JAMJournal of Applied Mathematics1687-00421110-757XHindawi Publishing Corporation94204210.1155/2012/942042942042Research ArticleSome Oscillation Results for Linear Hamiltonian SystemsWangNanMengFanweiSezerMehmet School of Mathematical SciencesQufu Normal UniversityQufu 273165Chinaqfnu.edu.cn20121922012201208092011071220112012Copyright © 2012 Nan Wang and Fanwei Meng.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.

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)UA(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,tt0, 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 tt0. 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,tt0. 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 , Yang and Cheng , and Wang  used the substitution W1(x)=a(x)[V(x)U-1(x)+f(x)En],a(x)=exp{-2x0xf(s)ds}, to study the oscillation of system (1.1). One of the main results in  is as follows.

Theorem A.

Let D={(x,s)x0sx} and D0={(x,s)x0s<x}. Let the functions HC(D,) and hC(D0,) satisfy the following three conditions:

H(x,x)=0, for xx0,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 fC1[x0,) such that limsupx1H(x,x0)λ1[x0x{H(x,s)T(s)+F(x,s)}ds]=, where T(x)=a(x)[-C-f(A+A*)+f2B-fEn](x),a(x)=exp{-2x0xf(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 , Wang , and Zheng and Zhu  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{-2x0xf(s)ds}. One of the main results in  is as follows.

Theorem B.

Let the functions HC(D,) and hC(D0,) satisfy (i)–(iii) in Theorem A and, for all sufficiently large s,liminfxH(x,s)1. Assume that there exist a function fC1[x0,) and a monotone subhomogeneous functional q of degree c on 𝒮 such that limsupx1H(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  also studied the oscillation of system (1.1). One of the main results in  is as follows.

Theorem C.

Let H,h be as in Theorem A, and suppose that 0<infst0{liminftH(t,s)H(t,t0)}+. If there exist a function fC1[t0,) and a positive linear functional g on such that liminft1H(t,t0)t0tg[-H(t,s)(C1+A*B1-1A+(B1-1A))(s)]ds>-,limsupt1H(t,t0)t0th2(t,s)g[B1-1(s)]ds<, and suppose also that there exists a function mC[t0,) such that limsupt1H(t,T)Ttg[H(t,s)(C1+A*B1-1A+(B1-1A))(s)-14h2(t,s)B1-1(s)]ds,m(T), for all Tt0 and t0m+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.  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)],tt0, where a(t) is as in (1.5). One of the main results in  is as follows.

Theorem D.

Let H,h be as in Theorem A, and suppose that there exist a function fC1[t0,) and a positive linear functional g on , for some β1, such that limsupt1H(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{-2tf(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 . Let D={(t,s)t0<st<+} and D0={(t,s)t0<s<t<+}. We say that a continuous function H(t,s):D+ belongs to the class if

H(t,t)=0 for tt0, 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  in defining the space 𝒮 as the real linear spare of all real symmetric n×n matrices. Let g be a linear functional on , g is said to be positive if g(A)>0 whenever A𝒮 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 , then, for all A,B, one has |g[A*B]|2g[A*A]g[B*B].

Theorem 2.2.

Let H(t,s). If there exist a function bC1([t0,+),+), a matrix function ψC1([t0,+),𝒮), and a positive linear functional g on , for some α1, 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 tt0. This allows us to make a Riccati transformation W(t)=-b(t)[V(t)U-1(t)+ψ(t)], for all tt0. 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 α1, 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+α4t0tg[B1-1(s)](h(t,s)-H(t,s)k(s)b(s)b(s))2ds-α-1αt0tH(t,s)k(s){g[B1-1(s)]}-1{g[P(s)]}2ds-H(t,t0)k(t0)g[P(t0)]+α4t0tg[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 limsupt1H(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<infst0{liminftH(t,s)H(t,t0)}+. If there exists a function ϕC([t0,)), such that, for all tTt0, and for some α1, limsupt1H(t,T)Ttg{-H(t,s)k(s)T1(s)-α4B1-1(s)T2(t,s)2}dsϕ(T),t0ϕ+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 tTt0, and for some α1, 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, limsupt1H(t,T)Ttg{-H(t,s)k(s)T1(s)-α4B1-1(s)T2(t,s)2}ds-k(T)g[P(T)]-α-1αliminft1H(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-1αliminft1H(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 liminft1H(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 infst0{liminftH(t,s)H(t,t0)}>ɛ>0. And according to the above ɛ, there exists t1t0 such that t0t{g[B1-1(s)]}-1{g[P(s)]}2ds>1ɛ2,tt1. 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))st0s{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 t2t1 such that, for all tt2, 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 liminft1H(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 t0ϕ+2(t)g[B1-1(t)]k2(t)dtt0g[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/Tϕ(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{-2tf(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,h2Lloc(D,).

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,),+), matrix function ψC1([t0,),𝒮), H satisfies (i), (3.1) and (3.2), and a positive linear functional g on , one has, for some α1, 1H(a2,a1)a1a2g{-H(t,a1)T1(s)-α4B1-1(t)(h1(t,a1)+H(t,a1)b(t)b(t))2}dtg[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 α1, 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)]dtH(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+α4sa2g[B1-1(t)](h1(t,s)+H(t,s)b(t)b(t))2dt-α-1αsa2H(t,s){g[B1-1(t)]}-1{g[P(t)]}2dtH(a2,s)g[P(a2)]+α4sa2g[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}dtg[P(a2)]. Let sa1, 1H(a2,a1)a1a2g{-H(t,a1)T1(t)-α4B1-1(t)(h1(t,a1)+H(t,a1)b(t)b(t))2}dtg[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,),+), matrix function ψC1([t0,),𝒮), H satisfies (i), (3.1) and (3.2), and a positive linear functional g on , one has, for some α1, 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 ta3, 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,),+), matrix function ψC1([t0,),𝒮), H satisfies (i), (3.1) and (3.2), and a positive linear functional g on such that, for some α1, 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 Tt0, there exist b(t)C1([t0,),+), matrix function ψC1([t0,),𝒮), H satisfies (i), (3.1), (3.2), a positive linear functional g on and a1,a2,a3, such that Ta1<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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