Abstract differential equation

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an abstract differential equation is a <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a> in which the unknown <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 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 <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equations</a>: if to one of the variables is given a privileged position (e.g. time, in <a href="/facts/Heat_equation/2JPPr5GG">heat</a> or <a href="/facts/Wave_equation/J2ksD2Az">wave</a> 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 <a href="/facts/Boundary_condition/QEfmUqmP">boundary conditions</a> 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

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

where the unknown function 
 
 
 
 u
 =
 u
 (
 t
 )
 
 
 {\displaystyle u=u(t)}
 
 belongs to some <a href="/facts/Function_space/lLUE2R1t">function space</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
, 
 
 
 
 0
 ≤
 t
 ≤
 T
 ≤
 ∞
 
 
 {\displaystyle 0\leq t\leq T\leq \infty }
 
 and 
 
 
 
 A
 :
 X
 →
 X
 
 
 {\displaystyle A:X\to X}
 
 is an <a href="/facts/Operator_(mathematics)/1VWgEdjS">operator</a> (usually a linear operator) acting on this space. An exhaustive treatment of the homogeneous (
 
 
 
 f
 =
 0
 
 
 {\displaystyle f=0}
 
) case with a constant operator is given by the theory of <a href="/facts/C0-semigroup/GrcTHhj0">C0-semigroups</a>. 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 <a href="/facts/Einar_Hille/56qNTKUA">Einar Hille</a> in several papers and in his book Functional Analysis and Semi-Groups. Other main contributors were <a href="/facts/K%C5%8Dsaku_Yosida/SPeekQPq">Kōsaku Yosida</a>, <a href="/facts/Ralph_S._Phillips/JW4EJ7v3">Ralph Phillips</a>, Isao Miyadera, and Selim Grigorievich Krein.

Abstract differential equation open-in-new

Abstract differential equation