Multiple Solutions Of Second-order Nonlinear Differential Equation

This thesis, which is divided into two parts, is concerned with two point boundary value problems(BVP) of nonlinear second-order differential equation.In the first part, the author studies the multiplicity of solutions for the second-order differential equation ( p (t)u′)′+h(t)f(u)=0, 0 < t <1,with the boundary condition u ( 0)= u(1)=0. A generalized Prüfer transformation, Sturm Comparison Theorem and some new techniques, which are different from the ordinary reasonings, show the main results.The second part gives a lower bound for the number of solutions to the boundary value problem (φ(u′) )′+a(x)f(u)=0, u ( 0)= u(1)=0, whereφ(u )=up ?1u, 0 < x <1. The author establishs the condition concerning the behavior of the ratio f ( s)/φ(s) at infinity and zero for the existence of solutions with prescribed nodal properties. Then the author proves the multiplicity result for this problem by using shooting method and generalized Prüfer transformation Sturm's comparison theorem.