The purpose of this paper is to develop a generalized matrix Riccati technique for the selfadjoint matrix Hamiltonian system U′=A(t)U+B(t)V, V′=C(t)U−A∗(t)V. By using the standard integral averaging technique and positive functionals, new oscillation and interval
oscillation criteria are established for the system. These criteria extend and improve some results that have been required before. An interesting example is included to illustrate the importance of our results.

1. Introduction

In this paper, we consider oscillatory properties for the linear Hamiltonian systemU′=A(t)U+B(t)V,V′=C(t)U-A*(t)V,t≥t0,
where A(t),B(t), and C(t) are real n×n matrix-valued functions, B,C are Hermitian, and B is positive definite. By M*, we mean the conjugate transpose of the matrix M, for any n×n Hermitian matrix M.

For any two solutions (U1(t),V1(t)) and (U2(t),V2(t)) of system (1.1), the Wronski matrix U1*(t)V2(t)-V1*(t)U2(t) is a constant matrix. In particular, for any solution (U(t),V(t)) of system (1.1), U*(t)V(t)-V*(t)U(t) is a constant matrix.

A solution (U(t),V(t)) of system (1.1) is said to be nontrivial if detU(t)≠0 is fulfilled for at least one t≥t0. A nontrivial solution (U(t),V(t)) of system (1.1) is said to be conjoined (prepared) if U*(t)V(t)-V*(t)U(t)≡0,t≥t0. A conjoined solution (U(t),V(t)) of system (1.1) is said to be a conjoined basis of system (1.1) if the rank of the 2n×n matrix (U(t),V(t))T is n.

In 2000, Kumari and Umamaheswaram [1], Yang and Cheng [2], and Wang [3] used the substitution W1(x)=a(x)[V(x)U-1(x)+f(x)En],a(x)=exp{-2∫x0xf(s)ds},
to study the oscillation of system (1.1). One of the main results in [1] is as follows.

Theorem A.

Let D={(x,s)∣x0≤s≤x} and D0={(x,s)∣x0≤s<x}. Let the functions H∈C(D,ℝ) and h∈C(D0,ℝ) satisfy the following three conditions:

H(x,x)=0, for x≥x0,H(x,s)>0 on D0;

H has a continuous and nonpositive partial derivative on D0 with respect to the second variable;

-(∂/∂s)H(x,s)=h(x,s)H(x,s), for all (x,s)∈D0.

If there exists a function f∈C1[x0,∞) such that
limsupx→∞1H(x,x0)λ1[∫x0x{H(x,s)T(s)+F(x,s)}ds]=∞,
where T(x)=a(x)[-C-f(A+A*)+f2B-f′En](x),a(x)=exp{-2∫x0xf(s)ds},En is the n×n identity matrix, and
F(x,s)=H(x,s)[af(A+A*)-aA*B-1A](s)-a(s)[12h(x,s)H(x,s)+f(s)H(x,s)][A*B-1+B-1A](s)-a(s)[(12h(x,s)+f(s)H(x,s))B-1/2(s)-f(s)H(x,s)B(s)]2,
then, system (1.1) is oscillatory.

In 2003, Meng and Mingarelli [4], Wang [3], and Zheng and Zhu [5] studied the oscillation of system (1.1) by using the substitution W2(x)=a(x)[V(x)U-1(x)+f(x)B-1(x)],a(x)=exp{-2∫x0xf(s)ds}.
One of the main results in [4] is as follows.

Theorem B.

Let the functions H∈C(D,ℝ) and h∈C(D0,ℝ) satisfy (i)–(iii) in Theorem A and, for all sufficiently large s∈ℝ,liminfx→∞H(x,s)≥1. Assume that there exist a function f∈C1[x0,∞) and a monotone subhomogeneous functional q of degree c on 𝒮 such that
limsupx→∞1H(x,x0)cq[∫x0x{H(x,s)R1(s)+14a(s)h2(x,s)B1-1(s)}ds]=∞,
where R1(x)=ϕ*(x)R(x)ϕ(x),B1(x)=ϕ-1(x)B(x)[ϕ*(x)]-1, ϕ(x) is a fundamental matrix of the linear equation v′=A(x)v, and
R(x)=a(x)[-C-f(A*B-1+B-1A)+f2B-1-(fB-1)′](x).
Then, system (1.1) is oscillatory.

