Studies On Oscillation Of Second-order Differential Systems

With the rapid development of science and technology , all sorts of nonlinear problems have derived from mathematics, physics, chemistry, biology and so on. On one hand, many nonlinear problems come from all kinds of applied subject which attracts many scholars to study .On the other hand, great changes of nonlinear differential equations have taken place in several decades It's powerful and fruitful theoretical tools and advanced methods have become ripe step by step.The theory of matrix differential systems is one of the important branches of differential equations. In the field of modern applied mathematics,it has made considerable headway in recent years, because all the structure of its emergence has deep physical background and realistic mathematical model.The present paper employs a generalized Riccati transformation and integral average technique to investigate the oscillation criteria and interval oscillation criteria of some kinds of second-order differential equations.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 main contents of this paper.In chapter 2, we mainly study the interval oscillation of second order nonlinear differential equations(P(t)Y'(t))' + F{t, Y(t), Y'(t)) = 0, t ∈ [t₀, ∞), (2.1.1)(P(t)Y(t)')' + D(t)Y(t)' + F(t, Y(t), Y'(t)) = 0 , t ∈ [t₀, ∞) , (2.1.2)where P(t), D(t), Y(t), and F(t, Y(t), Y'(t)) are n × n real continuous matrix-valued functions with P{t), D(t) symmetric , and P(t) positive definite , i.e. , P(t) > 0 , for t∈ [t₀,∞) .We obtained the following results for the study of interval oscillation of Eq.(l):Lemma 2.2.1 Assume that (A) holds . Y(t) is a prepared solution of (2.1.1) and Y(t) is nonsingular for t G [c, b]. For any / G Cl[c, b], definea(t) = exp{-2 f f{s)ds},V(t) = a(t){P(t)Y'(t)Y^l(t) + f(t)P(t)}, (2.2.1)Then, for any H E Q,uf (if (6, s)R(s) - la(s)h%(b, s)P(s)j ds < H(b, c)V(c), (2.2.2)whereR(t) = a(t){G(t) + f\t)P{t) - (f{t)P(t))'}. (2.2.3)Lemma 2.2.2 Assume that (A) holds and Y(t) is a prepared solution of (2.1.1) and Y(t) is nonsingular for t ï¿¡ [a,c\. Then , for any / e Cl[a,c\ andt (h(s, a)R{s) - ^a{s)hl(s, a)P(s) J ds < -H{c, a)V(c),where a(t), V(t), R(t) are same as Lemma 2.2.1Theorem 2.2.1 Assume that (^4) holds , q : 5 t0, there exist / G C1^, oo),H G fii, and a,b,c e R such that T < a < c < 6 and (2.2.5) holds.Remark If P(t) = p(t),F(?,r(t),r(t)) = /(t,y(i),2/'(i))9W = * are scalar functions. Let f(i) = 0,then a(t) 1, R(t) = G(t) — g(t) = d(t) and Theorem 2.2.1 generalize and include the main results in [20].Theorem 2.2.2 Assume that (A) holds , q : Snxn —> R is a nondecreasing function . Then system (2.1.1) is oscillatory provided that for each T > t0, there exist / G Cl[a, 2c — a], H G fl'l, and a,b,c G i? such that T < a < c and/* (iJ(s-a)[i2(s) + fl(2c-s)])ds|r\a ,c J n (2-2.11)- / [a(s)P(s) + a(2c - s)P{2c - s)] h2(s - a)ds .. J a JLemma 2.2.4 Assume that (A) holds , and 3/ G C1^, oo), H G Cl[, a >lim Xi \ I (is- r)a[R(s) + R(2t - s)}-i?2(s - r)a-2[a(s)P(s) + a(2c - s)P(2c - s)])ds\ > 0,where a(t) and R(t) are same as Lemma 2.2.1. Then, system (2.1.1) is oscillatory.Lemma 2.2.3 Assume that (A) holds ,Y(t) is a prepared solution of (2.1.1), and detF(i) ^ 0,t ï¿¡ [a,b], q : Snxn -? R is a nondecreasing function . If V/ G C1[a,6],Vu G C^a.ftl.