Convergence of solutions of implicit differential equations (Q1328918)

The convergence of solutions of the implicit differential equations (1) \(d(B_ n y)(t)/dt+ (A_ n y)(t)= f_ n(t)\), \((B_ n y)(0)= z^ n_ 0\), \(t\in [0,T]\), is studied. Here \(\{A_ n\}\), \(\{B_ n\}\) are two families of operators in a Banach space convergent in the sense of graph to operators \(A\) and \(B\), respectively. Both the case of nonlinear subpotential operators \(A_ n\), \(B_ n\) and linear ones is investigated. The techniques used in considerations are highly nontrivial and involve such tools as maximal monotone operators, subdifferentials, semigroups of operators and Sobolev spaces, for example. Applications of the results obtained to nonlinear degenerate parabolic equations, to optimal control theory and to the theory of linear operators are discussed.