In 2004, Sun and Meng [6] also studied the oscillation of system (1.1). One of the main results in [6] is as follows.

Theorem C.

Let H,h be as in Theorem A, and suppose that
0<infs≥t0{liminft→∞H(t,s)H(t,t0)}≤+∞.
If there exist a function f∈C1[t0,∞) and a positive linear functional g on ℝ such that
liminft→∞1H(t,t0)∫t0tg[-H(t,s)(C1+A*B1-1A+(B1-1A)′)(s)]ds>-∞,limsupt→∞1H(t,t0)∫t0th2(t,s)g[B1-1(s)]ds<∞,
and suppose also that there exists a function m∈C[t0,∞) such that
limsupt→∞1H(t,T)∫Ttg[H(t,s)(C1+A*B1-1A+(B1-1A)′)(s)-14h2(t,s)B1-1(s)]ds,≥m(T),
for all T≥t0 and
∫t0∞m+2(t)g[B1-1(t)]dt=+∞,
where m+(t)=max{m(t),0} and B1(t),C1(t) are the same as in Theorem A, then, the system (1.1) is oscillatory.

Recently, Li et al. [7] also studied the oscillation of system (1.1) by using the standard integral averaging technique and the substitution W3(t)=-a(t)[Y(t)X-1(t)+f(t)B-1(t)],t≥t0,
where a(t) is as in (1.5). One of the main results in [7] is as follows.

Theorem D.

Let H,h be as in Theorem A, and suppose that there exist a function f∈C1[t0,∞) and a positive linear functional g on ℝ, for some β≥1, such that
limsupt→∞1H(t,t0)∫t0tg[-H(t,s)(C1+A*B1-1A+(B1-1A)′)(s)-β4h2(t,s)B1-1(s)]ds=∞,
where
B1(t)=a-1(t)B(t),a(t)=exp{-2∫tf(s)ds},C1(t)=a(t){C(t)+f(t)[B-1A+A*B-1](t)+[f(t)B-1(t)]′-f2(t)B-1(t)}.
Then, system (1.1) is oscillatory.

The purpose of this paper is further to improve Theorems A, B, C, and D as well as other related results regarding the oscillation of the system (1.1), by refining the standard integral averaging technique and Riccati transformation.

Now we use the general weighted functions from the class ℋ. Let D={(t,s)∣t0<s≤t<+∞} and D0={(t,s)∣t0<s<t<+∞}. We say that a continuous function H(t,s):D→ℝ+ belongs to the class ℋ if

H(t,t)=0 for t≥t0, H(t,s)>0 on D0,

H has a continuous and nonpositive partial derivative on D0 with respect to the second variable,

-(∂/∂s)(H(t,s)k(s))=h(t,s)H(t,s)k(s), for all (t,s)∈D0, where k(t)∈C1([t0,+∞),(0,+∞)).

We now follow [8] in defining the space 𝒮 as the real linear spare of all real symmetric n×n matrices. Let g be a linear functional on ℝ, g is said to be positive if g(A)>0 whenever A∈𝒮 and A>0.

2. Main Results

In this paper, we need the following lemma.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B1">6</xref>]).

If g is a positive linear functional on ℝ, then, for all A,B∈ℝ, one has
|g[A*B]|2≤g[A*A]g[B*B].

Theorem 2.2.

Let H(t,s)∈ℋ. If there exist a function b∈C1([t0,+∞),ℝ+), a matrix function ψ∈C1([t0,+∞),𝒮), and a positive linear functional g on ℝ, for some α≥1, such that
limsupt→+∞1H(t,t0)∫t0tg{-H(t,s)k(s)T1(s)-α4B1-1(s)T2(t,s)2}ds=∞,
where B1(t)=(1/b(t))B(t),
D(t)=A(t)-b(t)B1(t)ψ(t),
F1(s)=b(s)[C+A*ψ+ψA-ψBψ+ψ′](s),
T1(s)=[F1+(B1-1D)′+D*B1-1D-(b′/b)B1-1D](s), and T2(t,s)=h(t,s)-H(t,s)k(s)(b′(s)/b(s)), then, system (1.1) is oscillatory.

Proof.

