| The oscillation theory of matrix differential equations which is one of important branches of differential equations arises in a variety of differential applied mathematics and physics. In the field of modern applied mathematics it has made considerable headway in recent years.In the present article a generalized Riccati transformation, integral average technique and positive linear functionals are used to investigate the oscillation criteria of linear matrix Hamiltonian system. Moreover, the oscillation criteria of linear second order matrix differential system with damping is studied. The obtained results generalize and improve some known oscillation criteria.The thesis is divided into three sections according to contents.In chapter 1 Preface.we introduce the importance of the main contents of this paper.In chapter 2 we are concerned with oscillation of linear matrix Hamiltonian systemwhere X(t), Y(t), A(t), B(t), C(t) are n×n real continuous matrix functions such that C(t) is symmetric and B(t) is symmetric and positive definite i.e.B(t) > 0 for t > t_o. By M~* we mean the transpose of the matrix M.In this chapter some sufficient conditions for oscillation of system (2.1.1) are established. We now recall for the sake of convenience of reference the following definitions.Definition 1 A solution (X(t), Y(t)) of (2.1.1) is said to be "nontrivial" if det X(t)≠0 for at least one t ∈ [to, ∞).4Definition 2 a nontrivial solution (X(t),Y(t)) of (2.1.1) is said to be prepared if for every t G [tQ, oo),X*(t)Y(t)-Y*(t)X{t)=O.Definition 3 System (2.1.1) is said to be oscillatory if every nontrivial prepared solution (X(t), Y(t))of (2.1.1) has the property that det X(t) vanishes on [T, oo) for sufficiently large T > t0.We state the main results as follows:Theorem2.1.1 Assume there exist a smooth and real-valued function /(￡) on [*o,oo) and a(t)tr{B~l(t)) < m (m > 0 is a constant),where a(t) = exp(-2j￡/(s)ds), andlimmf)- f tr W(t) + f E(s)dsdt > —oo.If one of the conditions(1) limsup- / Ai !>(*)+ / E(s)ds\dt = +oo,T-?oo J- Jto L JtQ J(2) limsup-/ \xl[D(t)+ E(s)ds)\ dt = +oo,r^oo J- Jt0 I V Jt0 / J(3) limapproxsupr_).ooAi ￡>(￡)+ / ^(sjds =+oo,L Jt0 Jr , x rl , i(4) limapproxinfr .^Ai \Dn)+ j E(s)ds\ = —oo,L Jto Jholds, where D(t) = -a(t)B-l(t)A(t) - f^ a(s)A*(s)B-l{s)A(s)dsand E(t) — —a(t) [C(t)'+ f(t){B~lA + A*B'l)(t) + (fB~l)'(t) — (j then (2.1.1) is oscillatory.Theorem2.1.2 Assume liminfr->oo f ft ^r \D(t) + Jt E(s)ds\ dt = — oc Then (2.1.1) is oscillatory if..A,,, ^D(t) + ^ ^(s)dsj = 0 > -oo,where a(t), D(t), E(t) are the same as in Theorem2.1.1 ' .Remark2.1.1 Theorem2.1.1 and Theorem2.1.2 are generalizations and improvements of [3,Theorem2.1 and Theorem2.2], respectively.' In fact , when A{t) = 0,B(t) = I,C{t) = -Q{t), we let. f(t) = 0,a(t) = 1 then .Theo-rem2.1.1 and Theorem2.1.2 are reduced toTheorem2.1 and Theorem2.2 of [3] respectively.Remark2.1.2 Compared with Theorem B ,Theorem 2.1.1 has the following advantages. First, Theorem 2.1.1 removes the fundamental matrix of the linear equation V = A(t)V.Therefore, Theorem2.1.1 can be conveniently applied to (2.1.1).Second, in some cases the assumption a~l(t)tr(B~l(t)) < m is weaker than the assumption