View source for 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]].

====Comments====
The term  "well-posed"  (also properly posed or correctly set) was coined by the French mathematician J. Hadamard at the beginning of the 19th century [[#References|[a1]]]. In particular, he stressed the importance of continuous dependence of solutions on the data (i.e. property 3). Practical problems (e.g. in hydrodynamics, seismology) lead not seldom to formulations that are ill-posed (cf. [[Ill-posed problems]]).

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Hadamard,   "Lectures on Cauchy's problem in linear partial differential equations" , Dover, reprint  (1952)  (Translated from French)  {{ZBL|0049.34805}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Garabedian,   "Partial differential equations" , Wiley  (1964)  {{ZBL|0124.30501}}</TD></TR>
</table>

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