Studies On Oscillation Of Second-order Differential Equations

With the rapid development of science and technology, all sorts of nonlinear problems have derived from mathematics, physics, chemistry, biology and so on. On the one hand, many nonlinear problems come forth all kinds of applied subject which attracts many scholars to study them.On the other hand, great changes of nonlinear differential equations have taken place in several decade years. It's powerful and fruitful theoretical tools and advanced methods have become ripe step by step.The theory of normal differential equation is one of 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 kind of second-order differential equations.The obtained results generalized and improved 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 study the oscillation of differential equations with damping. In this chapter, we give the following definitions:Definition 1. A function x : [t₀,t₁) —> (—∞,+∞), t₁ ≥ t₀ is called a solution of Eq.(l) if x(t) satisfies Eq.(l) for all t ∈ [t₀, t₁).Definition 2. A solution x(t) of Eq.(l) is called oscillatory if it has arbitrarily large zeros, otherwise it is called non-oscillatory.Firstly, we study differential equation with the form(a(t)x'(t))' + p(t)x'(t) + q(t)f(x(t))g(x'(t)) = O,t>to (1)where(#1) a(t) e Cl([to, oo);(0, oo));(#2) The function q : [io, 00) —>? [0, 00) is continuous and not identicallyzero on any [T, 00) for some T > to;(#3) peC([*0, 00). fl);(#4) The function g : i? -> i? is continuous and g(x) > (i > 0 for x ^ 0;(Jf5) The function / € C^R, i?) is continuous and f'(x) >K>0foxx^0;(H$) The function / e Cx(i?, /?) is continuous and f{x)/x > KQ > 0 forIn order to study the oscillation of Eq.(l), we introduce one particular function <ï¿¡(ï¿¡, s, r) denned as(t, s, r) = (t — s)a(s - r)0,a,l3 > - are constants , r > t0. (2)idWe state the main results as follows:Theorem 2.1.1 Let assumptions (Hi)-(H5) hold, $(ï¿¡, 5, r) be defined by (9). If there exists a positive nondecreasing function v € C1^, 00)) such that for every r > i0\ v(s)$2(t, s, r)q(s)fi - 1/ [$',(t, 5, r)lim sup(3)2a(s) V! ' ' ' 2v(s) then Eq.(l) is oscillatory.Remark: When p = 0,f(x) = x,g(x) = 1, equation(l)is the equation studied by H.J.Li [10];when a = l,p = 0, the equation (1) is the equationstudied by Yuri V.Rogovchenko [11];when a = l,f(x) = x,g(x) = 1, the equation(l) is the equation studied by Wong[5]. They use a function H — H(t,s), but in the proof of this theorem, we use a new function $(t,s,r) = (t — s)a(s — r)P, a,p > \ are constant and r > t0 , so we get some conclusions that are different from others.Theorem 2.1.1' Let assumptions {H2) - (H5) hold, a{t) e Cl{[t0, oo);(0,1]), $(i, s, r) be defined by (2). If for every r > t0Urn sup f {$2(t,s,r)g(s)/x- h&.(t,s,r) - lp{s)<$>{t,s,r)}2}ds > 0,t->oo Jr A Zthen Eq.(l) is oscillatory.Theorem 2.1.2 Let assumptions (H2) - {H$) hold, a(t) G Cx([tQ, oo);(0, 1]), Eq.(l) is oscillatory provided that for some a > \ and for enery r > tosL {A* - s)2(s - r)2t->oo I (Jrr + at- (a + l)s,l \ a>] *j >Theorem 2.1.3 Let assumptions (H2) — (H5) hold, a(t) E (0, 1]), Eq.