Positive Solutions Of Nonlinear Singular Third-order Boundary Value Problems

Recently, the existence of positive solutions of nonlinear singular boundary value problems attracts close attention. In most work, authors study the existence of positive solutions for third-order multi-point boundary value problems by the method of upper and lower solutions, Schauder’s fixed point theorem or the fixed point index in cone. The first eigenvalue and the spectral radius of linear completely continuous operators are very important and significant indexes. In chapter2, we devote ourselves to give the eigenvalue criteria for the existence of positive solutions of nonlinear singular third-order three-point boundary value problems. That is, the nonlinear function is given the conditions concerning the eigenvalue corresponding to the relevant linear operator.In chapter2, the nonlinear singular third-order three-point boundary value problem is considered under some conditions concerning the first eigenvalue corresponding to the relevant linear operator, where and h(t) is allowed to be singular at t=0and t=1and f(s) may be singular at s=0,include lim f(s)=+∞and lim f(s) is nonexistent. The existence results of positive solutions are given by using the method of topological degree.In chapter3, the existence of positive solutions of the singular third-order two-point boundary value problem is discussed. The result is given by fixed point theorem concerning cone expansion and compression of norm type. Where andh(t) may be singular at t=0and t=land f(t,u) may be singular at t=0, t=1and u=0.