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17 pages, 334 KB

Open AccessFeature PaperArticle

A Penalty Method for Elliptic Variational–Hemivariational Inequalities

by Mircea Sofonea and Domingo A. Tarzia

Axioms 2024, 13(10), 721; https://doi.org/10.3390/axioms13100721 - 17 Oct 2024

Viewed by 953

We consider an elliptic variational–hemivariational inequality

P

in a real reflexive Banach space, governed by a set of constraints K. Under appropriate assumptions of the data, this inequality has a unique solution

u \in K

. We associate inequality

P

to a [...] Read more.

We consider an elliptic variational–hemivariational inequality

P

in a real reflexive Banach space, governed by a set of constraints K. Under appropriate assumptions of the data, this inequality has a unique solution

u \in K

. We associate inequality

P

to a sequence of elliptic variational–hemivariational inequalities

{P_{n}}

, governed by a set of constraints

\tilde{K} \supset K

, a sequence of parameters

{λ_{n}} \subset R_{+}

, and a function

ψ

. We prove that if, for each

n \in N

, the element

u_{n} \in \tilde{K}

represents a solution to Problem

P_{n}

, then the sequence

{u_{n}}

converges to u as

λ_{n} \to 0

. Based on this general result, we recover convergence results for various associated penalty methods previously obtained in the literature. These convergence results are obtained by considering particular choices of the set

\tilde{K}

and the function

ψ

. The corresponding penalty methods can be applied in the study of various inequality problems. To provide an example, we consider a purely hemivariational inequality that describes the equilibrium of an elastic membrane in contact with an obstacle, the so-called foundation. Full article

(This article belongs to the Special Issue Recent Developments in Stability and Control of Dynamical Systems)

14 pages, 288 KB

Open AccessArticle

The Solvability of Generalized Systems of Time-Dependent Hemivariational Inequalities Enjoying Symmetric Structure in Reflexive Banach Spaces

by Lu-Chuan Ceng, Yi-Xuan Fu, Jie Yin, Liang He, Long He and Hui-Ying Hu

Symmetry 2021, 13(10), 1801; https://doi.org/10.3390/sym13101801 - 27 Sep 2021

Cited by 10 | Viewed by 2095

Abstract

In real reflexive Banach spaces, let the GSTDHVI, SHVI, DVIP, VIT, and KKM represent a generalized system of time-dependent hemivariational inequalities, a system of hemivariational inequalities, a derived vector inclusion problem, Volterra integral term, and Knaster–Kuratowski–Mazurkiewicz, respectively, where the GSTDHVI consists of two parts which are of symmetric structure mutually. By virtue of the surjectivity theorem for pseudo-monotonicity mappings and the Banach contraction mapping principle, instead of the KKM theorems exploited by other authors in recent literature for a SHVI, we consider and study a GSTDHVI with VITs. Under quite mild assumptions, it is shown that there exists only a solution to the investigated problem via demonstrating that a DVIP with VIT is solvable. Full article

(This article belongs to the Special Issue Nonlinear Analysis and Its Applications in Symmetry)

24 pages, 1106 KB

Open AccessArticle

Halpern-Subgradient Extragradient Method for Solving Equilibrium and Common Fixed Point Problems in Reflexive Banach Spaces

by Annel Thembinkosi Bokodisa, Lateef Olakunle Jolaoso and Maggie Aphane

Mathematics 2021, 9(7), 743; https://doi.org/10.3390/math9070743 - 31 Mar 2021

Cited by 1 | Viewed by 2465

Abstract

In this paper, using the concept of Bregman distance, we introduce a new Bregman subgradient extragradient method for solving equilibrium and common fixed point problems in a real reflexive Banach space. The algorithm is designed, such that the stepsize is chosen without prior knowledge of the Lipschitz constants. We also prove a strong convergence result for the sequence that is generated by our algorithm under mild conditions. We apply our result to solving variational inequality problems, and finally, we give some numerical examples to illustrate the efficiency and accuracy of the algorithm. Full article

(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)

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24 pages, 423 KB

Open AccessArticle

A Parallel Hybrid Bregman Subgradient Extragradient Method for a System of Pseudomonotone Equilibrium and Fixed Point Problems

by Annel Thembinkosi Bokodisa, Lateef Olakunle Jolaoso and Maggie Aphane

Symmetry 2021, 13(2), 216; https://doi.org/10.3390/sym13020216 - 28 Jan 2021

Cited by 4 | Viewed by 2110

Abstract

We introduce a new parallel hybrid subgradient extragradient method for solving the system of the pseudomonotone equilibrium problem and common fixed point problem in real reflexive Banach spaces. The algorithm is designed such that its convergence does not require prior estimation of the Lipschitz-like constants of the finite bifunctions underlying the equilibrium problems. Moreover, a strong convergence result is proven without imposing strong conditions on the control sequences. We further provide some numerical experiments to illustrate the performance of the proposed algorithm and compare with some existing methods. Full article

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16 pages, 309 KB

Open AccessArticle

Convergence Theorems for Modified Implicit Iterative Methods with Perturbation for Pseudocontractive Mappings

by Jong Soo Jung

Mathematics 2020, 8(1), 72; https://doi.org/10.3390/math8010072 - 2 Jan 2020

Viewed by 1996

Abstract

In this paper, first, we introduce a path for a convex combination of a pseudocontractive type of mappings with a perturbed mapping and prove strong convergence of the proposed path in a real reflexive Banach space having a weakly continuous duality mapping. Second, we propose two modified implicit iterative methods with a perturbed mapping for a continuous pseudocontractive mapping in the same Banach space. Strong convergence theorems for the proposed iterative methods are established. The results in this paper substantially develop and complement the previous well-known results in this area. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Its Applications)

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