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Search Results (4)

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Keywords = limiting subdifferentials

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22 pages, 352 KB  
Article
On Approximate Variational Inequalities and Bilevel Programming Problems
by Balendu Bhooshan Upadhyay, Ioan Stancu-Minasian, Subham Poddar and Priyanka Mishra
Axioms 2024, 13(6), 371; https://doi.org/10.3390/axioms13060371 - 30 May 2024
Cited by 2 | Viewed by 2047
Abstract
In this paper, we investigate a class of bilevel programming problems (BLPP) in the framework of Euclidean space. We derive relationships among the solutions of approximate Minty-type variational inequalities (AMTVI), approximate Stampacchia-type variational inequalities (ASTVI), and local ϵ-quasi solutions of the BLPP, [...] Read more.
In this paper, we investigate a class of bilevel programming problems (BLPP) in the framework of Euclidean space. We derive relationships among the solutions of approximate Minty-type variational inequalities (AMTVI), approximate Stampacchia-type variational inequalities (ASTVI), and local ϵ-quasi solutions of the BLPP, under generalized approximate convexity assumptions, via limiting subdifferentials. Moreover, by employing the generalized Knaster–Kuratowski–Mazurkiewicz (KKM)-Fan’s lemma, we derive some existence results for the solutions of AMTVI and ASTVI. We have furnished suitable, non-trivial, illustrative examples to demonstrate the importance of the established results. To the best of our knowledge, there is no research paper available in the literature that explores relationships between the approximate variational inequalities and BLPP under the assumptions of generalized approximate convexity by employing the powerful tool of limiting subdifferentials. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
18 pages, 374 KB  
Article
Optimal Control Problems for Hilfer Fractional Neutral Stochastic Evolution Hemivariational Inequalities
by Sivajiganesan Sivasankar, Ramalingam Udhayakumar, Velmurugan Subramanian, Ghada AlNemer and Ahmed M. Elshenhab
Symmetry 2023, 15(1), 18; https://doi.org/10.3390/sym15010018 - 21 Dec 2022
Cited by 15 | Viewed by 1969
Abstract
In this paper, we concentrate on a control system with a non-local condition that is governed by a Hilfer fractional neutral stochastic evolution hemivariational inequality (HFNSEHVI). By using concepts of the generalized Clarke sub-differential and a fixed point theorem for multivalued maps, we [...] Read more.
In this paper, we concentrate on a control system with a non-local condition that is governed by a Hilfer fractional neutral stochastic evolution hemivariational inequality (HFNSEHVI). By using concepts of the generalized Clarke sub-differential and a fixed point theorem for multivalued maps, we first demonstrate adequate requirements for the existence of mild solutions to the concerned control system. Then, using limited Lagrange optimal systems, we demonstrate the existence of optimal state-control pairs that are regulated by an HFNSEHVI with a non-local condition. In order to demonstrate the existence of fixed points, the symmetric structure of the spaces and operators that we create is essential. Without considering the uniqueness of the control system’s solutions, the best control results are established. Lastly, an illustration is used to demonstrate the major result. Full article
(This article belongs to the Special Issue Symmetry in System Theory, Control and Computing)
17 pages, 287 KB  
Article
Primal Lower Nice Functions in Reflexive Smooth Banach Spaces
by Messaoud Bounkhel and Mostafa Bachar
Mathematics 2020, 8(11), 2066; https://doi.org/10.3390/math8112066 - 19 Nov 2020
Cited by 6 | Viewed by 2372
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 [...] Read more.
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 epigraphs in the sense proposed recently by the authors in [J. Math. Anal. Appl. 2019, 475, 699–29]. Full article
(This article belongs to the Special Issue Nonlinear Optimization, Variational Inequalities and Equilibrium Problems)
11 pages, 284 KB  
Article
Generalized Nonsmooth Exponential-Type Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problems in Asplund Spaces
by Syed Shakaib Irfan, Mijanur Rahaman, Iqbal Ahmad, Rais Ahmad and Saddam Husain
Mathematics 2019, 7(4), 345; https://doi.org/10.3390/math7040345 - 10 Apr 2019
Cited by 4 | Viewed by 2611
Abstract
The aim of this article is to study new types of generalized nonsmooth exponential type vector variational-like inequality problems involving Mordukhovich limiting subdifferential operator. We establish some relationships between generalized nonsmooth exponential type vector variational-like inequality problems and vector optimization problems under some [...] Read more.
The aim of this article is to study new types of generalized nonsmooth exponential type vector variational-like inequality problems involving Mordukhovich limiting subdifferential operator. We establish some relationships between generalized nonsmooth exponential type vector variational-like inequality problems and vector optimization problems under some invexity assumptions. The celebrated Fan-KKM theorem is used to obtain the existence of solution of generalized nonsmooth exponential-type vector variational like inequality problems. In support of our main result, some examples are given. Our results presented in this article improve, extend, and generalize some known results offer in the literature. Full article
(This article belongs to the Special Issue Applied Functional Analysis and Its Applications)
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