Applications of Generalized Monotonicity to Variational-Like Inequalities and Equilibrium Problems

Author(s):  
N. K. Mahato ◽  
R. N. Mohapatra
2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
N. K. Mahato ◽  
C. Nahak

We introduce the notion of relaxed (ρ-θ)-η-invariant pseudomonotone mappings, which is weaker than invariant pseudomonotone maps. Using the KKM technique, we establish the existence of solutions for variational-like inequality problems with relaxed (ρ-θ)-η-invariant pseudomonotone mappings in reflexive Banach spaces. We also introduce the concept of (ρ-θ)-pseudomonotonicity for bifunctions, and we consider some examples to show that (ρ-θ)-pseudomonotonicity generalizes both monotonicity and strong pseudomonotonicity. The existence of solution for equilibrium problem with (ρ-θ)-pseudomonotone mappings in reflexive Banach spaces are demonstrated by using the KKM technique.


Author(s):  
Habib ur Rehman ◽  
Aviv Gibali ◽  
Poom Kumam ◽  
Kanokwan Sitthithakerngkiet

2020 ◽  
Vol 10 (1) ◽  
pp. 450-476
Author(s):  
Radu Ioan Boţ ◽  
Sorin-Mihai Grad ◽  
Dennis Meier ◽  
Mathias Staudigl

Abstract In this work we investigate dynamical systems designed to approach the solution sets of inclusion problems involving the sum of two maximally monotone operators. Our aim is to design methods which guarantee strong convergence of trajectories towards the minimum norm solution of the underlying monotone inclusion problem. To that end, we investigate in detail the asymptotic behavior of dynamical systems perturbed by a Tikhonov regularization where either the maximally monotone operators themselves, or the vector field of the dynamical system is regularized. In both cases we prove strong convergence of the trajectories towards minimum norm solutions to an underlying monotone inclusion problem, and we illustrate numerically qualitative differences between these two complementary regularization strategies. The so-constructed dynamical systems are either of Krasnoselskiĭ-Mann, of forward-backward type or of forward-backward-forward type, and with the help of injected regularization we demonstrate seminal results on the strong convergence of Hilbert space valued evolutions designed to solve monotone inclusion and equilibrium problems.


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