Assume to the contrary that system (1.1) is nonoscillatory. Then, there exists a nontrivial prepared solution of (U(t),V(t)) such that U(t) is nonsingular for all sufficiently large t. Without loss of generality, we assume that detU(t)≠0 for all t≥t0. This allows us to make a Riccati transformation
W(t)=-b(t)[V(t)U-1(t)+ψ(t)],
for all t≥t0. Then, W(t) is well defined, Hermitian, and solves the Riccati equation
W′(t)-b′(t)b(t)W(t)+W*(t)(A-Bψ)(t)+(A-Bψ)*(t)W(t)-1b(t)W*(t)B(t)W(t)+F1(t)=0,
on [t0,∞).

Let B1(t)=(1/b(t))B(t),D(t)=A(t)-b(t)B1(t)ψ(t). So, from (2.4), we have
W′(t)-b′(t)b(t)W(t)+W*(t)D(t)+D*(t)W(t)-W*(t)B1(t)W(t)+F1(t)=0.
Now by the substitution P(t)=W(t)-B1-1(t)D(t) in (2.5), we obtain
P′(t)-b′(t)b(t)P(t)-P*(t)B1(t)P(t)+T1(t)=0.
By rearranging the terms, we get
T1(t)=-P′(t)+b′(t)b(t)P(t)+P*(t)B1(t)P(t).
Multiplying (2.7), with t replaced by s, by H(t,s)k(s) and integrating from t0 and t, we obtain
-∫t0tH(t,s)k(s)T1(s)ds=∫t0tH(t,s)k(s)[P′(s)-b′(s)b(s)P(s)-P*(s)B1(s)P(s)]ds=-H(t,t0)k(t0)P(t0)+∫t0tP(s)[h(t,s)H(t,s)k(s)-H(t,s)k(s)b′(s)b(s)]ds-∫t0tH(t,s)k(s)P*(s)B1(s)P(s)ds.
Taking the linear functional g on both sides of the above equation, we have, for some α≥1,
∫t0tg{-H(t,s)k(s)T1(s)}ds=-H(t,t0)k(t0)g[P(t0)]+∫t0tg[P(s)][h(t,s)H(t,s)k(s)-H(t,s)k(s)b′(s)b(s)]ds-∫t0tH(t,s)k(s)g[P*(s)B1(s)P(s)]ds≤-H(t,t0)k(t0)g[P(t0)]+∫t0tg[P(s)][h(t,s)H(t,s)k(s)-H(t,s)k(s)b′(s)b(s)]ds-∫t0tH(t,s)k(s){g[B1-1(s)]}-1{g[P(s)]}2ds=-H(t,t0)k(t0)g[P(t0)]-∫t0t[H(t,s)k(s)αg[B1-1(s)]g[P(s)]-αg[B1-1(s)]2(h(t,s)-H(t,s)k(s)b′(s)b(s))]2ds+α4∫t0tg[B1-1(s)](h(t,s)-H(t,s)k(s)b′(s)b(s))2ds-α-1α∫t0tH(t,s)k(s){g[B1-1(s)]}-1{g[P(s)]}2ds≤-H(t,t0)k(t0)g[P(t0)]+α4∫t0tg[B1-1(s)]T2(t,s)2ds.
So,
∫t0tg{-H(t,s)k(s)T1(s)-α4B1-1(s)T2(t,s)2}ds≤-H(t,t0)k(t0)g[P(t0)].
Taking the upper limit in both sides of (2.10) as t→∞, we obtain
limsupt→∞1H(t,t0)∫t0tg{-H(t,s)k(s)T1(s)-α4B1-1(s)T2(t,s)2}ds≤-k(t0)g[P(t0)],
which contradicts (2.2). This completes the proof of Theorem 2.2.

Theorem 2.3.

Let the functions H,h and b,g be as in Theorem 2.2, and suppose that
0<infs≥t0{liminft→∞H(t,s)H(t,t0)}≤+∞.
If there exists a function ϕ∈C([t0,∞)), such that, for all t≥T≥t0, and for some α≥1,
limsupt→∞1H(t,T)∫Ttg{-H(t,s)k(s)T1(s)-α4B1-1(s)T2(t,s)2}ds≥ϕ(T),∫t0∞ϕ+2(t)g[B1-1(t)]k2(t)dt=+∞,
where ϕ+(t)=max{ϕ(t),0},B1(t),F1(t),D(t),T1(t), and T2(t,s) are the same as in Theorem 2.2, then, system (1.1) is oscillatory.

