Michel Théra

Monotonicity of pairs of operators is an extension of monotonicity of operators, which plays an important role in solving non-monotone inclusions. One of challenging problems in this new tool is how to design the associated mappings to obtain the monotone pairs. In this paper, we solve this problem and propose a Generalized Inertial Proximal Point Algorithm (GIPPA) using warped resolvents under the monotonicity of pairs. The weak, strong and linear convergence of the algorithm under some mild assumptions are established. We also provide numerical examples illustrating the implementability and effectiveness of the proposed method.

Error bounds are central objects in optimization theory and its applications. They were for a long time restricted only to the theory before becoming over the course of time a field by itself. This paper is devoted to the study of error bounds of a general inequality defined by a proper lower semicontinuous function on an Asplund space. If one drops the convexity assumption, the dual characterization of error bounds for a general inequality may not be valid, but even in this case, several dual necessary conditions are still obtained in terms of Fréchet/Mordukhovich subdifferentials of the concerned function at points in the solution set. Moreover, for an inequality defined by a convex-composite function that is to say by a function which is the composition of a convex function with a smooth mapping, such dual conditions also turn out to be sufficient to have the error bound property. Our work is an extension of the results on dual characterizations of convex inequalities to the possibly non-convex case.

The authors give an overview of the mathematical life of Hedy Attouch (1949–2023).

The current paper addresses non-local results on approximate and fixed points theorems for set-valued mappings in the context of symmetric $ (q_1, q_2) $-quasi-metric spaces, considering the regularity of the mappings involved. From these results, we derive non-local coincidence and double fixed point theorems for a pair of set-valued mappings, as well as non-local (Hölder) approximate, Milyutin, and Lyusternik-Graves theorems, along with (Hölder) approximate and implicit multifunction theorems in the same context. Our results extend those recently presented in the literature [ 1 , 25 , 23 , 2 , 26 ], which encompass classical results such as Nadler's multi-valued fixed point theorem [ 22 ] and Dontchev-Rockafellar's fixed point theorem [ 11 ].