(l) is oscillatory provided that for some a > i and for every r > t0limsup -^ { [\s - r)\t - sfa \Kq{a)v - \ip\s)\ a "U) > (2a-l)(2Q + l)-t-HX {rt + ar- (a+l)s/| , ] daTheorem 2.1.4 Let assumptions (Hi) — (7?4) and (Hq) hold, $(t,s,r) be defined by (2). If there exists a positive nondecreasing function v G C^Qto, oo)) such that for every r > to, we havelimsup/ {v(s)$2(t,s,r)q{s)KoLt-a{s)v(s)[$'s(t,s,r)Then Eq.(l) is oscillatory.Theorem 2.1.4' Let assumptions (H2) - (HA) and (H6) hold, a(t) € io, oo);(0,1]), $(*, s, r) be defined by (2).If for every r > t0, we havelimsup./" i$2(t,s,r)g(s)K0^- $'s(t,s,r)--p(s)$(t,s,r)ds > 0.Then Eq.(l) is oscillatory.Theorem 2.1.5 Let assumptions (H2) — (#4) and (He) hold, a(i) € C^^o, 00);(0, 1]), Eq.(l) is oscillatory provided that for some a > j, and for every r > t0lim supt-KX t1ft\2a^+TK0q(s)fi--(p2(s)(7) r + at- (a + l)s.l ,1a(t,)(,r) >J "S| >Theorem 2.1.6 Let assumptions (H2) - {H4) and (H6) hold, a(t) e io, 00);(0, 1]), Eq.(l) is oscillatory provided that for some a > Jji and for every r >tolimsupt ' (8)a(2a-l)(2a + l)'At the same time, we study the interval oscillation of differential equations with the following form(r(t)kl(x(t),xl(t)))'+p(t)k2(x(t),x'(t))x'(t) + q(t)f(x(t)) = 0, * > t0 > 0, (9)where(i) the function r : [tQ, oo) —)? (0, oo) is continuously differential,to > 0;(ii) p : [t0, oo) -> i? is continuous, p(t) > 0 for all ï¿¡ > t0, to > 0;(iii) 5 : [t0, oo) —?? R is continuous, g(t) > 0 for all t > t0, t0 > 0;(iv) / : R —> R is continuous and satisfies. f(x)/x > iT for some positiveconstant K and for all x ^ 0;(v) k\ : i?2 —>â–  i?2 is continuously differentiable and satisfies A;2(n, ?;) R2 is continuous and has the sign of v for all v € R/{0} andall ue R;(vii) / e C1^, R] and x/(x) > 0 for x ^ 0, there exists }'{x) for x e Rand /'(x) > /i > 0 for x ^ 0.We state the main results as follows:Lemma [33,Lemma 1] Suppose that the assumptions (i)-(vi) are satisfied. If x(t) is a non-oscillatory solution of Eq.(9),then x(t)x'(t) < 0 for all large t.Theorem 2.2.1 Suppose that the main assumptions (i)-(vi) are satisfied, and assume that there exists a function p e Cl([t0, oo), (0, oo)) such that for some H G X and for each sufficiently large To > to, there exist a, b, c with To < a < c < b such that^i^y ï¿¡ H{8, a)Kp{s)q{s)ds + j^ J H(b, s)Kp(s)q(s)dswhere= h2(t,s) -Then Eq.(9) is oscillatory.Theorem 2.2.2 Suppose that the main assumptions (i)-(vi) are satisfied, and assume that there exists a function p G Cl([t0, oo), (0, oo)) such that/* r ( \ ( \ iH(s, l)Kp(s)q(s) - an ,'QUs, I) Ids > 0 (11)L 4 Jandlim sup / H{t, s)Kp(s)q(s) - ar,Q22(t, s) Ids > 0 (12)t^oo Jl I 4 Jfor some H G X, and for each / > ^o, then Eq.(9) is oscillatory.Theorem 2.2.3 Suppose that the main assumptions (i)-(vi) are satisfied, and assume that there exists a function p G C^fto, oo), (0, oo)) such that for any u G C[a, b] satisfying u'(t) G L2[a, b] and u(a) = u(b) = 0, we haveJau2{s)Kq(s)p{s) - ar(s)p(s) (u'ds > 0, (13)then Eq.