Research On A Class Of The Asymptotic Behavior Of Solutions Of Differential Equations With Deviation And The Existence Of Multiple Positive Solutions Of Boundary Value Problems

In this paper , first,we consider the integro-diferential equations with deviating argumentt∈[0,∞). With the aid of the integral inequality , we obtain some sufficientconditions under which ensure all solutions y(t) of equation (1) are efined on allof [0,∞),and these satisfy the equation as t→∞,y^k(t)=(?)(δ_i+o(1))z_i^k(t),k=0,1,2,...,n-1,whereδ_i,i=1,2,...,n, are contants,{z_i(t)}_i=1ⁿ is a standard fundamental system of solutions of the equationSecond , Using the fixed point theorem in cone ,this paper deals with the existence of multiple positive solutions for a class of singular boundary value problems of impulsive differential equationsin Banach space.If: (H₃) for any R>0,[a,b](?)(0,1),f in [a,b]×T_R is uniformly continuous ,(H₄)α(f(t,D))≤Lα(D),α(I_k(D))≤L_kα(D),k=1,2,…,m;(H'₅)‖x‖→0 in (H₅) is exchanged by‖x‖→+∞.discussed:Lemma 3. 2.1 x∈PC[J,P] is a non-negtive solution of (21) if and only if(?)has a fixed pointx in PC[J,P].Lemma 3.2.2 let conditions (H₁)-(H₄)be satisfied, then for any r>0 , A isa bounded and continuous operator from PC[J,P]∩B_r into PC[J,P].Lemma 3. 2. 3 let conditions (H₁),(H₃),(H₄)be satisfied, then for any R>0 ,A is a strict set contraction operator from PC[J,P]∩B_r into PC[J,P].Theorem3.1 let conditions (H₁)-(H₅) be satisfied, (21) has at least one positive solution.Theorem 3.2 let conditions (H₁),(H'₂),(H₃),(H₄),(H'₅)be satisfied, (21)has at least one positive solution.Theorem 3.3 let conditions (H₁),(H"₂),(H₃),(H₄),(H₅),(H'₅)be satisfied,(21) has at least two positive solutions x₁ and x₂ satisfied 0<‖x₁‖_PC2‖_PC.