Proof.

Assume to the contrary that system (1.1) is nonoscillatory. Similar to the proof of Theorem 2.2, we can obtain, for all t≥T≥t0, and for some α≥1,
1H(t,T)∫Ttg{-H(t,s)k(s)T1(s)-α4B1-1(s)T2(t,s)2}ds≤-k(T)g[P(T)]-α-1α1H(t,T)∫TtH(t,s)k(s){g[B1-1(s)]}-1{g[P(s)]}2ds.
Taking the upper limit of the above inequation as t→∞,
limsupt→∞1H(t,T)∫Ttg{-H(t,s)k(s)T1(s)-α4B1-1(s)T2(t,s)2}ds≤-k(T)g[P(T)]-α-1αliminft→∞1H(t,T)∫TtH(t,s)k(s){g[B1-1(s)]}-1{g[P(s)]}2ds.
By (2.13), we obtain
-k(T)g[P(T)]≥ϕ(T)+α-1αliminft→∞1H(t,T)∫TtH(t,s)k(s){g[B1-1(s)]}-1{g[P(s)]}2ds,-k(T)g[P(T)]≥ϕ(T).
Besides, we have
liminft→∞1H(t,t0)∫t0tH(t,s)k(s){g[B1-1(s)]}-1{g[P(s)]}2ds≤-αα-1[ϕ(t0)+k(t0)g[P(t0)]]<∞.
Now, we claim that
∫t0+∞{g[B1-1(s)]}-1{g[P(s)]}2ds<∞.
Suppose to the contrary that
∫t0+∞{g[B1-1(s)]}-1{g[P(s)]}2ds=+∞.
By (2.12), there exists a positive constant ɛ satisfying
infs≥t0{liminft→∞H(t,s)H(t,t0)}>ɛ>0.
And according to the above ɛ, there exists t1≥t0 such that
∫t0t{g[B1-1(s)]}-1{g[P(s)]}2ds>1ɛ2,t≥t1.
Thus,
1H(t,t0)∫t0tH(t,s)k(s){g[B1-1(s)]}-1{g[P(s)]}2ds=1H(t,t0)∫t0tH(t,s)k(s)d(∫t0s{g[B1-1(ξ)]}-1{g[P(ξ)]}2dξ)=-1H(t,t0)∫t0t∂(H(t,s)k(s))∂s∫t0s{g[B1-1(ξ)]}-1{g[P(ξ)]}2dξds>1ɛ21H(t,t0)∫t1t-∂(H(t,s)k(s))∂sds=k(t1)ɛ2H(t,t1)H(t,t0).
From (2.22), there exists a t2≥t1 such that, for all t≥t2,
H(t,t1)H(t,t0)>ɛ.
So,
1H(t,t0)∫t0tH(t,s)k(s){g[B1-1(s)]}-1{g[P(s)]}2ds>k(t1)ɛ.
Since ɛ is arbitrary, we get
liminft→∞1H(t,t0)∫t0tH(t,s)k(s){g[B1-1(s)]}-1{g[P(s)]}2ds=∞,
which contradicts (2.19). So, (2.20) holds; then, by (2.18) and (2.20), we can obtain
∫t0∞ϕ+2(t)g[B1-1(t)]k2(t)dt≤∫t0∞g[P(t)]2g[B1-1(t)]dt<∞,
which contradicts (2.14). This completes our proof of Theorem 2.3.

Example 2.4.

Consider the linear Hamiltonian system (1.1), where B(t)=tI2,C(t)=-((1/t)cost+(3/4t3))I2,A(t)=(01/t-1/t0) are 2×2-matrices and B,C are Hermitian.

