Oscillation criteria for perturbed nonlinear dynamic equations

Keywords--Oscil lat ion, Second-order nonlinear dynamic equation, Time scale, Riccati transformation technique, Positive solution. 1. I N T R O D U C T I O N The theory of t ime scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. thesis [1] in order to unify continuous and discrete analysis. Not only can this theory of so-called "dynamic equations" unify the theories of differential equations and of difference equations, but also it is able to extend these classical cases to cases "in between", e.g., to so-cMled q-difference equations. A time scale T is an arbi t rary closed subset of the reals, and the cases when this t ime scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. Many other interesting t ime scales exist, and they give rise to plenty of applications, among them the s tudy of population dynamic models (see [2]). A book on the subject of time scales by Bohner and Peterson [2] summarizes and organizes much of the t ime scale calculus (see also [3]). For the notions used below, we refer to [2] and to the next section, where we recall some of the main tools used in the subsequent sections of this paper. While oscillation theories for differential equations and for difference equations (see, e.g., [4]) are well established, the discrepancies in some of the results in these two theories are not well understood. In the last years there has been much research activity concerning the oscillation and nonoscillation of solutions of some dynamic equations on t ime scales, and we refer the reader to 0895-7177/04/$ see front matter © 2004 Elsevier Ltd. All rights reserved. doi: 10.1016/j.mcm.2004.03.002 Typeset by .AA4S-TF_ ~ 250 M. BOHNER AND S. H. SAKER the papers [5-13]. Following this trend, in this paper we shall provide some sufficient conditions for oscillation of second-order nonlinear perturbed dynamic equations of the form (a( t ) (xA)~)a+F(t ,x~)=G(t ,x~ ,xa) , fortE[a,b], (1.1) where 7 is a positive odd integer and a is a positive, real-valued rd-continuous function defined on the time scales interval [a, b] (throughout a, b E T with a < b). Since we are interested in oscillatory behavior, we suppose that the time scale under consideration is not bounded above, i.e., it is a time scales interval of the form [a, co). By a solution of (1.1) we mean a nontrivial real-valued function x satisfying (1.1) for t > a. A solution x of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory. Our attention is restricted to those solutions of (1.1) which exist on some half line Its, co) and satisfy sup{lx(t)l : t > to} > 0 for any to _> tx. In this paper, we obtain some oscillation criteria for (1.1). The paper is organized as follows. In the next section, we present some basic definitions concerning the calculus on time scales. In Section 3, we give some sufficient conditions for oscillation of (1.1) by using elementary calculus on time scales. In Section 4, we will use Riccati transformation techniques to give some sufficient conditions in terms of the coefficients which guarantee that every solution of (1.1) is oscillatory or converges to zero. To the best of our knowledge, nothing is known regarding the qualitative behavior of (1.1) on time scales up to now. 2. S O M E P R E L I M I N A R I E S O N T I M E SCALES A time scale T is an arbitrary nonempty closed subset of the real numbers ]~. In this paper, we only consider time scales that are unbounded above. On ql" we define the forward jump operator a and the graininess # by o ( t ) : = i n f { s e v : s > t } a n d , ( t ) : = t. A point t E ~" with a(t) = t is called right-dense while t is referred to as being right-scattered if a(t) > t. The backward jump operator p and left-dense and left-scattered points are defined in a similar way. A function f : T ~ l~ is said to be rd-continuous if it is continuous at each rightdense point and if there exists a finite left limit in all left-dense points. The (delta) derivative f a of f is defined by f(a(t)) fA(t) = lim a ( t ) s ' seu(t) where U(t) = qF \ {a(t)}. The derivative and the forward jump operator are related by the useful formula f a = f 4-/zf A, where f a := f o or. (2 .1 ) We will also make use of the following product and quotient rules for the