Monotonic iterative processes for operator equations in semiordered spaces (Q1385910)

We investigate the nonlinear operator equation of the first kind \(A(x)=y\). Generally speaking, it does not satisfy the Hadamard conditions of correctness; thus, it is possible that the inverse operator \(A^{-1}\) does not exist or is discontinuous. Assume that \(A\) is an isotonic operator or an antitonic operator that acts in a semiordered \(B\)-space \(X\), in which the relation of order is defined by a regular cone \(K\). The purpose of this work is to investigate convergence of explicit and implicit iterative processes for approximate data, as well as for exact ones, which may be either explicit (the method of a simple iteration) or implicit (the Newton-Chaplygin method); that is, we deal with constructing methods for the iterative regularization of the original ill-posed problem. For explicit and implicit three-parametric iterative schemes the following alternative was proved earlier: either iterations converge, or the initial approximation is made more precise. In this work, instead of the alternative, we establish that iterations converge monotonically for some versions of the above-mentioned schemes and show that the Volterra equations form a natural class of nonlinear equations that satisfy the conditions of theorems of convergence.