CONVERGENCE CRITERIA, WELL-POSEDNESS CONCEPTS AND APPLICATIONS

Abstract

We consider an abstract problem P in a metric space X which has a unique solution u G 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} C X which guarantee the convergence un ^ u in the space X; second, to find a Tyknonov triple T such that a sequence {un} C 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.