The Existence Of Positive Solution And Oscillation Of Neutral Differential System With Multi-delays And Asymptotic Behavior Of Corresponding Impulsive System

In recent years, delay differential equations of neutral type have been concerned by many authors. Many results have been gain about the equation's oscillation, asymptotic behavior and stability, etc. But most of them are devoted to linear neutral equations, the study about nonlinear equations is not much.. Especially for the existence of positive solutions of equations with many delays. In this paper, we first study the existence of positive solution and oscillation of neutral differential equation with multi-delays. After that, we study asymptotic behavior of neutral delay differential equation with impulses. This paper is organized as follows. In Chapter one, we refer to paper [4] and obtain some sufficient conditions for the existence of positive solution of the multi-delays differential equation by the Banach contraction mapping principle. Furthermore, when we discuss the oscillation of above equation, we divide three situations to gain the result. In Chapter two, we discuss the asymptotic behavior of solutions of corresponding neutral delay differential equation with impulses and obtain some sufficient conditions about the nonoscillatory solutions and oscillatory solution tend to zero. With the extensively of the function f in the equation, all these results can also be utilized for linear equations, thus make the application of the equation more extensively. Chapterâ… The Existence of Positive Solution and Oscillation of Neutral Delay Differential System with Multi-delays In this chapter, we discuss the existence of positive solution and oscillation of neutral differential equation: where p , r , qi (i = 1,2, …, n ) ∈C ([t₀, ∞),R),and p ,r is derivable; σ, ρ, Ï„_i (i = 1,2, L…n) ∈(O, ∞), O < Ï„₁ < Ï„₂<…<Ï„_n, f ∈C ( R , R), f (O) = 0,and let m = max{σ, ρ, Ï„_n}. Main results: Theorem1 Suppose p (t), p′(t), r(t ), r′(t), q_i (t)(i=1,2,…, n) is boundary on [t₀, ∞),furthermore, the following conditions hold ( ) ( )01( ) ( ) ( ) ( ) ( ) ,iniip t μp t e μσr t μr t e μρq t e μτμt t=′+ + ′+ + ∑≤≥(1.2) then equation(1.1)has positive solution. Theorem2 Assume p (t ), p ′( t ), r (t ), r ′( t ), qi (t )(i = 1,2, L , n) is boundary on [t 0, ∞),and condition ( A1 )hold.If there exist t ? ≥t 0, μ> 0,for any λ∈Λ= {λ: λ(t ) = 0, t ∈(t ? ? m, t ?), λ(t )is continuous on[t ? , ∞)and 0 ≤λ(t )≤μ}, the following inequalities hold: [ ] ( ) [ ] ( )( ) ( ( ))10 ( ) ( ) ( ) exp ( ) ( ) ( ) ( ) exp ( )( )exp ( ) exp ( ) ,it tt tn t tii t tp t p t t s ds r t r t t s dsq t s ds f s ds t tσρτλσλλρλλλμ? ?? ?? ?=≤? ′+ ? ? ? ′+ ? ?? ? ≤≥∫∫∑∫∫then equation(1.1)has positive and increase solution on [t ? , ∞). Suppose equation(1.1)satisfy: 1 1( B ) y R , f ( y)( y0)? ∈α≤y≠, α1 is a positive constant; ( B2 ) qi (t )(i = 1,2, L , n) satisfy: qi (t ) ≥0 Situation(â… ) p (t ) ≤0, r (t ) ≤0, ? p (t ) ? r (t ) ≤1 Theorem3 Assume there exist some i1 ∈{1,2, L , n},such that 111liminf ( )1itt →∞∫t?ταqi s ds >e, then all the solution of equation(1.1)are oscillatory. Theorem4 Assume ( H 1) there exist i0 ∈{1,2, L , n},such that 00liminf ( ) 0itt →∞∫t?Ï„qi s ds> if there exist a I , I is a subset of{1,2, L , n}and I is not empty,and i0 of ( H 1)is in I ,for some large enough T ≥t0,when t ≥T,we have ( )1 1limt →i∞n f ??? ??1 + mi∈iI n ? p (t ? Ï„i ) ? r (t ? Ï„i ) ??∫tt?τα∑i∈Iqi ( s )ds ???