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.??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 ?>0 is a parameter,?>-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 ?*>0 such that the problem has exactly one solution for ???0,?*?.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 ?=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 ??t??1,k=1,but in this paper we get the uniqueness results.??When f₀,f_? are not exist,we prove the existence and uniqueness of positive solutions of problem?P1?by using Time-map method.??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 ?>0 is a parameter,?>-1 is a constant,???0,1?is a impulsive point.?u???=u??+?-u??-?,u??+?,u??-?denote the right and left limit of u at t= ?,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]. ??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 ?i>-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].