Generalized Inverse Operators and Fredholm Boundary-Value Problems

The problems of development of constructive methods for the analysis of linear and weakly nonlinear boundary-value problems for a broad class of functional differential equations traditionally occupy one of the central places in the qualitative theory of differential equations. The authors of this monograph suggest some methods for the construction of the generalized inverse (or pseudo-inverse) operators for the original linear Fredholm operators in Banach (or Hilbert) spaces for boundary-value problems regarded as operator systems in abstract spaces.

They also study basic properties of the generalized Green's operator. In the first three chapters some results from the theory of generalized inversion of bounded linear operators in abstract spaces are given, which are then used for the investigation of boundary-value problems for systems of functional differential equations.

Subsequent chapters deal with a unified procedure for the investigation of Fredholm boundary-value problems for operator equations; analysis of boundary-value problems for standard operator systems; and existence of solutions of linear and nonlinear differential and difference systems bounded on the entire axis.