a~l{t){^TlB(j)*~l){t) > /(an nxn identity matrix) for t > to For example, for the case when a(t) = 1 and 1, then we have that tr{B-l(t)) - 2 + \ < 3.However, B(t) > I does not hold for t > 1.We introduce the definitions of a function class T and the positive linear functional g.Definition 4[23] we say that a function H = H(t, s) belongs to a function class JF, denoted by H E T. If H eC (D, R+), whereD = {(t,s):t>s>t0},satisfiesH{t, t) = 0, H(t, s)>0 for t > s,and has continuous derivatives dH/dt and dH/ds on D such that—- = h1(t,s)y/H(t,s) and -^~ = -h2{t,s)y/H(t,s), at oswhere hu h2 6 C(D,R).Definition 5[10] Let R be the linear space ofnxn matrices with real entries, p C 5R be the subspace ofnxn symmetric matrices, and g be a linearfunctional on K. g is said to be "positive" if g(A) > 0* whenever A G p and A>0.We obtained the following results of system (2.1.1):Lemma2.2.3 Assume (X(t),Y(t)) is a prepared solution of (2.1.1) and X(t) is nonsingular for t G [c, b]. Then, for any / G Cl[c, 6] and H G T,f ^ ^^ > -H(b,c)g[P(c)]. (2.2.11)jf ^Lemma2.2.4 Assume (X(t),Y(t)) is a prepared solution of (2.1.1) and X(i) is nonsingular for t G [a, c]. Then, for any / G Cl[a, c] and H e T,, a)g [Di{s)\ + ^(s, a)g [B{l(s)] \ ds >H(c, a)g [P(c)}. (2.2 14)Lemma2.2.5 Assume (X(t), Y(t)) is a prepared solution of (2.1.1). Suppose also that for some c G (a,b), there exist / e Cl[a,b] and H € J- such thatH-l(c, a) ￡ jff(a, aMAW] + \hlis, a+ H~\b,c) I^H(b,s)g[Dl(s)} + ^hl(b,s)g[B^(s)}^ ds < 0. (2.2.16)Then, detX(t) has at least one zero on [a, b].Theorem 2.2.1 System (2.1.1) is oscillatory provided that for each T > t0, there exist a, b, c ￡ R such that T < a < c t0,Km sup jf (#(s,r) !(*)] + \h\{s,r)g [B^(s)]\ ds < 0, (2.2.17)lim sup f .(H{t,s)g[D1{8)] + ]h22{t,s)g {B^l(s)]\ ds < 0. (2.2.18)*->00 Jr V 4 /andThen system (2.1.1) is oscillatory.Theorem2.2.2 Assume that for each T > to, there exist a, c € R such that T < a < c and / G C1^, 2c - a], He T, such thatf{H(s - a) [g (D^s)) + g (Dl(2c - s))} +(2.2.21) ± - a) [g (B^(s)) + g (B^l(2c - s))] | ds < 0.+h |Then system (2.1.1) is oscillatory.Remark2.2.2 With an appropriate choice of the positive linear functional g such as g[M] = ma for M = (rriij)(i,j = 1, 2, ..., n), g-[M] = tr[M] and ^[M] = c*Mc where c is an arbitrary but fixed vector in Rn, we may derive many possibilities for oscillation criteria of (2.1.1) . Because of the limited space, we omit them here.In Chapter 3, we consider the oscillation criteria of linear second order matrix differential system with dampingIn the first section ,we study oscillatory properties for the linear second order matrix differential system with dampingX" + P(t)X' + Q(t)X = Q, te[tQ,oo), (3.1.1)where P{t), Q(t) and X(t) are n x n continuous matrix-valued functions, P(t) and Q(t) are symmetric. In this section, we present some new oscillation criteria of system (3.1.1). We recall the definition of prepared solution of system (3.1.1) as follows:Definition 6. a solution X(t) of (3.1.1) is said to be prepared if it is a nontrivial solution of system (3.1.1) andx*(t)x'(t) - (x*(t))'x(t) = o, t e [to.oo).We know that for any nxn symmetric matrix M, its eigenvalues are real numbers. We