The Existence And Multiplicity Of Positive Solutions For Several Kinds Of Nonlinear Differential Equations

Nonlinear functional analysis has become one of the important research direction in modern mathematics. Especially the integer order differential equation. Many experts and scholars have launched a widely research. On the other hand, for Sturm-Liouville eigenvalue problems And impulsive differential equations with integral boundary condi-tions also are widely applied in many areas of physics and mathematics. Such as thermal conduction problems, semiconductor problems, hydrodynamic problems and so on. In recent years, the existence and multiplicity of positive solutions have become hot spot.In this paper, by using the fixed point index theory and Krasnosel'skill's fixed point theorem, respectively, investigated that existence and multiplicity of positive solutions for Sturm-Liouville eigenvalue problems and nth-order impulsive differential equation with integral boundary conditions.The thesis is divided into two chapters.The chapter 1. by using the fixed point index theory, we consider the existence and multiplicity of positive solutions for the following fourth-order nonlinear Sturm-Liouville boundary value problem (BVP)(?)where ? > 0 is positive real parameter,?i,?i,?i,?i?0 and ?i?i > 0(i =1,2) are given constants with ?2=?2?2 + ?3?2B(0,1) +?2?2 > 0, B(0,1) = ?011/p(T)dT, p?C1((0,1), (0,+?)). The chapter 2, by means of Krasnosel'skill's fixed point theorem,we discuss the existence, nonexistence and multiplicity of positive solutions for nth-order impulsive differential equation with integral boundary conditions:(?)where J ? [0, 1], g? C((0,1),R+), g(t) may be singular at t = 0,1 ? is a positive parameter, 0 <t1 < t2 < … < tm < 1,f?C(J×R+ × R ×…×R,R?), Ii?C(R+ × R × R ×…×R,R+)(i = 1(?),where un-1(tk+) and un-1(tk-) respectively denote the right limit and left limit of un-1 (t) at t ?tk;?01a(t)u(t)d?(t),?01b(t)u(t)d?(t) are the Riemann-Stieltjes integrals, a,6 ? L1[0, 1] are nonnegative; moreover a and ? are right continuous on [0,1), left continuous at t=1,and nondecreasing on [0,1], with (?). In which R+ = [0,+?).