Generalized $ (f, \lambda) $-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spaces

Messaoud Bounkhel; Messaoud Bounkhel

doi:10.3934/math.20231513

AIMS Mathematics

2023, Volume 8, Issue 12: 29555-29568. doi: 10.3934/math.20231513

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Generalized $ (f, \lambda) $-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spaces

Messaoud Bounkhel ^,

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh11451, Saudi Arabia

Received: 25 August 2023 Revised: 18 October 2023 Accepted: 26 October 2023 Published: 01 November 2023
MSC : 34A60, 49J53

In this paper, we expanded from the convex case to the nonconvex case in the setting of reflexive smooth Banach spaces, the concept of the $ f $-generalized projection $ \pi^{f}_S:X^*\to S $ initially introduced for convex sets and convex functions in ^[19,20]. Indeed, we defined the $ (f, \lambda) $-generalized projection operator $ \pi^{f, \lambda}_S:X^*\to S $ from $ X^* $ onto a nonempty closed set $ S $. We proved many properties of $ \pi^{f, \lambda}_S $ for any closed (not necessarily convex) set $ S $ and for any lower semicontinuous function $ f $. Our principal results broaden the scope of numerous theorems established in ^[19,20] from the convex setting to the nonconvex setting. An application of our main results to solutions of nonconvex variational problems is stated at the end of the paper.
Citation: Messaoud Bounkhel. Generalized $ (f, \lambda) $-projection operator on closed nonconvex sets and its applications in reflexive smooth Banach spaces[J]. AIMS Mathematics, 2023, 8(12): 29555-29568. doi: 10.3934/math.20231513

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Abstract

In this paper, we expanded from the convex case to the nonconvex case in the setting of reflexive smooth Banach spaces, the concept of the $ f $-generalized projection $ \pi^{f}_S:X^*\to S $ initially introduced for convex sets and convex functions in ^[19,20]. Indeed, we defined the $ (f, \lambda) $-generalized projection operator $ \pi^{f, \lambda}_S:X^*\to S $ from $ X^* $ onto a nonempty closed set $ S $. We proved many properties of $ \pi^{f, \lambda}_S $ for any closed (not necessarily convex) set $ S $ and for any lower semicontinuous function $ f $. Our principal results broaden the scope of numerous theorems established in ^[19,20] from the convex setting to the nonconvex setting. An application of our main results to solutions of nonconvex variational problems is stated at the end of the paper.

References

[1]	Y. Alber, I. Ryazantseva, Nonlinear Ill-posed problems of monotone type, Dordrecht: Springer, 2006. http://dx.doi.org/10.1007/1-4020-4396-1
[2]	Y. Alber, Generalized projection operators in Banach spaces: properties and applications, arXiv: funct-an/9311002.
[3]	Y. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, arXiv: funct-an/9311001.
[4]	H. Ben-El-Mechaiekh, W. Kryszewski, Equilibria of set-valued maps on nonconvex domains, Trans. Amer. Math. Soc., 349 (1997), 4159–4179.
[5]	M. Bounkhel, Regularity concepts in nonsmooth analysis, theory and applications, New York: Springer, 2012. http://dx.doi.org/10.1007/978-1-4614-1019-5
[6]	M. Bounkhel, generalized Projections on closed nonconvex sets in uniformly convex and uniformly smooth Banach spaces, J. Funct. Space., 2015 (2015), 478437. http://dx.doi.org/10.1155/2015/478437 doi: 10.1155/2015/478437
[7]	M. Bounkhel, M. Bachar, Generalized prox-regularity in reflexive Banach spaces, J. Math. Anal. Appl., 475 (2019), 699–729. http://dx.doi.org/10.1016/j.jmaa.2019.02.064 doi: 10.1016/j.jmaa.2019.02.064
[8]	M. Bounkhel, L. Tadj, A. Hamdi, Iterative schemes to solve nonconvex variational problems, Journal of Inequalities in Pure and Applied Mathematics, 4 (2003), 1–14.
[9]	M. Bounkhel, Dj. Bounekhel, Iterative schemes for nonconvex quasi-variational problems with V-prox-regular data in Banach spaces, J. Funct. Space., 2017 (2017), 8708065. http://dx.doi.org/10.1155/2017/8708065 doi: 10.1155/2017/8708065
[10]	J. Chen, A. Pitea, L. Zhu, Split systems of nonconvex variational inequalities and fixed point problems on uniformly prox-regular sets, Symmetry, 11 (2019), 1279. http://dx.doi.org/10.3390/sym11101279 doi: 10.3390/sym11101279
[11]	F. Clarke, Y. Ledyaev, R. Stern, R. Wolenski, Nonsmooth analysis and control theory, New York: Springer-Verlag, 1998. http://dx.doi.org/10.1007/b97650
[12]	F. Clarke, Y. Ledyaev, R. Stern, Fixed points and equilibria in nonconvex sets, Nonlinear Anal.-Theor., 25 (1995), 145–161. http://dx.doi.org/10.1016/0362-546X(94)00215-4 doi: 10.1016/0362-546X(94)00215-4
[13]	J. Li, The generalized projection operator on reflexive Banach spaces and its applications, J. Math. Anal. Appl., 306 (2005), 55–71. http://dx.doi.org/10.1016/j.jmaa.2004.11.007 doi: 10.1016/j.jmaa.2004.11.007
[14]	J. Li, On the existence of solutions of variational inequalities in Banach spaces, J. Math. Anal. Appl., 295 (2004), 115–126. http://dx.doi.org/10.1016/j.jmaa.2004.03.010 doi: 10.1016/j.jmaa.2004.03.010
[15]	M. Noor, On an implicit method for nonconvex variational inequalities, J. Optim. Theory Appl., 147 (2010), 411–417. http://dx.doi.org/10.1007/s10957-010-9717-y doi: 10.1007/s10957-010-9717-y
[16]	R. Tyrrell Rockafellar, R. Wets, Variational analysis, Berlin: Springer-Verlag, 1998. http://dx.doi.org/10.1007/978-3-642-02431-3
[17]	W. Takahashi, Nonlinear functional analysis, Yokohama: Yokohama Publishers, 2000.
[18]	D. Wen, Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators, Nonlinear Anal.-Theor., 73 (2010), 2292–2297. http://dx.doi.org/10.1016/j.na.2010.06.010 doi: 10.1016/j.na.2010.06.010
[19]	K. Wu, N. Huang, The generalized $f$-projection operator with an application, Bull. Austral. Math. Soc., 73 (2006), 307–317. http://dx.doi.org/10.1017/S0004972700038892 doi: 10.1017/S0004972700038892
[20]	K. Wu, N. Huang, Properties of the generalized $f$-projection operator and its application in Banach spaces, Comput. Math. Appl., 54 (2007), 399–406. http://dx.doi.org/10.1016/j.camwa.2007.01.029 doi: 10.1016/j.camwa.2007.01.029

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