always denote by X\[M] > A2[M] > ... > An[M], andas usual, tr[M] = ￡"=1Ai[M].we state the main results as follows:Lemma3.1.2 Assume that (3.1.1) is non-oscillatory on [t0, oo), and a(t) < m* (m* > 0 is a constant), for t € [to,oo). ThenrT0 < lim / a-1(s)W2(s)ds < +oo, t > t0 (3.1.8)T->°°Jtif and only if1 fT liminf-/ {tr [Z{t))) dt > -oo. (3.1.9)00 JtoTheorem3.1.1 Assume there exist a smooth and real-valued function f(t) on [t0, oo) and a{t) < m* (m* > 0 is a constant), where a(t) = exp(—2 Jt f(s)dsV andi rTliminf-/ tr [Z(t)]dt > -oo.If one of the conditions1 fT lim sup-/ Xl[Z(t)]dt = +oo;(3.1.19)T-kx, T Jtoi rTlim sup -/ {Xx[Z{t)]f dt = +oo, (3.1.20)—>oo J Jtolim approxsupr^^Ai [Z{T)\ = 4-oo, (3.1.21)lim approxinfj^^Ai [^(T)] = — oo, (3.1.22)holds, then (3.1.1) is oscillatory.TheoremS.1.2 Assume a(t) < m*, for t > to,1 fT liminf-/ tr [Z[t)\ dsdt - -oo, (3.1.33)Then (3.1.1) is oscillatory iflim approxsup(_>ooAn \Z(t)) = 0 > —oo. where a(t), Z(t) is the same as in Theorem 3.1.1.Remark 3.1.1 Theorem 3.1.1 and Theorem 3.1.2 are generalizations and improvements of [3,Theorem2.1 and Theorem2.2], respectively. In fact, when P{t) = 0, we let f(t) = 0, a(t) = 1 then Theorem 3.1.1 and Theorem 3.1.2 are reduced to Theorem2.1 and Theorem2.2 of [3] respectively:In the second section, we consider the oscillation criteria of linear second order matrix differential system with damping(P{t)X')' + R{t)X' + Q(t)X = 0, te[tQ,oo), (3.2.1)where P(t), R(t), Q(t) and X(t) are n x n real continuous matrix-valued functions such that P(t), R(t) and Q(t) are symmetric and P(t) is positive definite, i.e., P{t) > 0, for t e[t0,oo).Sufficient conditions have been obtained for the oscillation of the system (3.2.1) by using positive functional on a suitable matrix space. The concept of a prepared solution of system (3.2.1) are as follows:Definition 7. A solution X(t) of (3.2.1) is said to be prepared if it is a nontrivial solution of system (3.2.1) andX*{t)P{t)X'(t) - {X*(t))'P(t)X(t) = 0, t E [t0, oo).we state the main results as follows:Lemma3.2.1. Assume that (3.2.1.) is non-oscillatory on [io,oo). If there exists a positive linear functional g on SR such that a(t)g [P(t)] < m* ( m* > 0 is a constant ), then0< lim / a-1(s)glW(s)p-1(s)W(s)]ds<+oo, t>t0, (3.2.8)T-+oo Jtif and only if1 fTliminf- / g[J(t)}dt> -oo, (3.2.9)T^°° T Jt0VllTheorem3.2.1. Assume there exist a smooth and real-valued function f(t) on [to, oo) and a positive linear functional g on 3ft such that a(t)g [P(t)\ < m* ( m* > 0 is a constant ), where a(t) — exp f — 2 JtQ /(s)dsj, and1 CT liminf- / # [J(t)] dt > -oo. (3.2.18)T^oo T Jt0If one of the conditionsi rT(Ai) limsup- / g[J(t)]dt =1 Jto1 fT{A2) lim sup - / g2 [J(t)] dt = +oo,lim approxsup^ooP [J(t)] = +oo, lim approxinf^oo^ [J(t)] = -oo,holds, then system (3.2.1) is oscillatory.Remark 3.2.1 With an appropriate choice of the positive linear functional g such as g[M] = ma for M = (mij)(i,j = 1, 2, ..., n), g[M] = tr[M] and g[M] = c*Mc where c is an arbitrary but fixed vector in Rn, we may derive many possibilities for oscillation criteria of (3.2.1). Because of the limited space, we omit them here.
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