Studies On Oscillation For Second-order Nonlinear Differential Equations

The oscillation of ordinary differential equation is ohe of important branche of differential equations. With the increasing development of science and technology , there are many problems relating to differential equation derived from lots of practical applications, such as whether differential equation has a oscillating solution or not, and whether all of its solutions are oscillatory or not. In very resent years, great changes of this field have taken place. Especially, the second order differential equation has been paid more attentions and investigated in various classes by using different methods(see [1]-[45]).The present paper employs a general Riccati transformation and integral average to investigate the oscillation criteria for some class of nonlinear differential equations, the results of which generalized and improved some known oscillation criteria.The thesis is divided into four sections according to contents.In Chapter 1, Preface.we introduce the main contents of this paper.In chapter 2, we consider the second-order nonlinear differential equationwhere r, and also satisfied the following conditions throughout this chapter:continuous and satisfyingfor some constants and for all x (A3) f'(x) exists, xf(x) > 0 for x 0 andfor some 72 > 0 and all x G K \ {0}.By a solution of (2.1), we mean that a function x G Cl\Tx, 00), Tx > t0, which has the property (x'(t)) G Cx\Tx,oo) and satisfies (2.1). We only restrict our attention to the nontrivial solutions of (2.1), i.e., to the solutions x(t) such that sup{|x(i)| : t > T} > 0 for all T >TX. A nontrivial solution of (2.1) is called oscillatory if it has arbitrarily large zeros, otherwise, it is said to be nonosdilatory. Equation (2.1) is called oscillatory if all of its solutions are oscillatory.In this chapter, we introduce some signals and definitions for the sake of the proofs of our main results.Let D = {(t, s) : to < s < t) denote a subset in R2.Definition 1. We say that a function H = H(t, s) belongs to a function class T, denoted by H G T. If H be continuous and sufficiently smooth on D, such that the following conditions are satisfied:(Hx) H(t,t) = 0 and H(t,s) > 0 for t0 < s < t;(H2) —gs's t0 lim inf 77—TT < °°;(H4) S(M) = S(M) for (t,s)eD;Definition 2. Let p e Cl[t0, 00) and p > 0. Define the integral operatorAPT in terms of H(t, s) and p(s) asApT(b;t)= H(t,s)b(s)p(s)ds, t > r > t0,(2.4)where b G C[￡o,oo). In the sequel, for the sake of simplicity, we will use the notation Ap{b) instead of Ap(b;t).As to the studies on oscillation of Eq. (2.1), we have the following main results.Theorem 2.1 Suppose that assumptions (Ai)-(A3) hold and H(t,s) satisfies (Hi) and (H2), and APT is defined by (2.4). If there exists a positive function p G Cl[to, oo) such that(try)*= oo,(2.8)where 7 := —, then Eq.(2.1) is oscillatory.Remark 2.1 For (x) = 1, a = p, where p > 0 be a constant, Theorem 2.1 reduce to Theorem 4 in [6]. The advantage of our results is without any restriction on the sign of p'(t).Theorem 2.2 Let assumptions (A1)-(A3) be fulfilled, and Ap be defined by (2.4). If there exist functions tpi,wherewhere a > 0, 71 > 0 are constants;By a solution of (3.1), we mean a function x G Cl\Tx, 00), Tx > to, which has the property r(t)i(;(x(t))(j)(x'(t)) G C^T^oo) and satisfies Eq. (3.1). The nontrivial solutions of Eq. (3.1) and a nontrivial solution of its called oscillatory are defined as in Chapter 2.We shall deal with the oscillation for Eq. (3.1) in two different cases, one of which is Eq. (3.1) satisfies assumptions (Aj)-(A2) and the following assumption:(Ci) f'(x) exists, xf(x) > 0, x ^ 0 and for a certain constant 72 > 0{,{xJ'^la_l)l/a > 72 > 0, x e E/{0}.The main results are:Theorem 3.1 Suppose that assumptions (Ai)-(A2) and (Ci) hold, and for any T > to, there exist T < ax < b\ < a2 < b2 such thate{t)\ -°' tG|ai'6lj . (3.9)[ >0, te [a2)b2]Denote D(ai,6j) = {H G Cl[ahbi] : Ha+l(t) > 0, ^ 0, t G (at,^), //(a,) = H(bi) = 0}, for i = 1, 2. If there exist H G D(a,i,bi) and a positive function p G Cx([t0, 00), E) such thatJ\°+I(t)q(t)p(t)dt > {a{^a+l ￡ r(t)p(t)\(a+l)H'(t)+H(t)^\a+1dt a' a' ' (3.10)for i = l, 2, where 7 := ^, then Eq. (3.1) is oscillatory.Remark 3.1 If the hypothesis in Theorem 3.1 is replaced by the following conditionp.