On the convergence of a certain class of iterative procedures under relaxed conditions with applications
The paper deals with a locally unique solution \(x^*\) of the equation \(F(x)+ G(x)= 0\) in Banach spaces, where \(F\) is Fréchet differentiable and \(G\) is continuous. Inexact Newton-like methods of the form \[ x_{n+1}= x_n- A^{-1}(F(x_n)+ G(x_n))- z_n,\quad z_n= z(x_n)\tag{1} \] are studied. At each step the inverse of the same linear operator is used. The vectors \(z_n\) are chosen so that iteration \(\{x_n\}\) converges to \(x^*\). For \(A= F'(x_0)\), \(G(x)= 0\) and \(z_n= 0\) for all \(n\) one obtains the modified Newton's method. The iterative procedure is studied under natural assumptions. The assumed Ptak-like conditions are weaker than earlier ones. A semilocal convergence result is presented based on the contraction mapping principle. The result is applied to solve special nonlinear integral equation of Uryson type with the integral operator \[ [F(x)](t)= x(t)- \int^1_0 p_1(t) p_2(s)p_3(x(s)) ds \] appearing in elasticity theory. This equation pertains to the buckling of a thin shallow spherical shell clamped at the edge and under uniform external pressure.
nonlinear integral equation of Uryson type
scientific article; zbMATH DE number 1326394