Remarks on compact pertubations of m-accretive operators

On compact perturbations and compact resolvents of nonlinear 𝑚-accretive operators in Banach spaces

“…Hirano [6, Theorem 2] gave a rather involved proof of his result, and several other results of his in [6] are dependent on that proof. The same is true for Morales's results in [13] in connection with Theorem 2 above. Our proof of Theorem 1 actually involves only the verification of the fact that a certain mapping is a homotopy of compact transformations.…”

“…This result improves, when C is uniformly continuous on bounded sets, a relevant result of Hirano [6,Theorem 2], a result of the author [8,Theorem 3] and the following theorem of Morales in [ 13] .…”

“…Since Morales considers in [13] only the possibility Bb(0) c R(T + C) or Bb(0) c R(T + C), our comments on his results refer to the conditions which are implied by his results to conclude that a single point p belongs to R(T + C) or R(T + C). Theorem 4 below provides an improvement of Corollary 1 and the Proposition in Morales [13].…”

“…Theorem 4 below provides an improvement of Corollary 1 and the Proposition in Morales [13]. In the Proposition of [13] T : X -* B(X) is continuous, where B(X) is the space of closed and bounded subsets of X associated with the Hausdorff metric. Also, a condition stronger than ours holds there on 3Bb(0).…”

“…Theorem 2 (Morales [13]). Let It is easy to see that condition (**) implies condition (*) but with the same j £ Jx working for all p £ Br(0).…”