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The Solutions of a Class of Hammerstein-type Integral Equations

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 Frechet derivative A′1θ are obtained by related conclusions of topological degree and fixed point index.