Direct gradient method for nonlinear integral equations (Q1375073)

Direct application of the gradient method to nonlinear integral equations in the Hilbert space \(L^2\) is described. It involves the generalization of the gradient method to the nonlinear case together with providing fast convergence with the quotient \((M-m)/M\). The proof by \textit{J. Necas} [``Introduction to the theory of nonlinear elliptic equations'' (1983; Zbl 0526.35003)] is modified by choosing the descent parameter as by \textit{L.V. Kantorovich} [Dokl. Akad. Nauk SSSR 48, 483-487 (1945)] for the linear case. Thus the optimal quotient \((M-m)/(M+m)\) is attained. A numerical realization is given on the equation \(F(u)(x):=u(x)+\int_\Omega K(x,y,u(x),u(y))d\lambda(y)=g(x)\) which allows to avoid approximate integration in the steps of the iteration.