^a) = u(6) = 0, T > io,we have2{s)R{s)ds\ to,3f G C^ooJ.w G C1[a,6],it(a) = w(6) = 0,where a,b E R,T < a < b, such that (2.2.12) holds, where a(ï¿¡) and R(t) are same as Lemma 2.2.1. Then, system (2.1.1) is oscillatory.Theorem 2.2.4 Assume that (A) holds and lim ini Hit, s) > 1. If therpi->ooexists a function / G Cl[to,oo),H G fii, and a monotone subhomogeneousfunctional q of degree m on Snxn such thatlimsup * q ^ (tf (*, s)R(s) - ^a(s)h22(t, s)P(?)) dsj = oo,(2.2.14)where a(t) and R(t) are same as Lemma 2.2.1. Then, system (2.1.1) is oscillatory.Remark In Theorem 2.2.4, choosing F(t, Y{t), Y'(t)) = D(t)Y(t),q(t) = t,f(t) = 0,then a(t) = l,choosing R(t) = D(t),Theorem 2.2.4 is same as Theorem 1 in [5].Lemma 2.3.1 Assume D(t)E(t) = E{t)D(t), where P"1^) = E2(t), then D{t)E-\t) = E-l(t)D{t), andJ1(t,s)E-1(t)=E-1(t)J1(t,s),Lemma 2.3.2 Assume that (A) holds and D(t)E(t) = E(t)D(t), where P-1(i) = E2(t). Y(t) is a prepared solution of (2.1.2) and Y(t) is nonsingular for t e [c, b]. For any / € Cl[c, b], definea(t) =exp{-2 I f{s)ds},V(t) = aitHPWWY-1? + f(t)P(t)}, (2.3.1)Then, for any H ï¿¡ Q2)I J^L ) 6, c)V(c), (2.3.2)where= a(t){Q(t) + f2(t)P(t) - f(t)D(t) - (f(t)P(t))'}. (2.3.3)Lemma 2.3.3 Assume that (A) holds and D(t)E(t) = E(t)D(t), where P-1(i) â– = E2(t).Y(t) is a prepared solution of (1.1) and y(t)-is nonsingular foi t G [a, c]. Then , for any / G Cl[a, c] and H G Q2,(h{s, a)R(s) - i-^Lp(s)F12(S, a)) ds < -H(c, a)V(c), (2.3.5)where a(t), V(t), R(t) are same as Lemma 2.3.2.Theorem 2.3.1 Assume that (A) holds and D{t)E{t) = E{t)D{t), where P^l{t) = E2(t). Then system (2.1.2) is oscillatory provided that for each T > t0, there exist / € C1^, oo), H € 0,2, and a,b,c € R such that T < a < c < b and (2.3.5) holds, where a(t),V(i) and R(t) are same as Lemma 2.3.2.Theorem 2.3.2 Assume that (A) holds and D(t)E(t) = E(t)D(t), where P^l(t) = E2(t). Furthermore , for all sufficiently large s e R,limMH{t.s) >t—>oo1. If there exists a function / G C1[to>oo) and a monotone subhomogeneous functional q of degree m on 5nx" such that(2.3.12)where a(t) and R(t) are same as Lemma 2.3.1. Then, system (2.1.2) is oscillatory.Theorem 2.3.3 Assume that (A) holds and D(t)E(t) = E(t)D(t), where P"1^) = E2(t). Let q(A) = Xi[A] be subhomogeneous of degree 1. If (2.3.12) is replaced by the two conditions_<sub>1 , \ fta(s)P(s)Jl(t,s)ds] ^ hmsup rT/± ±,Xi\ W'\------- oo n{tt) lJ JThen (2.1.2) is oscillatory.In Chapter 3, we study the oscillation of differential