique  solution.
And we can easily get0=Hμ′(t,0). Hence, we have |y|→0 when |x|→0. Otherwise, there exists ϵ0>0, xn, yn∈R2n, tn∈[0,1] s.t. |xn|→0. But |yn|≥ϵ0, xn=Hμ′(tn,yn). Hence, |yn| is bounded, and there exists y0, t0 such that yn⇀y0, tn→t0, xn→0. Thus, 0=Hμ′(t0,y0). This contradicts the uniqueness, which yields there exists ρ(ρ<δ) such that, for any |x|≤ρ, we have |y|≤δ. So, for any |x|≤ρ, we have
(28)Hμ*(t,x)=supy∈R2n{(x,y)-Hμ(t,y)}=sup|y|≤δ{(x,y)-Hμ(t,y)}≤sup|y|≤δ{(x,y)-12(((A1(t)-ϵI2n)+μI2n)y,y)}=12(((A1(t)-ϵI2n)+μI2n)-1x,x).
By (18),
(29)ψ(u)≤∫01‍(12Λ-1u(t),u(t))+12(((A1(t)-ϵI2n)+μI2n)-1u(t),u(t))dtforany|u|≤ρ,
that is,
(30)ψ(u)≤qμ,(A1-ϵ)(u,u)forany|u|≤ρ.
LetX1=Eμ-(A1-ϵ). Then, dimX1=iμ(A1-ϵ). For any u∈X1∩Sρ, we have supψ(u)<0. By Proposition 4, for ϵbeing small enough, iμ(A1-ϵ)=iμ(A1).Lemma 12.
The assumption (2) of Lemma6 is also satisfied.Proof.
From (H2), let h2(t,x)=H′(t,x)-B∞(t,x)x. Then, for any ϵ>0, there exists a constant M1 such that
(31)|h2(t,x)|≤ϵ|x|+M1,∀(t,x)∈[0,1]×R2n.
Combining (H2) and (31), there exists a constant M2 such that
(32)Hμ(t,x)≤12((B2(t)+4ϵI2n+μI2n)x,x)+M2,
for all (t,x)∈[0,1]×R2n. By the definition of Hμ*(t,x), we have
(33)Hμ*(t,x)≥supy∈R2n{(x,y)-12((B2(t)+4ϵI2n+μI2n)y,y)}-M2=12(((B2(t)+4ϵI2n)+μI2n)-1x,x)-M2.
Thus,
(34)ψ(u)≥∫01‍(12Λ-1u(t),u(t))+12((B2(t)+4ϵI2n)+μI2n)-1(u(t),u(t))dt-M2.
So, we infer
(35)ψ(u)≥qμ,(B2+4ϵ)(u,u)-M2.
Let X2=Eμ-(B2+4ϵ). We have dimX2=iμ(B2+4ϵ) and X2⊥=Eμ0(B2(t)+4ϵ)⊕Eμ+(B2(t)+4ϵ). So, we have
(36)infX2⊥ψ(u)≥-M2.
By Proposition4, we have iμ(A1)-iμ(B2)=i(A1)-i(B2), and we can also let ϵ be small enough such that iμ(B2+4ϵ)=iμ(B2). This completes the proof.Theorem 13.
Ifi(A1)<i(B1), then (2) has at least i(B1)-i(A1) pairs of solutions.The argument in Theorem8 can also be used here by using Lemma 7. More precisely, if H+=Eμ+(A2+ϵ), then codimH+=i(A1) and ψ(u)>c0>0 on H+∩Sρ. If H-=Eμ-(B1-4ϵ), then dimH-=i(B1) and ψ(u)≤c∞ on H-.



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