Generation theorem of semigroup for multivalued linear operators
The paper studies a multivalued version of the Hille-Yosida generation theorem of linear semigroup in a Banach space \(X\). The generation theorem is applied to abstract equations of the following three types \[ d(Mv)/dt + Lv= f(t),\quad 0\leq t\leq T,\quad Mv(0)= u_ 0,\tag{D.E.1} \] \[ M^* d(Mv)/dt+ Lv= M^* f(t),\quad 0\leq t\leq T,\quad Mv(0)= u_ 0,\tag{D.E.2} \] \[ Mdu/dt+ Lu= Mf(t),\quad 0\leq t\leq T,\quad Mu(0)= u_ 0.\tag{D.E.3} \] Here \(M\) and \(L\) denote linear operators in \(X\), \(f(t)\) denotes a given function, \(u_ 0\) is an initial value, and \(u\) and \(v\) are unknown functions. Changing the unknown functions to \(u= Mv(t)\), every equation is reduced to the form \[ du/dt+ Au = f(t),\quad 0\leq t\leq T,\quad u(0)= u_ 0,\tag{E} \] where \(A\) is a multivalued operator (\(A= LM^{-1}\), \(A= (M^*)^{-1} LM^{-1}\), \(A= M^{-1} L\), respectively). The paper investigates, in each case, sufficient conditions on \(M\) and \(L\) in order that the multivalued operator \(A\) generates a semigroup on \(X\) and (E) has a strict solution in \(X\). The paper gives also some examples of differential equations of degenerate type which can be studied with these methods.
scientific article; zbMATH DE number 59008