(9) is oscillatory.Theorem 2.2.4 Suppose that the main assumptions (i)-(v) and (vii) are satisfied, and assume that there exists a function p e C1^, oo), (0, oo)) such that for some H G X and for each sufficiently large TQ > to, there exist a, b, c with To < a < c < b such thatwhere Qi(s, t),Q2(t,s) are defined as Theorem 2.2.1 Then Eq.(9) is oscillatory.Theorem 2.2.5 Suppose that the main assumptions (i)-(v) and (vii) are satisfied, and assume that there exists a function p G Cl([to, oo), (0, oo)) such that"Urn sup f \h(s, l)p(s)q{s) - ar^p^' Q\(Sj I)] ds > 0 (15)t-xx Ji L 4/z Jandlim sup / H(t, s)p(s)q(s) - —^^-Ql(t, s) Ids > 0 (16)t-Kx> Jl [ 4/i Jfor some HEX, and for each / > to, then Eq.(9) is oscillatory.Theorem 2.2.6 Suppose that the main assumptions(i)-(v) and (vii) are satisfied, and assume that there exists a function p G Cl([t0, oo), (0, oo)) such that for any u G C[a, b] satisfying u'(t) G L2[a, b\ and u(a) = u(b) — 0, we have2"ds > 0, (17)then Eq.(9) is oscillatory.Remark: We have required in this paper that the function p(t) is non-negative and q(t) is strictly positive.Although positive damping is commonly encountered in applications,it would be desirable to consider also the cases where the damping term is non-positive and of variable sign.Careful examination if the proof of Lemma reveals that nonnegativity of p(t) can be replaced with non-positivity of this function provided that the assumption&2(u, v) has the sign opposite to that of v for all v G R/{0} and u G R is considered in place of condition (vi) for the function k2(u,v).At the last of this chapter, we study the interval oscillation of second-order nonlinear differential equation with damped term and forcing term4- P(t)x'(t) + q(t)f(x(t))g(x'(t)) = e(t), t > t0 (18)where(Ax) r{t)eCl{[t-o, oo);(0, oo));(A2) The function e(i), p(ï¿¡), q(t) ((A3) The function g : i? —>â–  R is continuous and ^(x) > /i > 0 for a;7^ 0;(A4) / G C^-R;H) and x/(x) > 0(x / 0);(A5) The function / G CX{R;R) is continuous and f'{x) >K>0forx^0;(A6) The function / G CJ(/?;i?) is continuous and /(x)/x > Ko > 0 for x / 0.We state the main results as follows:Theorem 2.3.1 Let assumptions (Ai) - (A5) hold. Suppose that for any T > t0, there exist T < sj < t\ < s2 < t2 such that e(t) < 0 for t G [si, tx] and e(t) > 0 for t G [s2, i2]. Let D(si, ti) = {u G C1^, ij] : u(t) is not identically zero, u(si) = u(ti) = 0} for i = 1, 2. If there exist u G D(si, ti) and a positive, nondecreasing function G C^fto, 00)) such that-[97/^ — -I- Tff"M2!/-/" >? fl HQifor i — 1, 2, then every solution of (18) is oscillatory.Remark: If p = 0, /(x) = 1, we can get the theorem in Wong [32].Theorem 2.3.2 Let assumptions (Ai) — (A4) and (Aq) hold. Suppose that for any T > to, there exist T < s\ < tx < s2 < t2 such that e(t) < 0 for t G [si, ti] and e(t) > 0 for t G [?2, fe]- Let Z?(3,-, U) - {u G C1^, **] : ?