>1e, then all the solution of equation(1.1)are oscillatory.Theorem5 Assume ( H 1)and ( H 2)there exist T ≥t0, when t ≥Thave ? p (t ? Ï„i ) qi (t ) ≥? p (t ) qi (t ? σ),? r (t ? Ï„i ) qi (t ) ≥? r (t ) qi (t ? ρ), i = 1,2, L ,n hold.If one of the following conditions hold: (1) Suppose I is a subset of{1,2, L , n}and I is not empty, ω= min{Ï„1 + σ, Ï„1+ ρ},we have 1limt →i∞n f ∫tt? ωα[ ? p ( s ) ∑i∈I qi ( s ? σ) ? r ( s ) ∑i∈Iqi ( s ? ρ)]ds >1e. (2)1 1limt →i∞n f ∫tt? τα[ ? p ( s ) ∑i∈I qi ( s ? σ) ? r ( s ) ∑i∈I qi ( s ? ρ) + ∑i∈Iqi ( s )]ds >1e, hold, then all the solution of equation(1.1)are oscillatory. Situation(â…¡) p (t ) > 0, r (t ) < 0 Theorem 6 Assume ρ> Ï„n,Ï„1> σ, p ′( t ) ≤0,if there exist k1 , k2∈{1,2,L,n},such that 1qk (t ) > 0is a periodic function with period σ,we have 1111( )liminf( ) 11 ( (2 ))kt kttkq sdsτσp s eα→∞∫? ?+ ? Ï„?σ> and 2221( )liminf( ) 1( )kt kttkq sdsρτr s eα→∞∫? ?? + ρ?Ï„> hold,then all the solution of equation(1.1)are oscillatory. Corollary1 Assume p (t ) < 0, r (t ) > 0,σ> Ï„n,Ï„1> ρ,,r ′( t ) ≤0,if there exist l1 , l2∈{1,2,L,n},such that 1ql (t ) > 0 is a periodic function with period ρ,we have 1111( )liminf( ) 11 ( (2 ))lt lttlq sdsτρr s eα→∞∫? ?+ ? Ï„?ρ> and 2221( )liminf( ) 1( )lt lttlq sdsστp s eα→∞∫? ?? + σ?Ï„> hold,then all the solution of equation(1.1)are oscillatory. Situation(â…¢) p (t ) > 0, r (t ) > 0 Theorem7 Assume Ï„1 >ρ>σ, p ′( t ) ≤0, r ′( t ) ≤0,if there exist somek ∈{1 ,2,L,n},such that qk ′(t ) ≤0,and 1( )liminf( ) 11 ( (2 )) ( (2 ))kt kttk kq s dsτρp s r s eα→∞∫? ?+ ? Ï„? σ+ ? Ï„?ρ> then all the solution of equation(1.1)are oscillatory. Chapterâ…¡Asymptotic Behavior of Solutions of Neutral Delay Differential System with Impulses In this chapter, we discuss the asymptotic behavior of solutions of corresponding impulses differential equation of the equation (1.1) in Chapter â… : [ ] ( )1( ) ( ) ( ) ( ) ( ) ( ) ( ) 0, 0, ,( ) ( ) ( ), 1,2,ni i kik k k ky t p t y t r t y t q t f y t t t ty t y t b y t kσρτ=+ ???? + ? + ? ′+ ? = ≥≠?? ? = =∑L (1) we obtain some sufficient conditions about the nonoscillatory solutions and oscillatory solution of equation (1) tend to zero. Where p , r , qi (i = 1,2, L , n ) ∈C ( [0, ∞], R ); σ, ρ, Ï„i(i =1,2, L , n) and f is the same to the equation (1.1) in chapter â… .Furthermore, 0 < t1 < t 2< L < tk max{σ, ρ} Main results: Theorem1 Assume for any β> 0, there exist η> 0satisfy f ( y )≥η,for y ≥β, (3) 1kk∞b+=∑< ∞,where bk + = max{b k,0}, (4) 1( ) 0ni iiq t Ï„=∑+ ≥, ∫0∞∑i =n1 qi ( s + Ï„i)ds= ∞, (5) and there exist constant γ> 0, such that? p (t ) ? r (t ) < 1? α2γand ? p ? r≠1. (6) 2( ) ( )1ii iii it tτ∑<θ∫t ??θτq ? s + Ï„i ds + τ∑>θ∫t??τθq +s + Ï„ids≤γ<α, (7) where θ∈[0, Ï„n], ( ) max { ( ),0}q i+ s = qi s, ( ) max { ( ),0}q i? s = ? qi s, p = lti→m∞p (t ),lim ( )r = t→∞r t, p ,r ∈R?.Then the nonoscillatory solutions of equation (1) tend to zero. In Theorem1,if θ= 0 or θ= Ï„n,then we have: Corollary1 Suppose equation (1) satisfy the conditions in Theorem1 except (7), and one of the following conditions hold: 1 2( )1iin t∑i = ∫t?Ï„q +s + Ï„ids≤γ<α, 11 ( )12inin t∑i =? ∫t??Ï„Ï„q ?s + Ï„ids≤γ<αThen the nonoscillatory solutions of equation (1) tend to zero. Theorem2 Assume condition (3)hold,and 1kk∞b=∑< ∞, (11) 1( ) 0ni iiq t Ï„=∑+ ≠,for large enought, (12) 1 22limsup ( ) limsup ( )1 2 2t t→∞D t + →∞D t < + αp +r, (13) where 11( ) ( )in tD t = ∑i= ∫t?Ï„qi s +Ï„ids, 2 ( ) 1sgn( ) ( )n tiD t = ∑i= ∫t??θτθ? Ï„i qi s +Ï„ids, p and r is the minimum of p (t ) and r (t ),respectively. and ?2 p ? 2 r< 1.Then the oscillatory solutions of equation (1) tend to zero. In Theorem2, if θ= Ï„næ—¶,we have Corollary2 If the condition (13)in Theorem2 is replaced by 1 1 2limsup ( ) limsup ( )1 2 2ii nn t ntt i t i i titi i→∞∑= ∫?Ï„q s + Ï„ds + →∞∑= ∫??Ï„Ï„q s + Ï„ds<+ αp +r, Then the oscillatory solutions of equation (1) tend to zero.