(t.) < ~ 0,which is in accordance with the (a + l)-degree functional Ja : U —> R, where U = {qe Cl[a: b) : 77(0) = Tj(b) = 0} in [29].Remark 3.3 For (x') = x', f(x) = x, ijj(x) = 1 and a = 1, 71 = 1, 72 = 1, pit) = 1, Theorem 3.1 reduce to the result in [16].Theorem 3.2 Suppose (Ai)-(A2), (Ci) be fulfilled and for any T > i0, there exist T < ai < bi < a2 < b 0 for t > s;(ii) H has partial derivatives ^ and ^ on D such thatdHwhere hi, h2 G Lioc(D,M+). If there exist some c, € (a,,^), i = 1, 2, and a positive function p € C1([io)oo),lR) such that(3.18)for i = 1, 2, where 7 := —Hi (t, s)= (a +H2{t,s)= {a + I)h2(t, s)y/H(t, s) - H(t, s)-PP(s) then Eq. (3.1) is oscillatory.The other case is that Eq. (3.1) satisfies assumptions (Aj)-(A2) and the following additional conditions:(C2) 0 < ip(x) < C, for some positive constants C > 0 and for x ^ 0.(C3) f(x) satisfies— > K\xf~\ xfor x ^ 0, where K' > 0 and (3 > a be constant.The main results are:Theorem 3.3 Suppose that assumptions (Ai)-(A2) and (C2)-(Ca) be satisfied, and for any T > t0, there exist T < ai < b\ < a2 < b2 such that (3.9) holds and q(t) > 0 ^ 0 for t G [a\, bi] U [a2, b2]. Denote D(ai, bi) as in Theorem 3.1. If there exists H G D(ai,bi) and some positive function p G C^QicbOO^R) such that■biHa+1(t)Q(t)p(t)dt^ , (3-3°)'1 / ?/'i^ 14-\ I , -1 \ TTl I i\- , TTfj.\ P \ IO-ifor i = 1, 2, where Q(t) = [^g(i)]°/'3|e(i)|^-Q)^, then Eq. (3.1) is oscillatory. Theorem 3.4 Suppose (Ai)-(A2) and (C2)-(C3) be valid, and for any T > t0, there exist T < ai < bx < a2 < b2 such that (3.9) holds and q(t) > 0 ^ 0 for t E [ai, bi] U [a2, b2}. Further, let D — {(t, s) : t0 < s < t < oo} and the functions if, hi, h2 G C(D, R) are similar to ones in Theorem 3.2. If there exist some Ci G (di,bi) for i — 1, 2 and some positive function p G C1([to,oo),Rjsuch that^) f(3.37) for i = I, 2, where ifi, if2 are defined as in Theorem 3.2 andthen Eq. (3.1) is oscillatory.Remark 3.4 The Remark 3.1 is also valid for Eq. (3.1) under assumptions (C2) - (C3):Remark 3.5 For(A3) the function ip(x) is bounded, that is, there exist two positive constants C and Cx such that for all x € R, C < ip(x) < C\\f'(x) (A4) f'(x) exists and , ., > /32 > 0, for some positive constantj32 and for all x t0, which satisfies Eq. (4.1). The nontrivial solutions of Eq. (4.1) and a nontrivial solution of its called oscillatory are defined as in Chapter 2.Before giving the main results, we introduce some denotations.Let D = {(t,s) : t > s > t0} denote a subset in R2. We say that a function H = H(t, s) belong to a function class T', denote by H e T. If H be continuous such that the following conditions are satisfied:(Hi) H(t, t) = 0 and H{t, s) > 0 for t > s > t0;OS OSwhere h € C[D,R);(H3) 0 < inf [liminf H^s\ < ool < oo.s>t0 L t-^oo H(t,t0) jTheorem 4.1 Suppose that assumptions (Ai)-(A4) hold. If there exist H e C(D) satisfying (Hi)-(H2) and a positive function p € Cl[tQ, oo) such thatlimsup / \H{t,s)q(s)p(s)ds = °°'where p := %-, h(t, s) = \-^dL - ^| + gg|, then Eq. (4.1) is oscillatory. Theorem 4.2 Suppose assumptions (Ai)-(A4) be valid. If there exist functions H ￡ T, ip E C([to, oo),R) and a positive function p e Cl\ta, oo) suchthat1 rl limsup / H(t, s)r{s)p(s)h°+1(t, s)ds < oo , (4.17)i-^oo H(t,to) JtQl/a( I/a )ds = °° (4J8)and for every r > t0 limsupH{t,r)Jr [i/3)a H(t,s)r(s)p(s)h°+1(t,s)\ ds > v{t),where /3 and hi(t,s) are defined as in Theorem 4.1,...