Hammerstein型非线性积分方程正解的个数

<正> 本文是作者工作[8]、[9]的继续.在[9]中作者利用Leray-Schauder拓扑度理论研究了多项式型Hammerstein非线性积分方程的固有值,即设

THE NUMBER OF POSITIVE SOLUTIONS OF HAMMERSTEIN NONLINEAR INTEGRAL EQUATIONS

In this paper we use the Leray-Schauder degree to investigate the number of positive solutions of Hammerstein integral equation where G is a bounded closed domain in Euclidean space R~N and f(x,u)=a_i>0(i=1,2,…, n).Theorem 1. Suppose that (i) the non-negative continuous kernel k(x,y) satisfies and among numbers a_i (i=1, 2,…, n) there exist a_(i_0)<1 and a_(i_1)>1 such that and Then equation (*) has at least two positive (i.e.≥0 and ) continuous solutions.Theorem 2. Let the hypothesis (i) of Theorem 1 be satisfied. Suppose that a_i(x) ≥0, a_i(x)∈L, a_i<1 (i=1,2,…, n) and among functions a_i(x) (i=1,2, …,n) there exists α_(i_0)(x) such that a_(i_0) Then equation (*) has at least one positive continuous solution.Theorem 3. Let the hypothesis (i) of Theorem 1 be satisfied. Suppose that α_i(x)≥0, α_i(x)∈L, α_i,>1 (i=1,2,…, n) and among functions α_i(x) (i=1,2,…,n) there exists α_(i_l)(x) such that a_(i_1) Then equation (*) has at least one positive continuous solution.If we assume, in addition, that there exists σ>0 such that,then equation (*) has exactly one positive continuous solution in the case of Theorem 2 and cannot have two comparable positive continuous solutions in the case of Theorem 3.The method mentioned above can also be used to establish the existence of positive solutions of the Dirichlet boundary value problem where L is a uniformly elliptic differential operator:Theorem 4. Let a_i(x)∈C~(o,λ)(Ω) (0 <λ<1),a_i(x)≥0 (i= 1,2,…, n) and among numbers a_i (i=1, 2,…,n) there exists a_(io)<1 such that a_(io)(x)>0 for all x∈ Ω. Suppose that where G(x,y) is the Green function for the Dirichlet problem on Ω of the operator L. Then the problem (**) has at least one solution u(x) such that u(x)∈C~(2,μ)(Ω) (μ=min{λ, α_1,…, α_n}) and 00 for all x∈Ω. Then the problem (**) has exactly one positive solution u(x)∈C~(2,μ)(Ω) (μ=min{λ,α_1,…,α_n}) and u(x)>0 for every x∈Ω.