Let H(t,s)=(t-s)2,h(t,s)=2,b(t)=t,ψ(t)=-(1/2t2)I2, and g[A]=a11, where A=(aij) is a 2×2-matrix. Then, limt→∞(H(t,s)/H(t,t0))=(t-s)2/(t-t0)2=1,B1(t)=I2, D(t)=(1/2t1/t-1/t1/2t), F1(t)=-costI2,T(t)=(1/4t2-cost-2/t22/t21/4t2-cost), limsupt→∞(1/t2)∫Ttg{-(t-s)2T(s)-(α/4)B1-1(s)[2-(t-s)(1/s)]2}ds>1/T≐ϕ(T), and ∫t0∞(ϕ+2(t)/g[B1-1(t)]k2(t))dt=∫t0∞(1/t)dt=∞. According to Theorem 2.3, we get that this linear system is oscillatory.

Remark 2.5.

In Theorem 2.2, let b(t)=exp{-2∫tf(s)ds},ψ(t)=f(t)B-1(t), k(t)=1. Theorem 2.2 reduces to Theorem D. In Theorem 2.3, we obtain the same result in which we remove the two assumptions (1.9) in Theorem C. Therefore, Theorems 2.2 and 2.3 are generalizations and improvements of [7, Theorem 2.1] and [6, Theorem 3].

Remark 2.6.

The above theorems give rather wide possibilities of deriving different explicit oscillation criteria for system (1.1) with appropriate choices of the functions H(t,s),k(s), and f(s). For example, we can obtain some useful oscillation criteria if we choose H(t,s)=(x-s)m,[ln(x/s)]m,[∫sxdz/θ(z)]m, or ρ(x-s), and so forth.

3. Interval Oscillation Criteria

Now we establish interval oscillation criteria of system (1.1), that is, criteria given by the behavior of system (1.1) only on a sequence of subinterval of [t0,∞). We assume that a function H=H(t,s) satisfying (i). Further, we assume that k(t)=1 and H(t,s) has partial derivatives ∂H/∂t and ∂H/∂s on D such that ∂∂tH(t,s)=h1(t,s)H(t,s),∂∂sH(t,s)=-h2(t,s)H(t,s),
where h1,h2∈Lloc(D,ℝ).

We first prove two lemmas.

Lemma 3.1.

Suppose that (U(t),V(t)) is a nontrivial prepared solution of system (1.1) such that detU(t)≠0 on (a1,a2]⊂[t0,∞). Then, for any b(t)∈C1([t0,∞),ℝ+), matrix function ψ∈C1([t0,∞),𝒮), H satisfies (i), (3.1) and (3.2), and a positive linear functional g on ℝ, one has, for some α≥1,
1H(a2,a1)∫a1a2g{-H(t,a1)T1(s)-α4B1-1(t)(h1(t,a1)+H(t,a1)b′(t)b(t))2}dt≤g[P(a2)],
where W(t) is defined by (2.3) on (a1,a2], B1(t),D(t),F1(s), and T1(s) are the same as in Theorem 2.2.

Proof.

Since (U(t),V(t)) is a nontrivial prepared solution of system (1.1) such that U(t) is nonsingular on (a1,a2], then, W(t) by (2.3) is well defined and solves the Riccati equation (2.7) on (a1,a2].

On multiplying (2.7) by H(t,s) and integrating with respect to t from s to a2 for s∈(a1,a2], we can find
-∫sa2H(t,s)T1(t)dt=∫sa2H(t,s)P′(t)dt-∫sa2H(t,s)b′(t)b(t)P(t)dt-∫sa2H(t,s)P*(t)B1(t)P(t)dt=H(a2,s)P(a2)-∫sa2P(t)(h1(t,s)H(t,s)+H(t,s)b′(t)b(t))dt-∫sa2H(t,s)P*(t)B1(t)P(t)dt.
Taking the linear functional g on both sides of the above equation, we have, for some α≥1,
∫sa2g{-H(t,s)T1(t)}dt=H(a2,s)g[P(a2)]-∫sa2g[P(t)](h1(t,s)H(t,s)+H(t,s)b′(t)b(t))dt-∫sa2H(t,s)g[P*(t)B1(t)P(t)]dt≤H(a2,s)g[P(a2)]-∫sa2g[P(t)](h1(t,s)H(t,s)+H(t,s)b′(t)b(t))dt-∫sa2H(t,s){g[B1-1(t)]}-1{g[P(t)]}2dt=H(a2,s)g[P(a2)]-∫sa2[H(t,s)αg[B1-1(t)]g[P(t)]+αg[B1-1(t)]2(h1(t,s)+H(t,s)b′(t)b(t))]2dt+α4∫sa2g[B1-1(t)](h1(t,s)+H(t,s)b′(t)b(t))2dt-α-1α∫sa2H(t,s){g[B1-1(t)]}-1{g[P(t)]}2dt≤H(a2,s)g[P(a2)]+α4∫sa2g[B1-1(t)](h1(t,s)+H(t,s)b′(t)b(t))2dt.
That is,
1H(a2,s)∫sa2g{-H(t,s)T1(t)-α4B1-1(t)(h1(t,s)+H(t,s)b′(t)b(t))2}dt≤g[P(a2)].
Let s→a1,
1H(a2,a1)∫a1a2g{-H(t,a1)T1(t)-α4B1-1(t)(h1(t,a1)+H(t,a1)b′(t)b(t))2}dt≤g[P(a2)].

