On solution sets of nonlinear equations with nonsmooth operators in Hilbert space and the quasi–solution method

Abstract

We investigate nonlinear irregular equations in Hilbert space with a priori constraints. The differentiability of the problem's operator is not assumed. The constraints are described by a bounded closed set D that is part of an extended source representation class in terms of a given linear operator. The unique solvability of the problem is not assumed. It is established that solutions to the problem form a cluster of diameter ρdiam(D) with ρ∈(0,1). The value ρ depends on the problem parameters and can be arbitrarily small. We also justify the approximation properties of the quasi–solution method in relation to the solution set of the original problem.