Journal of Optimization Theory and Applications

, Volume 94, Issue 2, pp 425–448

# Finding a Zero of The Sum of Two Maximal Monotone Operators

• A. Moudafi
• M. Théra
Article

## Abstract

In this paper, the equivalence between variational inclusions and a generalized type of Weiner–Hopf equation is established. This equivalence is then used to suggest and analyze iterative methods in order to find a zero of the sum of two maximal monotone operators. Special attention is given to the case where one of the operators is Lipschitz continuous and either is strongly monotone or satisfies the Dunn property. Moreover, when the problem has a nonempty solution set, a fixed-point procedure is proposed and its convergence is established provided that the Brézis–Crandall–Pazy condition holds true. More precisely, it is shown that this allows reaching the element of minimal norm of the solution set.

Maximal monotone operators variational inequalities Weiner–Hopf equation Dunn property Yosida approximate regularization fixed-point methods proximal point algorithm Brézis–Crandall–Pazy condition

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• A. Moudafi
• 1
• 2
• M. Théra
• 3