derivative of the product fg and the quotient f /g (where gg~ # 0) of two differentiable functions f and g: (f g)A = fag + fa g~ and ( f ) A = fAg fg/' gg~ (2.2) By using the product rule from (2.2), the derivative of f(t) = ( t a) m for m E N and a Eb [~ can be calculated (see [2, Theorem 1.24]) as m--1 fZx(t) ---E (~(t) -cx) v (t a)m-v-1. (2.3) v=0 Osci l la t ion C r i t e r i a 251 For a, b E T and a differentiable function f , the Cauchy integral of f~x is defined by b f fA(t) At = f(b) f(a). The integration by parts formula follows from (2.2) and reads b f[ f ff'(t)g(t) At = f(b)g(b) f(a)g(a) ff(t)gA(t) At, and infinite integrals are defined as ~ f ( s ) As= lim [t t--,~ Ja f ( s ) As. Note that rd-continuous functions possess antiderivatives and, hence, are integrable. EXAMPLE 2.1. In case T = 1~, we have a(t) = p(t) = t, #(t) =_ O, and in case T = Z, we have a(t)=tA-1, p ( t )=t -1 , b fzx = if, and f~ f(t) At = f(t) dr, b b--1 #(t) ----1, fA = A f , and fa f(t) At = E f(t). t-----a (2.4) 3. O S C I L L A T I O N C R I T E R I A In this section, we give some oscillation criteria for (1.1). Throughout this paper, we shall assume tha t (H1) a : T --* R is a positive and rd-continuous function; (H2) 7 E N is odd; (H3) p, q : T --* R are rd-continuous functions such that q(t) p(t) > 0 for all t e qF; (H4) f : ~ ~ IR is continuously differentiable and nondecreasing such that uf(u) > 0, for all u e ~ \ {0}; (Hs) F : T x R ~ IR and G : T x R 2 --* ]R are functions such that uF(t,u)>O and uG(t, u, v) > O, f o r a l l u e R \ { 0 } , v C R , t E T ; (H6) F(t, u)/f(u) > q(t) and G(t, u, v)/f(u) < p(t) for all u, v e R \ {0} and all t e T. For simplicity, we list the conditions used in the main results as follows (to > a): f , ? At (3.1) (~(t))l/~ = ~ ' f o ~ At (a(t))l/.y < c¢, (3.2) f t ? [ q ( t p(t)] At = (3.3) OO, t 1 p(r)] AT As c¢, lim [q(~-) = (3.4) £--*~ ? [q ( t ) p(t)] At > 0, (3.5) f t~ { M e ft ~ -p(t)]At} -c~, a(s) a(s) [q(t) As = for all M > 0, (3.6) ~ [q(t) p(t)] At a(s) J As = c~, for all M > 0. (3.7) 252 M. BOHNER AND S. H. SAKEt~ THEOREM 3.1. Assume (H1)-(H6). Suppose that (3.1) and (3.3) hoId. Then every solution o~ (1.1) is oscilUtory on [a, ~). PROOF. Let x be a nonoscillatory solution of (1.1), say, x(t) > 0 for t > to for some to _> a. We consider only this case, because the proof for the case that x is eventually negative is similar. Prom (1.1), (2.2), and the chain rule [2, Theorem 1.87], we have for t _> to ( a ( X A ) ~ zx G(t, xa(t),xA(t)) F(t, xa(t)) f '(x(~))a(t) (xh(t)) ~+1 -f--~x ) (t)---f(xa(t)) f(x#(t)) f(x(t))f(x(cr(t))) where ~ is a number in the real interval [t, a(t)]. In view of (H2), (Ha), (H5), and (H6), we have for all t > to f o x ] (t) <_p(t)-q(t). (3.8) Because of (H6) and (Ha), from (1.1) we obtain for all t > to ( a (x ~)~) ~ (t) < -f(x(~r(t)))[q(t) p(t)] < 0, (3.9) which implies that a(xZX) ~ is decreasing on [to, c~). We claim that xA(t) >_ 0 for all t > tl > to. If not, then there exists t2 _> tl such that a(t)(xh(t)) ~ < oL(t2)(xA(t2)) ~ =: C < 0. Hence, cl/'y xA(t) < (3.10) ( ~ ( t ) ) ~ / ~ Integrating (3.10) from t2 to t provides ft. t AS (3.1) z(t) < x(t2) + C 1/7 (3.11) ~, (a(~))~/~ ~ ~ , as t ~ ~ , while the left-hand side of (3.11), i.e., x(t), is eventually positive. This contradiction implies that xA(t) >_ 0 for all t > t~. Then, integrating (3.8) from tl to t gives a ( t ) (xA( t ) ) "Y < a ( t l ) (xA(t~)) ~ _ [q(s) p(s)] A s (a.a~ c o (3.12) f ( x ( t ) ) f ( x ( t l ) ) as t --+ c~, while the left-hand side of (3.12) is always nonnegative, a contradiction. Therefore, every solution of (1.1) oscillates. The proof is complete. | EXAMPLE 3.2. If "IF = R, then a(t) -t and it(t) 0. Then (3.1) and (3.3) become (the Leighton-Wintner-type criteria) (c~(t))V'Y = ~ and [q(t) p(t)] d t = ~ . If T = Z, then a(t) = t + 1 and #(t) = 1. Then (3.1) and (3.3) become (the discrete analogue of Leighton-Wintner-type criteria) t=t E o (a(t))W'~ = ~ and E [q(t) p(t)] = c<D. t=tO EXAMPLE 3.3. Let "~" C [1, oc) be any time scale that is unbounded above. Some of the examples included are T = [1, c¢), "117 = N, and T = {2 k : k E No}. On T, we consider the perturbed nonlinear dynamic equation ( I 1 ) (x#) ' (3.13) (txA)A + X~ + ~ + t2(x#)2 = 2t((x#) a + 1)((X~X) 2 + 1)" Oscillation Criteria 253