Abstract
Solving some systems of operator equations, new kinds of generalized inverses are introduced.
 Since these new inverses can be expressed by inner and gMP inverses, they are called inner-gMP and gMP-inner inverses.
 In this way, the concepts of gMP, 1MP and MP1 inverses are generalized. Various representations and characterizations of inner-gMP and gMP-inner inverses are presented. Using the inner and *gMP inverse, we define the inner-*gMP and *gMP-inner inverses which are new extensions of 1MP, MP1 and *gMP inverses.
 We apply inner-gMP and gMP-inner inverses as well as inner-*gMP and *gMP-inner inverses to solve several kinds of linear equations. Consequently, we obtain solvability of the normal equation which is connected to the least--squares solution. Numerical examples are given to illustrate our results.
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