摘要
迁移理论中的一类非线性积分方程u(x)=ψ(x)+au(x)∫~1_0(k(x)u(y))/(k(x)+k(y))dy(*),其中ψ∈L’,ψ(x)≥0;0≤k(x)≤1,k∈L’且mes{x│k(x)=c}=0,c∈[0,1];α为正的实参数。本文的主要结果是下面的定理。 定理、方程(*)在L’空间里有且仅有两个非负解的充要条件是:0<a‖ψ‖<1/2而且存在β>0使∫~1_0(ψ(x))/(1-β~2k^2(x))dx=(2a)^-。
We consider the nonlinear integral equationarising in the transport theory, where ψ∈L',ψ(x)≥0; k ∈L' , 0≤k(x)≤1 and mes {X|k(x)=c| =0 c ∈[0,1] and a is a positive parameter. Our main result is the following theorem.Theorem. The equation has two non-negative solutions in Lt if and only if 0 α||ψ||<1/2 and there exists β>0 such that