equations with damping.we mainly consider differential equations of the form(P(t)Y'(t))' + D(t)Y'(t) + Q(t)Y(t) = 0, t G [*?, oo), (3.1.1)where P(t), D(t), Y(t), are n x n real continuous matrix-valued functions with P(t), D(t),Q(t) symmetric , and P(t) positive definite . In this chapter we establish some oscillation criteria of the equation of the form (3.1.1) in the first section, in the second section we generalize and improve many known conclusions and get some new oscillation criteria.We state the main results as follows:Lemma 3.2.1 Assume that D(t)E(t) = E(t)D(t), where P^l{t) = E2(t). Y(t) is a prepared solution of (3.1.1) and Y(t) is nonsingular for t G [c,b]. For any / G Cl[c, b], define a(t) — exp{-2 /t* f(s)ds},V(t) = a{t){P{t)Y'{t)Y-\t) + f(t)P(t)}, (3.2.1)Then, for any H G fi2,( H(b, s)R{s) - \^f) P(s)F%{b, s) j ds < H(b, c)V(c), (3.2.2) whereR(t) = a(t){Q(t) + f{t)P{t) - f(t)D(t) - (f{t)P{t))'}. (3.2.3)Lemma 3.2.2 Assume that D(t)E(t) - E{t)D(t), where P'1^) = E2(t).Y(t) is a prepared solution of (1.1) and Y(t) is nonsingular for t G [a, c]. Then , for any / G Cx[a, c] and H G f^,ï¿¡ (H(8,a)R{8) - i^7^)P(s)Jx(s'a)) ds < -H(c,a)V(c), where a(t), V(t), R(t) are same as Lemma 3.2.2.Theorem 3.2.1 Assume that D{t)E(t) = E(t)D(t), where P^l{t) = E2(t). Then system (1.1) is oscillatory provided that for each T > to, there exist / G Cx[t0, oc), H G O2, and a,b,c G R such that T < a < c < b and (3.2.5) holds.Remark If P(t) = p(t),Q(t) = q(t), Y(t) = y(t),D(t) = 0 are scalar functions,then Theorem 3.2.1 generalize and include the main results in [20].Let q(A) = Xi(A),A G Snxn, then we haveTheorem 3.2.2 Assume that D(t)E(t) = E(t)D(t), where P^l{t) = E2(t). q : Snxn —> R is a nondecreasing function . Then system (2.1.1) is oscillatory provided that for each T > to, there exists / G Cl[a, 2c—a), H G Q'2, and a,c G R such that T < a < c and>q/"1 /*c 1 1^a \ W(-------T (a{s)P{s)J2(s,a)+a(2c-s)P{2c-s)J%(2c-a,2c-s))ds\L^ Ja, H{S — a) J(3.2.10) where a(t) and i?(i) are same as Lemma 3.2.2.Lemma 2.1 Assume D(t)E(t) = E(t)D(t), where P'^s) = E2(s), then D{s)E-l{s) = E-1(s)D(s), andTheorem 3.3.1 Assume that D(s)E(s) = E(s)D(s),where P-1(s) = E2(s),J(t, s) is defined as before, and liminf H(t, s) > 1. If there exists a func-t—>ootion / G C1[ï¿¡o,oo),.ff'(ï¿¡, s) G ^3, and a monotone subhomogeneous functional q of degree m on 5nxn such that1 f /"' / 1 a(s) \ 1(3.3.1)wherea(t) = exp{-2 /â–  f(s)ds},JtoR(t) = a(t){Q(t) + f(t)P(t) - f(t)D(t) - (f(t)P(t))'}. (3.3.2)Then, system (3.1.1) is oscillatory.Theorem 3.3.2 Assume that D(s)E(s) = E(s)D(s), where P^l{s) = E2(s),J(t,s),a(t),R(t)is defined as Theory 3.3.1 , also0 < inf [liminf ?, , < oo (3.3.5)s>t0 [ S->oo H(t,tO)\ ^andlimsup——■< oo (3.3.6)hold,where'toIf there exists a function A € C([to, oo), i?),such thata(s) where A+(t) = max(0,A(t)), furthermore VT > to,Then system (3.1.1) is oscillatory.(3.3.7)(3.3.8)...