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Let $X, Y$ be Banach spaces and $T : X \to Y$ be a bounded linear operator. In this paper, we initiate the study of the perturbation problems for bounded homogeneous generalized inverse $T^h$ and quasi--linear projector generalized inverse $T^H$ of $T$. Some applications to the representations and perturbations of the Moore--Penrose metric generalized inverse $T^M$ of $T$ are also given. The obtained results in this paper extend some well--known results for linear operator generalized inverses in this field., Comment: 14 pages

In this paper, the problems of perturbation and expression for the Moore--Penrose metric generalized inverses of bounded linear operators on Banach spaces are further studied. By means of certain geometric assumptions of Banach spaces, we first give some equivalent conditions for the Moore--Penrose metric generalized inverse of perturbed operator to have the simplest expression $T^M(I+ \delta TT^M)^{-1}$. Then, as an application our results, we investigate the stability of some operator equations in Banach spaces under different type perturbations., Comment: 15 pages

In this paper, we first study the perturbations and expressions for the generalized inverses $a^{(2)}_{p,q}$, $a^{(1, 2)}_{p,q}$, $a^{(2, l)}_{p,q}$ and $a^{(l)}_{p,q}$ with prescribed idempotents $p$ and $q$. Then, we investigate the general perturbation analysis and error estimate for some of these generalized inverses when $p,\,q$ and $a$ also have some small perturbations., Comment: 18 pages

In this paper, we investigate the various different generalized inverses in a Banach algebra with respect to prescribed two idempotents $p$ and $q$. Some new characterizations and explicit representations for these generalized inverses, such as $a^{(2)}_{p,q}$, $a^{(1,2)}_{p,q}$ and $a^{(2,l)}_{p,q}$ will be presented. The obtained results extend and generalize some well--known results for matrices or operators., Comment: 17 pages