Abstract differential equation

In mathematics, an abstract differential equation is a differential equation in which the unknown function and its derivatives take values in some generic abstract space (a Hilbert space, a Banach space, etc.). Equations of this kind arise e.g. in the study of partial differential equations: if to one of the variables is given a privileged position (e.g. time, in heat or wave equations) and all the others are put together, an ordinary "differential" equation with respect to the variable which was put in evidence is obtained. Adding boundary conditions can often be translated in terms of considering solutions in some convenient function spaces.

The classical abstract differential equation which is most frequently encountered is the equation^[1]

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=Au+f}$

where the unknown function ${\displaystyle u=u(t)}$ belongs to some function space ${\displaystyle X}$, ${\displaystyle 0\leq t\leq T\leq \infty }$ and ${\displaystyle A:X\to X}$ is an operator (usually a linear operator) acting on this space. An exhaustive treatment of the homogeneous (${\displaystyle f=0}$) case with a constant operator is given by the theory of C₀-semigroups. Very often, the study of other abstract differential equations amounts (by e.g. reduction to a set of equations of the first order) to the study of this equation.

The theory of abstract differential equations has been founded by Einar Hille in several papers and in his book Functional Analysis and Semi-Groups. Other main contributors were^[2] Kōsaku Yosida, Ralph Phillips, Isao Miyadera, and Selim Grigorievich Krein.^[3]