# Difference between revisions of "Well-posed problem"

The problem of determining a solution $z=R(u)$ in a metric space $Z$ (with distance $\rho_Z({\cdot},{\cdot})$) from initial data $u$ in a metric space $U$ (with distance $\rho_U({\cdot},{\cdot})$), satisfying the following conditions: a) for any $u \in U$ there exists a solution $z \in Z$; b) the solution is uniquely defined; c) the problem is stable with respect to the spaces $(Z,U)$: For any $\epsilon>0$ there exists a $\delta(\epsilon)>0$ such that, for any $u_1,u_2 \in U$, the inequality $\rho_U(u_1,u_2) < \delta(\epsilon)$ implies $\rho_Z(z_1,z_2) < \epsilon$, where $z_1 = R(u_1)$, $z_2 = R(u_2)$.

Problems not satisfying one of these conditions for well-posedness are called ill-posed problems.