Lemma 3.2.

Suppose that (U(t),V(t)) is a nontrivial prepared solution of system (1.1) such that detU(t)≠0 on (a2,a3]⊂[t0,∞). Then, for any b(t)∈C1([t0,∞),ℝ+), matrix function ψ∈C1([t0,∞),𝒮), H satisfies (i), (3.1) and (3.2), and a positive linear functional g on ℝ, one has, for some α≥1,
1H(a3,a2)∫a2a3g{-H(a3,s)T1(s)-α4B1-1(s)(h1(a3,s)+H(a3,s)b′(s)b(s))2}ds≤-g[P(a2)],
where W(t) is defined by (2.3) on (a2,a3], B1(t),D(t),F1(s), and T1(s) are the same as in Theorem 2.2.

Proof.

Since (U(t),V(t)) is a nontrivial prepared solution of system (1.1) such that U(t) is nonsingular on (a2,a3], then, W(t) by (2.3) is well defined and solves the Riccati equation (2.7) on (a2,a3].

On multiplying (2.7) by H(t,s), integrating with respect to s from a2 to t for t∈(a2,a3], and following the proof of Lemma 3.1, we can find
1H(t,a2)∫a2tg{-H(t,s)T1(s)-α4B1-1(s)(h1(t,s)+H(t,s)b′(s)b(s))2}ds≤-g[P(a2)]<+∞.
Let t→a3,
1H(a3,a2)∫a2a3g{-H(a3,s)T1(s)-α4B1-1(s)(h1(a3,s)+H(a3,s)b′(s)b(s))2}ds≤-g[P(a2)].

Theorem 3.3.

Suppose that there exist some a2∈(a1,a3)⊂[t0,∞),b(t)∈C1([t0,∞),ℝ+), matrix function ψ∈C1([t0,∞),𝒮), H satisfies (i), (3.1) and (3.2), and a positive linear functional g on ℝ such that, for some α≥1,
g{1H(a2,a1)∫a1a2-H(t,a1)T1(t)-α4B1-1(t)(h1(t,a1)+H(t,a1)b′(t)b(t))2dt+1H(a3,a2)∫a2a3-H(a3,s)T1(s)-α4B1-1(s)(h1(a3,s)+H(a3,s)b′(s)b(s))2ds}>0,
where B1(t),D(t),F1(s), and T1(s) are defined as in Theorem 2.2. Then, for any nontrivial prepared solution (U(t),V(t)) of system (1.1), detU(t) has at least one zero in (a1,a3).

Theorem 3.4.

If, for each T≥t0, there exist b(t)∈C1([t0,∞),ℝ+), matrix function ψ∈C1([t0,∞),𝒮), H satisfies (i), (3.1), (3.2), a positive linear functional g on ℝ and a1,a2,a3∈ℝ, such that T≤a1<a2<a3 and condition (3.1) holds, where B1(t),D(t),F1(s), and T1(t) are defined as in Theorem 2.2, then, system (1.1) is oscillatory.

In conclusion, we note that the results given here can extend, improve and complement Theorems A–D, and deal with some cases not covered by known criteria by choosing the functions H,b,ϕ, and g. From our results, we can derive a number of easily verifiable oscillation criteria.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (11171178), the National Ministry of Education under Grant (20103705110003), and the Natural Sciences Foundation of Shandong Province under Grant (ZR2009AM011).

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