2020
DOI: 10.3390/math8112066
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Primal Lower Nice Functions in Reflexive Smooth Banach Spaces

Abstract: In the present work, we extend, to the setting of reflexive smooth Banach spaces, the class of primal lower nice functions, which was proposed, for the first time, in finite dimensional spaces in [Nonlinear Anal. 1991, 17, 385–398] and enlarged to Hilbert spaces in [Trans. Am. Math. Soc. 1995, 347, 1269–1294]. Our principal target is to extend some existing characterisations of this class to our Banach space setting and to study the relationship between this concept and the generalised V-prox-regularity of the… Show more

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Cited by 3 publications
(3 citation statements)
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“…Unfortunately, we do not obtain the generalized V -prox-regularity of the epigraph in the sense of Definition 2 since epi f is a generalized V -prox-regularity at ðx 0 , f ðx 0 ÞÞ if and only if it is V -prox-regularity at ðx 0 , f ðx 0 ÞÞ for ð0, 0Þ ∈ N π ðepi f ; ðx 0 , f ðx 0 ÞÞ which cannot be the case (since ðx * 0 ,−1Þ ≠ ð0, 0Þ) under the V-prox-regularity of f at x 0 . To ensure the generalized V -prox-regularity of the epigraph, we need another kind of regularity called the V-primal lower nice function introduced and studied in [5].…”
Section: Journal Of Function Spacesmentioning
confidence: 99%
See 1 more Smart Citation
“…Unfortunately, we do not obtain the generalized V -prox-regularity of the epigraph in the sense of Definition 2 since epi f is a generalized V -prox-regularity at ðx 0 , f ðx 0 ÞÞ if and only if it is V -prox-regularity at ðx 0 , f ðx 0 ÞÞ for ð0, 0Þ ∈ N π ðepi f ; ðx 0 , f ðx 0 ÞÞ which cannot be the case (since ðx * 0 ,−1Þ ≠ ð0, 0Þ) under the V-prox-regularity of f at x 0 . To ensure the generalized V -prox-regularity of the epigraph, we need another kind of regularity called the V-primal lower nice function introduced and studied in [5].…”
Section: Journal Of Function Spacesmentioning
confidence: 99%
“…Another proximal subdifferential ∂ π G f ðxÞ is defined (see [5]) geometrically via the V-proximal normal cone of the epi-graph as follows:…”
Section: Introductionmentioning
confidence: 99%
“…In 2023, Al-Omari, Acharjee, and Özkoç [26] defined and studied operator Ψ by using primal topological spaces as Ψ(L) = X − (X − L) ♢ . The authors in [27] expand the class of primal lower pleasant functions to the setting of reflexive smooth Banach spaces. Furthermore, generalized primal topological spaces are a new category of generalized topology that Al-Saadi and Al-Malki recently introduced with the concept of the primal [28].…”
Section: Introductionmentioning
confidence: 99%