On the convergence of sequences of ordinary differential equations (Q1889679)

The complex differential expression \[ l(y)=\sum_{k=0}^\infty(-1)^{n-k}(p_ky^{(n-k)})^{n-k}, \quad 0\leq x\leq 1, \] with bounday condition \[ By^\wedge+Cy^\vee=0 \] is considered in this paper, where \(p_k\in C^{n-k}[0,1]\) with \(\operatorname {Re} p_0(x)>\varepsilon>0\) for all \(x\in [0,1], \) \(B\) and \(C\) are in \(M_{2n}(\mathbb C),\) \(y^\wedge \) and \(y^\vee \) are some kind of boundary data of \(y.\) The author shows that a singular differential expression of type (1) with condition (2) can be regarded as a limit of a regular differential expression of type (1) with condition (2) if the requirement \[ B^{-1}\operatorname {im}C= \mathbb C^{2n} \circleddash \operatorname {ker}C \] on the boundary condition is satisfied, where \(B^{-1}\) denotes the preimage of \(B.\)