| Abstract: |
| We consider a minimization problem $\cal P$ in a reflexive Banach space for which we introduce the concept of $\cal T$-well posedness. It is based on a new mathematical object, the so-called Tykhonov triple, and it extends the classical Tykhonov and Levitin-Polyak well-posedness concepts. We provide several examples and counter-examples, then we state and prove various well-posedness results. The proofs are based on convexity, compactness and lower semicontinuous arguments. We also discuss the problem of describing an optimal $\cal T$-well-posedness concept. Next, we consider the case of differentiable functionals defined on a real Hilbert space. Under additional assumptions, we prove a convergence criteria, that is, we state necessary and sufficient condition that guarantee the convergence of any arbitrary sequence to the solution of problem $\cal P$, assumed to be unique. We apply these abstract results to prove the convergence of the solution of a penalty problem to the solution of the original minimization problem, as the penalty parameter converges to zero. |
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