Multipoint boundary value problems for transferable differential-algebraic equations. II: Quasilinear case
For part I see [ibid. 25, No. 4, 347-358 (1997; Zbl 0937.34016)]. The paper is the second part of the author's work on multipoint boundary value problems (BVPs) for differential-algebraic equations (DAEs). In this part the problem of solvability and approximate solution of BVPs for quasilinear DAEs of the form \[ Lx:=A(t)x'+ B(t)x=f(x,t), t\in J:=[t_0,T], \quad \Gamma x:=\int^T_{t_0} d\eta(t) x(t)=\gamma, \] are considered, where \(A,B\in C (J, \mathbb{R}^{n\times n})\) are continuous matrix-valued functions, \(\eta\in BV(J, \mathbb{R}^{n\times n})\) is a matrix-valued function of bounded variation, \(\gamma \in\mathbb{R}^n\), and \(f:\mathbb{R}^n\times \mathbb{R}^1\to\mathbb{R}^n\) is a nonlinear vector-valued function. If the corresponding linear multipoint BVPs for transferable DAEs are regular, then under certain hypotheses, the Schauder fixed-point principle ensures the solvability of multipoint BVPs for quasilinear DAEs. Otherwise, in irregular cases, a Tikhonov iterative regularization process is implemented for finding approximate solutions to quasilinear multipoint BVPs.
quasilinear differential-algebraic equations
scientific article; zbMATH DE number 1485047