Corrigendum | Open Access
Teffera M. Asfaw, "Corrigendum to “Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems”", Abstract and Applied Analysis, vol. 2017, Article ID 2739102, 1 page, 2017. https://doi.org/10.1155/2017/2739102
Corrigendum to “Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems”
In the article titled “Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems” , there was an error in Theorem . The operator is assumed to be linear, closed, densely defined, and monotone. However, it is required to replace this assumption on by the condition that is linear maximal monotone. It is known due to Brèzis (cf. Zeidler [2, Theorem . L, p.897]) that every linear maximal monotone operator is densely defined and closed. However, the converse is not generally true unless is monotone. In addition to conditions on in Theorem in , monotonicity assumption on (with ) is required. The condition for all is not required as it is automatically satisfied with because of monotonicity of and with . As a result, Theorem in  is restated and replaced by Theorem 1 as follows.
Theorem 1. Let be linear maximal monotone and be quasibounded demicontinuous and monotone of type with . Assume, further, that there exist and such that exactly (i) or (ii) of the following conditions holds.(i) for all .(ii)There exists such that as andThen is surjective.
The proof of Theorem 1 is completed by incorporating the following changes in the proof of Theorem in . For each , let denote the Yosida approximant of . It is well-known that is bounded, continuous, and monotone.
(a) In equation numbers (54) and (55), should be replaced with and should be replaced with in (55). In equation numbers (57), (58), (59), (60), (62), (63), and (64), should be replaced with .
(b) On lines numbers 8 and 9 from below (right column) on page 8, should be replaced with .
(c) The text on lines 1, 2, and 3 from below on page 8 (right column) should be deleted, Corollary and its proof in [1, p.9] should be deleted, and the text reading “The following corollary gives a characterization of linear maximal monotone operator in separable reflexive Banach space” should be deleted. In addition, the text reading “It is worth noticing that Brèzis proved (i) in arbitrary reflexive Banach space provided that is monotone and (ii) holds. As a result, Corollary is an improvement of the result of Brèzis when is separable” should be deleted.
In the abstract, the text reading “A new characterization of linear maximal monotone operator is given as a result of surjectivity of , where is of type with respect to ” should be deleted.
- T. M. Asfaw, “Noncoercive perturbed densely defined operators and application to parabolic problems,” Abstract and Applied Analysis, vol. 2015, Article ID 357934, 11 pages, 2015.
- E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer-Verlag, New York, NY, USA, 1990.
Copyright © 2017 Teffera M. Asfaw. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.