Inner-Φ-GCEP and Φ-GCEP-inner inverses

Abstract

Inspired by the concepts of Φ-GCEP and Φ-*GCEP inverses as generalizations of the gMP, *gMP and MP inverse, the aim of this paper is to extend notions of inner-gMP, gMPinner, inner-*gMP, *gMP-inner, 1MP and MP1 inverses using Φ-GCEP and Φ-*GCEP inverses. Thus, four new types of generalized inverses are introduced and called inner- Φ-GCEP, Φ-GCEP-inner, inner-Φ-*GCEP and Φ-*GCEP-inner inverses. As particular kinds of these new inverses inner-GCEP, GCEP-inner, inner-*GCEP and *GCEP-inner inverses are established. Numerous properties, representations and characterizations of new inverses are presented as well. Applying our new inverses, we solve some linear equations and express their solutions.