Convergence criteria, well-posedness concepts and Applications

SOFONEA, MIRCEA; D. A. TARZIA

Annals - Series on Mathematics and its applications

Academy of Romanian Scientits

Año: 2023

We consider an abstract problem $cP$ in a metric space $X$ which has a unique solution $uin X$. For such a problem the concept if well-posedness with a Tyknonov triple was introduced in cite{SX16}.Our aim in this current paper is two folds: first, to provide a convergence criterion to the solution of Problem $cP$, that is, to give necessary and sufficient conditions on a sequence ${u_n}subset X$ which guarantee the convergence $u_no u$ in the space $X$; second, to find a Tyknonov triple $cT$ such that a sequence ${u_n}subset X$ is a $cT$-approximating sequence if and only if it converges to $u$. The two problems stated above, associated to the original Problem $cP$, are closely related. We illustrate how they can be solved in three particular cases of Problem $cP$: 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 $cT$ which requires the condition stated above.We also show how the convergence criterion and the corresponding $cT$-well posedness concept can be used to deduce convergence and classical well-posedness results, respectively.