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Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems

Alakoya Timilehin Opeyemi, Jolaoso Lateef Olakunle, Mewomo Oluwatosin Temitope

Demonstratio Mathematica

2020

Vol. 53, nr 1

208--224

In this work, we introduce two new inertial-type algorithms for solving variational inequality problems (VIPs) with monotone and Lipschitz continuous mappings in real Hilbert spaces. The first algorithm requires the computation of only one projection onto the feasible set per iteration while the second algorithm needs the computation of only one projection onto a half-space, and prior knowledge o fthe Lipschitz constant of the monotone mapping is not required in proving the strong convergence theorems for the two algorithms. Under some mild assumptions, we prove strong convergence results for the proposed algorithms to a solution of a VIP. Finally, we provide some numerical experiments to illustrate the efficiency and advantages of the proposed algorithms.

The Nemitskij operator on Lipk-type and BVk-type spaces

Giménez J., López L., Merentes N.

Demonstratio Mathematica

2013

Vol. 46, nr 3

543--558

In this paper, we discuss and present various results about acting and boundedness conditions of the autonomous Nemitskij operator on certain function spaces related to the space of all real valued Lipschitz (of bounded variation, absolutely continuous) functions defined on a compact interval of R. We obtain a result concerning the integrability of products of the form (…) and a generalized version of the chain rule for functions a.e differentiable, in the sense of Lebesgue. As an application, we get a generalization of a theorem due to V. I. Burenkov for the case of functions of bounded Riesz-p-variation.