(t) is not identically zero, u(si) = u(ti) = 0} for i = 1, 2. If there exist u G ï¿¡>(si, ï¿¡*) and a positive, nondecreasing function G C1^, 00)), we have> 0, (20)for i — 1, 2, then every solution of (18) is oscillatory.In chapter 3, we first study the interval oscillation of second-order nonlinear delay differential equations(r(t)x'(t))' + q(t)f{x(T(t)))g{x'(t)) = 0,t > tQ > 0, (21)where q(t) is a positive continue function on [tQ, oo), r > 0 is an eventually positive function, r{t) is a positive continuously differentiable function on [t0, oo) such that r(t) < t, r'(t) > 0, t > to, lim^oo r(ï¿¡) = oo./ G C((-oo,+oo),(-oo,+oo)),5 G C((-oo, +oo);[0,oo)), f{x)/x > K > 0;g(x) > C > 0 for x ^ 0,where K and C are constants.We obtained the following results of Eq.(21):Theorem 3.1.1 Assume that there exists a positive,nondecreasing function p(t) G C1([io,oo)) such that for some H G X and for each sufficiently large To > to, there exist increasing divergent sequence of positive numbers {an}, {&?}, {cn} with To < an < cn < bn such thatf n H{s, an)KCp{s)q{s)ds + —±----r f " H(bn, s)KCp(s)q(s)dsJan ti{On,Cn) JCnH(cn,x r(r(s))p(s)\h2(bn,S)^niun) Jan>-r^-rl ---------^-----rr^---------+^ds4H(c1 ,bn r(r(s))p(s)+ 4H{bn,cn)JCn t'(s)(22)then Eq.(21) is oscillatory.Theorem 3.1.2 Assume that for each sufficiently large T > to there exist a, b G R such that T < b < a, and a positive,nondecreasing function p(t) G Cl([tQ,oo)) such that for any $ G C[a,b) satisfying $'(t) G L2[a,b] and $(a) = $(6) = 0,we havetJa\e{s)KCp{s)q{s) - r(r(yV(*) + Hs)^f\?]ds > 0. (23)Then every solution of Eq.(21) is oscillatory.Theorem 3.1.3 Assume that linv+oo R(t) = oo. Then every solution of Eq.(21) is oscillatory provided that for each / > t0 and for some A > l,the following two inequalities hold:lim sup -L f\R(s) - R(l)]xKCq(s) > —^— (24)t-too ti Jl 1\A — ijandlimsup -^ f\R(t) - R(s)}xKCq(s) > ttt^tt- (25)At the same time, we study the interval oscillation of another kind of second order nonlinear delay differential equations(r(t)i;(y(t))y'(t))' + q(t)f(y(r(t))) = 0, t > t0 > 0, (26)where(i) r(t) € C([t0, oo), R+);(ii) q G C([to, oo)), q(t) > 0;(iii) / G C\R, R) and yf{y) > 0 for y #0;(iv) ^ e C(i2, i?) and ^(y) > 0 for y ^0;(v) r(t) is a positive continuously differential function on [to, oo),such that r'(t) > 0 for t > tQ and lim^oo r(t) = oo, r(t) < t.We state the main results as follows:Theorem 3.2.1 Let assumptions (i)-(v) hold, andAssume that there exists a positive function p(t) e Cl([t0, oo)) such that for a < b and [a, b] C [tQ, oo),u € Cl[a, b],u'(t) e L2[a,b] satisfying u(a) = u(b) = 0, we have---------KV ' . '----------------ds. (27)P Ja T\s)Then,Eq.(26) is oscillatory.Remark: In some papers, the authors use a function H = H(t, s) which have been mentioned in the last chapter and another positive, nondecreasing function p 6 Cl([to, oo)), and some results does not seem to be hold without a restriction on the sign of p'(t). But in the proof of this theorem, we use a new function u e Cl[a,b],u (t) e L2[a,b],u(a) = u(b) = 0, [a,b] C [t0,oo): and the Maxium Theory, so that the conclusion is hold without a restriction on the sign of p'(t). On this point we emphasize the importance of Theorem.