Generalized inverse operators and Fredholm boundary-value problems. Translated from the Russian by P. V. Malyshev (Q263396)

This monograph covers extensively the theory of generalized inverse operators and Fredholm boundary value problems (BVPs), from the basic notions of functional analysis to the applications to various kind of differential equations, including systems of ordinary differential equations, equations with delay and equations with impulses.NEWLINENEWLINEThe authors investigate problems of Fredholm type, namely, BVPs in which the main linear part is an operator without inverse. In particular, they focus on the case of Fredholm linear operators of nonzero index, corresponding to either underdetermined or overdetermined BVPs, for which they present the notion of generalized inverse operators. In this way, the authors describe necessary as well as sufficient conditions for the existence of solutions of the investigated BVPs, both in resonance and in nonresonance. Moreover, in some cases, iterative schemes are obtained yielding the solutions analytically.NEWLINENEWLINEThis second edition presents, in addition to the contents of the first one [Utrecht: VSP (2004; Zbl 1083.47003)], some recent results for differential-algebraic systems with singular matrix of derivatives and new results for delay differential systems. More than 30 references have been added to the bibliography.