Existence And Multiplicity Of Positive Solutions And Sign-changing Solutions Of Dirichlet Problems Of Second-order Impulsive Differential Equations

In this paper, by using the Robinowitz’s global bifurcation theorem, we study the existence of positive solutions and sign-changing solutions of Dirichlet problem-s for two classes of second-order impulsive differential equations with linear and nonlinear impulsive functions, respectively. We describe them in detail as follows.1. By using the Robinowitz’s global bifurcation theorem, we prove the multi-plicity of sign-changing solutions of Dirichlet problems for second-order impulsive differential equations with linear impulsive functions where λ>0is a parameter, ak>-1and0<t1<t2<…<tm<1are given impulsive points,â–³u’(tk)=u’(tk+)-u’(tk-),u’(tk+) and u(tk-) denote the right and left limit of u at t=tk, respectively; a:[0,1]â†'[0,∞) is continuous and a(·)(?)0on any subinterval of [0,1];f:Râ†'R is continuous, the limits f0exist, and there exist two constants s2<0<s1such that f(s)=0for s E [βs2,αs2]∪{0}∪[αs1,βs1], and f(s)s>0for s∈R\{[βs2,αs2]∪{0}∪[αs1,βs1]}. Here α and β are two constants satisfying The main results extend and improve the corresponding ones of Ruyun Ma [Appl. Math. Lett.,2008] and Yansheng Liu, O’Regan [Comm. Nonlinear Sci. Numer. Simulat.,2011].2. By using the Topological degree theory, Krein-Rutman theorem and Robi-nowitz’s global bifurcation theorem, we prove the existence of positive solutions of Dirichlet problems for second-order impulsive differential equations with nonlinear impulsive functions where λ>0is a parameter,0<t1<t2<…<tm<1are given impulsive points,â–³u’(tk)=u’(tk+)-u’(tk-),u’(tk+)and u’(tk-)denote the right and left limit of u’at t=tk,respectively:f∈C([0,1]×[0,∞),[0,∞)),Jk∈C([0,∞),[0,∞)),k1,2,…,m,f and Jk are not necessarily linearizable near0and infinity.The main results improve the corresponding ones of Xiaoning Lin,Daqing Jiang[J.Math. Anal.Appl.,2006].