一类Hammerstein型积分方程的解

The Solutions of a Class of Hammerstein-type Integral Equations

利用变分方法,在Hilbert空间中,研究了一类带正定核的Hammerstein型积分方程φ(x)=∫Gk(x,y)f(y,φ(y))dy=Aφ解的存在性问题,通过对涅梅茨基算子fφ=f(x,φ(x))加条件,利用它的拟可加性,证明了泛函Φ(ψ)=21‖ψ‖2-Ψ(Hψ)具有强制性,根据已有结论证明了泛函临界点的存在性,进而等价地得到了积分方程解的存在性.进一步,利用拓扑度及不动点指数的相关结论,得到了算子A1=H*fH及其Fréchet导数A1'θ不动点的存在性.

In this paper,by using the variational method,we discuss the existence of the solutions of a class of Hammerstein-type integral equations with positive kernel φ（x）=∫G k（x,y）f（y,φ（y））dy=Aφ in Hilbert spaces.By adding some conditions to Немвщкий operator fφ=f（x,φ（x）） and using its quasiadditive property,the coercivity of the function Φ（ψ）=1 2‖ ψ‖2-Ψ（Hψ） is obtained.Therefore,we prove the existence of critical points of the function,which is equivalent to the existence of the solutions of the integral equation.Moreover,the existences of fixed points of operator A1=H＊fH and its Fréchet derivative A′1θ are obtained by related conclusions of topological degree and fixed point index.