We consider an abstract problem PK in the framework of a normed space, associated to a set of constraints K. We use the concept of T-well-posedness to deduce various convergence results to the unique solution of PK. We illustrate our abstract results in the study two representative nonlinear problems: a minimization problem and a second kind elliptic variational inequality, both in the framework of reflexive Banach spaces. Finally, we consider a static contact model with elastic materials which leads to a weak formulation in terms of a variational inequality for the displacement field. We specify the corresponding convergence results, give their numerical validation and provide various mechanical interpretations.