M. Sofonea†, D.A. Tarzia‡
Abstract: We consider an abstract problem P in a metric space X which has a unique solution u ∈ X. Our aim in this current paper is two folds: first, to provide a convergence criterion to the solution of Problem P, that is, to give necessary and sufficient conditions on a sequence {un} ⊂ X which guarantee the convergence un → u in the space X; second, to find a Tykhonov triple T such that a sequence {un} ⊂ X is a T-approximating sequence if and only if it converges to u. The two problems stated above, associated to the original Problem P, are closely related. We illustrate how they can be solved in three particular cases of Problem P: a variational inequality in a Hilbert space, a fixed point problem in a metric space and a minimization problem in a reflexive Banach space. For each of these problems we state and prove a convergence criterion that we use to define a convenient Tykhonov triple T which requires the condition stated above. We also show how the convergence criterion and the corresponding T-well posedness concept can be used to deduce convergence and classical well-posedness results, respectively.
MSC: 49J40, 47H09, 47H10, 49J45.
keywords: convergence criterion, Tykhonov triple, well-posedness, projection operator, contraction mapping, minimization problem, unilateral constraint.
DOI 10.56082/annalsarscimath.2023.1-2.308
†sofonea@univ-perp.fr, Laboratoire de Mathématiques et Physique, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France;
‡DTarzia@austral.edu.ar, Departamento de Matemática, FCE, Universidad Austral, Paraguay 1950, S2000FZF Rosario, Argentina and CONICET, Argentina.
PUBLISHED in Annals Academy of Romanian Scientists Series on Mathematics and Its Application, Volume 15 no 1-2, 2023
