一类奇异三阶微分方程的两点边值问题的正解

Positive Solutions to a Class of Two-point Boundary Value Problems of Singular Third-order Ordinary Differential Equations

考察了三阶两点边值问题u^(?)(t)+f(t,u(t))=0,0<t<1,u′(0)=u″(0)=u(1)=0的正解,其中非线性项f(t,u)可以在t=0,t=1及u=0处奇异.利用锥压缩与锥拉伸型的Guo-Krasnosel'skii不动点定理证明了正解的存在性与多解性.结论表明正解存在性依赖于非线性项的连续部分在某些有界集上的"高度".

The positive solutions are considered for the third-order two-point boundary value problem u^m（t）＋f（t,u（t）） =0,0t1,u＇（0）=u＂（0）=u（1） = 0,where the nonlinear term may be singular at t=0,t=1 and u=0.By applying the Guo-Krasnosel＇skii fixed point theorem of cone expansion-compression type,the existence and multiplicity of positive solutions are proved.The results show that the existence of positive solutions depends upon the ＂heights＂ of the continuous part of nonlinear term on some bounded sets.