On a Decomposition Formula for the Resolvent Operator of the Sum of Two Set-Valued Maps with Monotonicity Assumptions

Abstract

The aim of the present work is to provide an explicit decomposition formula for the resolvent operator $$\mathrm {J}_{A+B}$$ of the sum of two set-valued maps A and B in a Hilbert space. For this purpose we introduce a new operator, called the A-resolvent operator of B and denoted by $$\mathrm {J}^A_B$$ , which generalizes the usual notion. Then, our main result lies in the decomposition formula $$\mathrm {J}_{A+B}=\mathrm {J}_A\circ \mathrm {J}^A_B$$ holding true when A is monotone. Several properties of $$\mathrm {J}^A_B$$ are deeply investigated in this paper. In particular the relationship between $$\mathrm {J}^A_B$$ and an extended version of the classical Douglas–Rachford operator is established, which allows us to propose a weakly convergent algorithm that computes numerically $$\mathrm {J}^A_B$$ (and thus $$\mathrm {J}_{A+B}$$ from the decomposition formula) when A and B are maximal monotone. In order to illustrate our theoretical results, we give an application in elliptic PDEs. Precisely the decomposition formula is used to point out the relationship between the classical obstacle problem and a new nonlinear PDE involving a partially blinded elliptic operator. Some numerical experiments, using the finite element method, are carried out in order to support our approach.