Oscillation Of Second-order Delay Differential Systems With Oscillatory Potential And Second -order Delay Difference Systems With Impulses

Oscillation criteria of nonlinear DE with delay and interval oscillation criteria for a forced second-order nonlinear ODE with oscillatory potential are concerned by many authors ,they have obtained many good results[1-12,44] .now we discuss interval oscillation criteria for a forced second-order nonlinear DDE with oscillatory potential and obtain some sufficient oscillation criteria. Besides, it is well known that ODE with impulses and DE have been considered by many authors the theory of impulsive differential equations is emerging as an important investigation area, it is much richer and more complex than the corresponding theory of differential equations without impulses effects ,moreover, such equations may exhibit the real world's motions, changes and development. in the recent years, there is increasing interesting on oscillation/nonoscillation of impulsive delay differential equations and numerous papers have been published on this class of equations and good results were obtained[14-18,20-26] ,but fewer papers are on impulsive difference equations. The methods and techniques employed in this paper come from those employed in paper [13,14,22,33]],some interesting results are obtained. Chapter I Interval oscillation criteria for a forced second-order nonlinear delay differential system with oscillatory potential In this chapter, oscillation criteria which we discuss for a forced second-order nonlinear delay differential system with oscillatory potential (r(t) y′(t))′+ Q (t, y(t), y(t-Ï„)) = g(t) (1) are different from the most known ones that based on the information only on a sequence of subintervals [t₀, +∞), The forced term in this paper is oscillatory in [t₀, +∞)。We assume According to Philos[8]and Kong[9],We have a class of functions G ,if H (t ,s)satisfied: ( H₁ ):H(t,t)= 0,H(t,s)>0,(t>s); ( H 2): H ∈D,in D = {( t , s ) ?∞< s ≤t< +∞} ,there exist partials ??H tand ??H s, such that ??Ηt = 2 h1 (t , s ) H (t , s ) and ??Hs = ?2 h2 (t , s ) H (t , s), where h1 , h2 ∈Ll oc( D , R). Main results Theorem 1-2-1:Assume that (C 0 ' ), (C 0 ''), ( C1 ) and the follow conditions hold: ( C 2) :Let { [ ]}D1 (a i , bi ) = H ∈C′ai , bi , H (t ) ≥0 ≡/ 0, H (a i ) = H (b i ) = 0, ??Ht = 2h ( t ) H (t ) ,(i =1,2) Suppose there exists H (t ) ∈Di ( ai , bi),such that [ ]111 1 12Qi ( H ) = ∫a b ??? H ( s ) k1 k 2q ( s ) v g ( s ) ?v? r ( s ) h ( s ) ???ds>0, (4) Then (1) is oscillatory。Theorem 1-2-2:Assume (C 0 ),(C 0 ''),and (C 2) holds, then (1) is oscillatory。Theorem 1-2-3:Assume (C 0 ),(C 0 '')and the follow condition holds (C 3 ):Let { [ ]}D2 ( ai , bi ) = u ∈C ′ai , bi , u (t ) ≡/ 0, u ( ai ) = u (b i ) = 0 ,(i =1,2) Suppose there exists u (t )∈D2 (ai ,bi),such that [ ] [ ]1 1 12( ) ( ) 1 2( ) ( ) ( ) ( ) 0,( 1,2) (11)iib v vQi u = ∫a ??? H s k k q s g s ?? r s u ′s ???ds ≥i= then (1) is oscillatory。Theorem 1-2-4: (C 0 ), (C 0 '')and the following conditions hold: (C 4):There are some ci ∈( ai , bi ),(i = 1,2)and some H ∈G, such that T ≤a1 < b1 ≤a2 < b2and one of the following conditions hold:[ ]1 1 124 1 1 2 1( ) : 1( , ) ( ) ( ) ( ) ( , )( , )iic v vai ii iC H s a k k q s g s r s h s a dsH c a?? ????∫? ? [ ]1 1 121 2 11 ( , ) ( ) ( ) ( ) ( , ) 0( 1,2) (13)( , )i ii ib v vba i cii iH b s k k q s g s r s h b s ds iH b c+ ?? ?? ??> =∫? ∫? [ ] [ ]21 1 124 2 21 2( ) : 1( ) ( ) ( ) ( ) ( )( )iic v vai ii iC u s a k k q s g s r s u s a dsu c a? ∫??? ? ?? ′????[ ] [ ]21 1 1221 221 ( ) ( ) ( ) ( ) ( ) 0,( 1,2),( )( , ) ( ) ,( 1,2) (14)i ii ic v vba i cii iu b s k k q s g s r s u b s ds iu b cH t s u t s G i+ ? ??? ? ?? ′? ???≥== ? ∈=∫∫ then (1) is oscillatory。 Chapter II Oscillation criteria for second-order nonlinear delay difference equation with impulses In this chapter ,we investigate the oscillation of the second-order nonlinear impulsive delay difference equation 1 11 1( ( 1)) ( 1) ( , , ) 0,( )(2.1)k k k kn n n n l kn n k n na x n p x n f n x x n nB x M B x? ? ?? ?????? ? ? ? = ? + ?? ? + = ≠where ?denotes the forward difference operator ,i.e ? xn = ? x ( n ) = x ( n + 1) ? x ( n),l ∈N,N is the nature number set , 0 ≤n0 < n1 < ... < nk< ...and lk i→m∞nk= ∞, we note0nexp( n s 1 s1)s n sBp aa? ?== ∑+?. some interesting results and example are obtained ,the example which illustrates the that impulses play a very important role in giving rise to the oscillations of equations is also included. Throughout this chapter ,assume that the follow conditions hold: (I) uf ( n , u , v ) > 0,(u v> 0)and there exists a nonnegative {q n}and a function φ,such that f ( n, u , v ) φ( v ) ≥qn, v ≠0。where φsatisfies xφ( x ) > 0,( x≠0)and φ( u ) ? φ( v ) = g (u , v )(u ? v ), uv ≠0,gis a nonnegative function ;(II)there exist positive numbers d k ,dk?such that d k ≤M k ( x )x ≤dk?; (III) { }an n+0∞is a positive sequence; (IV) p ≤2 an + an ?1. Main results Lemma 1 . Let { x ( n )}be a solution of(2.1) ,Suppose that there exists some N 0 ≥n0, n ≥N,such that x ( n ) > 0.If ( ) (2.3)j j iA sis n n n s∞d e?= ≤≤∑∏= +∞hold for all sufficiently large n j( ≥n1),Then ?x ( nk ?1) ≥0 å’Œ? x ( n) ≥0 for n ∈N [ nk , nk +1), where nk ? 1≥N。Theorem 2-1 Assume that (2.3) holds,If ( )1,1 (2.8)j j jkiA ii n i n n iki n k Nq e= + a ≤≤d?≠∈∑∏= +∞ holds for all sufficiently large n j, Then every solution of (2.1) is oscillatory。...