非线性奇异Hammerstein积分方程的正解

Positive solutions for nonlinear singular integral equations of Hammerstein type

利用锥压缩和锥拉伸不动点定理研究下列非线性奇异Hammerstein积分方程正解及多重正解的存在性u(t)=∫_0~1k(t,s)a(s)f(s,u(s))ds其中f∈C([0,1]×R^+,R^+),a∈L(0,1),a在[0,1]上可奇异且非负,满足∫_0~1a(t)dt>0, k∈C([0,1]×[0,1],R^+).非线性项f的超线性和次线性增长条件都是用线性积分算子的第一特征值刻画的,从而本质推广了和改进了现有文献的结果.作为应用,还讨论了一个二阶奇异Sturm-Liouville问题的正解及多重正解的存在性问题.

By using the fixed point theorem of compression and expansion type on a cone,this paper studies the existence and multiplicity of positive solutions for nonlinear singular integral equation of the form u（t） =∫_0^1 k（t,s）a（s）f（s,u（s））ds where f∈C（[0,1]×R^＋,R^＋）,a∈L（0,1） may be singular and is nonnegative on[0,1]with∫_0^1 a（t）dt0,and k∈C（[0,1]×[0,1],R^＋）.We use the first eigenvalue of an associated linear integral operator to characterize growth behaviors of the nonlinearity f both for the superlinear case and for the sublinear case,extending and improving the results in the existing literature essentially.As applications,these results are used to discuss the existence and mutiplicity of positive solutions for a second-order singular Sturm-Liouville problem.