Existence And Uniqueness Of Positive And Nontrivial Solutions Of Dirichlet Problems For Second-order Impulsive Differential Equations

In this paper,we study the existence and uniqueness of positive solutions of Dirichlet problems for one class of second-order differential equation with lin-ear impulsive function by using Time-map method.And we study the existence of nontrivial solutions of Dirichlet problems for one class of second-order differential equation with linear impulsive function by using L（?）pez-G（?）mez’s bifurcation theo-rem.The main results are described as follows.一、We study the existence and uniqueness of positive solutions of Dirichlet problems for second-order differential equation which the derivative with pulse by using Time-map method where λ>0 is a parameter,α>-1 is a constant,△u（1/2）=u（1/2+）-u（1/2-）,△u’（1/2）= u’（1/2+）-u’（1/2-）.f∈C1（[0,∞）,[0,∞)),f（s）>0 for s>0.We show the existence of positive solutions under the assumptions f0= [0,∞].And we also show that there exist λ*>0 such that the problem has exactly one solution for λ∈（0,λ*）.The main results are the special case of Liu Yansheng et al. [Comm. Nonlinear Sci. Numer. Simulat.,2011],but we use different method and get the uniqueness results.When α=0,our results is the ones of Theodore Laetsch [Indian Univ.Math.J.,1970],moreover,our results is the ones of Ruyun Ma et al. [Nonlinear Anal.,2004]when α（t）≡1,k=1,but in this paper we get the uniqueness results.二、When f₀,f_∞ are not exist,we prove the existence and uniqueness of positive solutions of problem（P1）by using Time-map method.三、We study the existence and uniqueness of positive solutions of Dirichlet problems for second-order differential equation which the derivative without pulse by using Time-map method where λ>0 is a parameter,α>-1 is a constant,τ∈（0,1）is a impulsive point.△u（τ）=u（τ+）-u（τ-）,u（τ+）,u（τ-）denote the right and left limit of u at t= τ,respectively. f∈C1（[0,∞）,[0,∞)),f（s）>0 for s>0.We show the existence of positive solutions under the assumptions f0 [0,∞].The main results improve the corresponding ones of Liu Yansheng et al.[Com-m.Nonlinear Sci.Numer.Simulat.,2011]. 四、We prove the existence of nontrivial solutions of Dirichlet problems for second-order differential equation with linear impulsive function by using L（?）pez-G（?）mez’s bifurcation theorem where αi>-1,i=1,2,…,k are constants,0=t₀<t₁<t₂<…<t_k<t_k+1= 1 are impulsive points.△u|t=t_i=u（t_i⁺）-u（t_i^-）,u（t_i⁺）,u（t_i-） denote the right and left limit of u at t=t_i,respectively.f∈C（[0,1]×R,R）.We show the existence of nontrivial solutions under the assumptions The main results extend and improve the corresponding ones of Ma Ruyun et al.[Non-linear Anaglysis,2004],[Bound. Value Probl.,2012],Liu Yansheng et al.[Comm. Nonlinear Sci.Numer.Simulat.,2011].