Method of monotonization of nonlinear equations in Banach spaces (Q1119895)

Let E be a Banach space with a proper cone, A:E\(\to E\) be a linear bounded operator, and F:E\(\to E\) be a continuous operator. The author suggests a method of two-sided successive approximations for the equation \(u=AF(u)\) of the following form \[ u^{n+1}=(I+\lambda A)^{-1}A(F(u^ n)+\lambda u^ n),\quad u^ 0=\sigma^+,\quad u_{n+1}=(I+\lambda A)^{-1}A(F(u_ n)+\lambda u_ n),\quad u_ 0=\sigma^-, \] where \(<\sigma^-,\sigma^+>\) is a cone segment on which the operator F is \(\lambda\) I-monotonic.