On generalized $ f $-projection and generalized $ f $-prox-regularity on Banach spaces

Ali Al-Tane; See Keong Lee; Messaoud Bounkhel; Ali Al-Tane; See Keong Lee; Messaoud Bounkhel

doi:10.3934/math.2026029

AIMS Mathematics

2026, Volume 11, Issue 1: 661-683. doi: 10.3934/math.2026029

Previous Article Next Article

Research article

On generalized $ f $-projection and generalized $ f $-prox-regularity on Banach spaces

1.
School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia
2.
Department of Mathematics, College of Science, King Saud University, P. O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 04 September 2025 Revised: 12 December 2025 Accepted: 19 December 2025 Published: 09 January 2026
MSC : 34A60, 49J53

This paper investigates the connection between the generalized projection $ \pi_S $ and the generalized $ f $-projection $ \pi_S^f $. By introducing an $ f $-proximal normal cone, defined via the generalized $ f $-projection, we derive key properties of this set, generalizing established results on the classical $ V $-proximal normal cone to Banach spaces. Additionally, we propose and examine a new regularity condition called uniform $ f $-prox-regularity, which extends the standard notion of prox-regularity by leveraging the adaptability of $ f $-projections.
Citation: Ali Al-Tane, See Keong Lee, Messaoud Bounkhel. On generalized $ f $-projection and generalized $ f $-prox-regularity on Banach spaces[J]. AIMS Mathematics, 2026, 11(1): 661-683. doi: 10.3934/math.2026029

Related Papers:

Abstract

This paper investigates the connection between the generalized projection $ \pi_S $ and the generalized $ f $-projection $ \pi_S^f $. By introducing an $ f $-proximal normal cone, defined via the generalized $ f $-projection, we derive key properties of this set, generalizing established results on the classical $ V $-proximal normal cone to Banach spaces. Additionally, we propose and examine a new regularity condition called uniform $ f $-prox-regularity, which extends the standard notion of prox-regularity by leveraging the adaptability of $ f $-projections.

References

[1]	Y. Alber, Metric and generalized projection operators in Banach spaces: Properties and applications, 1993. Available from: https://arXiv.org/abs/funct-an/9311001.
[2]	M. Bounkhel, R. Al-Yusof, Proximal Analysis in reflexive smooth Banach spaces, Nonlinear Anal.: Theory, Methods Appl., 73 (2010), 1921–1939. http://doi.org/10.1016/j.na.2010.04.077 doi: 10.1016/j.na.2010.04.077
[3]	M. Bounkhel, M. Bachar, Generalised-prox-regularity in reflexive smooth Banach spaces with smooth dual norm, J. Math. Anal. Appl., 475 (2019), 699–729. http://doi.org/10.1016/j.jmaa.2019.02.064 doi: 10.1016/j.jmaa.2019.02.064
[4]	M. Bounkhel, M. Bachar, Primal lower nice functions in smooth Banach spaces, Mathematics, 8 (2020), 2066. http://doi.org/10.3390/math8112066 doi: 10.3390/math8112066
[5]	M. Bounkhel, Regularity Concepts in Nonsmooth Analysis, Theory and Applications, Springer, 2012.
[6]	M. Bounkhel, Generalized Projections on closed nonconvex sets in uniformly convex and uniformly smooth Banach spaces, J. Function Spaces, 2015 (2015), 478437. http://doi.org/10.1155/2015/478437 doi: 10.1155/2015/478437
[7]	M. Bounkhel, Calculus rules for V-proximal subdifferentials in smooth Banach spaces, J. Function Spaces, 2016 (2016), 1917387. http://doi.org/10.1155/2016/1917387 doi: 10.1155/2016/1917387
[8]	M. Bounkhel, Generalized $(f, \lambda)$-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spaces, AIMS Mathematics, 8 (2023), 29555–29568. http://doi.org/10.3934/math.20231513 doi: 10.3934/math.20231513
[9]	M. Bounkhel, V-Moreau envelope of nonconvex functions on smooth Banach spaces, AIMS Mathematics, 9 (2024), 28589–28610. http://doi.org/10.3934/math.20241387 doi: 10.3934/math.20241387
[10]	F. H. Clarke, Optimization and nonsmooth analysis, New York: John Wiley & Sons Inc., 1983.
[11]	F. H. Clarke, Y. Ledyaev, R. Stern, R. Wolenski, Nonsmooth analysis and control theory, Springer, 2008.
[12]	R. Deville, G. Godefroy, V. Zizler, Smoothness and Renormings in Banach Spaces, Harlow: Longman Scientific Technical, 1993.
[13]	J. Diestel, Geometry of Banach Spaces-Selected Topics, Lecture Notes in Mathematics, vol. 485, Springer-Verlag, 1975.
[14]	J. Li, The generalized projection operator on reflexive Banach spaces and its applications, J. Math. Anal. Appl., 306 (2005), 55–71. http://doi.org/10.1016/j.jmaa.2004.11.007 doi: 10.1016/j.jmaa.2004.11.007
[15]	K. Wu, N. Huang, The generalised $f$-projection operator with an application, Bull. Austr. Math. Soc., 73 (2006), 307–317. http://doi.org/10.1017/S0004972700038892 doi: 10.1017/S0004972700038892
[16]	K. Wu, N. Huang, Properties of the generalized $f$-projection operator and its applications in Banach spaces, Comput. Math. Appl., 54 (2007), 399–406. http://doi.org/10.1007/BF03022673 doi: 10.1007/BF03022673

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)