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2 pages, 143 KiB

Open AccessEditorial

Nonlinear Analysis and Its Applications in Symmetry

by Alexander Zaslavski

Symmetry 2022, 14(6), 1197; https://doi.org/10.3390/sym14061197 - 10 Jun 2022

Viewed by 955

Abstract

This Special Issue of Symmetry is devoted to recent advances in the nonlinear analysis and its applications [...] Full article

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

Research

Jump to: Editorial

16 pages, 331 KiB

Open AccessArticle

The Convergence Results of Differential Variational Inequality Problems

by Shih-Sen Chang, Salahuddin, Lin Wang and Zhaoli Ma

Symmetry 2022, 14(4), 760; https://doi.org/10.3390/sym14040760 - 06 Apr 2022

Cited by 2 | Viewed by 1205

Abstract

In this work, we suggest a differential variational inequality in reflexive Banach spaces and construct a sequence with a set of constraints and a penalty parameter. We use the penalty method to prove a unique solution to the problem and make suitable assumptions [...] Read more.

In this work, we suggest a differential variational inequality in reflexive Banach spaces and construct a sequence with a set of constraints and a penalty parameter. We use the penalty method to prove a unique solution to the problem and make suitable assumptions to prove the convergence of the sequence. The proof is based on arguments for compactness, symmetry, pseudomonotonicity, Mosco convergence, inverse strong monotonicity and Lipschitz continuity. Finally, we discuss the boundary value problem for the differential variational inequality problem as an application. Full article

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

27 pages, 17045 KiB

Open AccessArticle

2D and 3D Visualization for the Static Bifurcations and Nonlinear Oscillations of a Self-Excited System with Time-Delayed Controller

by Nasser A. Saeed, Jan Awrejcewicz, Mohamed A. Alkashif and Mohamed S. Mohamed

Symmetry 2022, 14(3), 621; https://doi.org/10.3390/sym14030621 - 20 Mar 2022

Cited by 7 | Viewed by 1609

Abstract

This research focuses on the nonlinear vibration control of a self-excited single-degree-of-freedom system. The integral resonant controller (IRC) is introduced to stabilize the unstable motion and suppress nonlinear oscillations of the considered system. The nonlinear dynamical equations that govern the vibratory behaviors of [...] Read more.

This research focuses on the nonlinear vibration control of a self-excited single-degree-of-freedom system. The integral resonant controller (IRC) is introduced to stabilize the unstable motion and suppress nonlinear oscillations of the considered system. The nonlinear dynamical equations that govern the vibratory behaviors of the proposed closed-loop control system are investigated using perturbation analysis, where loop delays have been included in the studied model. The system bifurcation behaviors have been visualized in both the two and three-dimensional spaces, and corresponding dynamical behaviors have been explored numerically using the bifurcation diagrams, Poincaré map, time-response, zero-one chaotic test algorithm, and frequency spectrum. The obtained analytical investigations revealed that the uncontrolled system can oscillate with one of four vibration modes depending on the excitation frequency, which are mono-stable periodic motion, bi-stable periodic motion, periodic and quasi-period motion, and quasi-periodic motion only. In addition, it is found that the existence of time delays in the control loop can either improve or degrade the control performance. Therefore, an objective function has been introduced to design the optimum control parameters. Based on the derived objective function, it is found that the performance of the proposed control strategy is proportional to the product of the control and feedback gains and inversely proportional to the internal loop feedback gain when the loop delays are neglected. Moreover, it is reported that the controller performance is a periodic function of the total sum of the loop delays. Accordingly, the optimal operating conditions of the time-delayed integral resonant controller have been explained. Finally, numerical validations for all obtained analytical results have been performed, where an excellent correspondence between the analytical and numerical investigations has been demonstrated. Full article

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

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7 pages, 229 KiB

Open AccessArticle

Two Generic Convergence Results for Infinite Products of Generalized Nonexpansive Mappings

by Simeon Reich and Alexander J. Zaslavski

Symmetry 2022, 14(3), 534; https://doi.org/10.3390/sym14030534 - 05 Mar 2022

Cited by 1 | Viewed by 1270

Abstract

In our 2014 work with M. Gabour, we introduced a metric space of generalized nonexpansive self-mappings of bounded and closed subsets of a Banach space and studied, using the Baire category approach, the asymptotic behavior of iterates of a generic operator belonging to [...] Read more.

In our 2014 work with M. Gabour, we introduced a metric space of generalized nonexpansive self-mappings of bounded and closed subsets of a Banach space and studied, using the Baire category approach, the asymptotic behavior of iterates of a generic operator belonging to this class. In the definition of a generalized nonexpansive mapping the norm is replaced by a general function which can be symmetric as a particular case. In this paper, we prove the convergence of infinite products of generalized nonexpansive self-mappings to a common fixed point in a generic setting. Full article

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

12 pages, 288 KiB

Open AccessArticle

A Modified Krasnosel’skiǐ–Mann Iterative Algorithm for Approximating Fixed Points of Enriched Nonexpansive Mappings

by Vasile Berinde

Symmetry 2022, 14(1), 123; https://doi.org/10.3390/sym14010123 - 10 Jan 2022

Cited by 7 | Viewed by 1523

Abstract

For approximating the fixed points of enriched nonexpansive mappings in Hilbert spaces, we consider a modified Krasnosel’skiǐ–Mann algorithm for which we prove a strong convergence theorem. We also empirically compare the rate of convergence of the modified Krasnosel’skiǐ–Mann algorithm and of the simple [...] Read more.

For approximating the fixed points of enriched nonexpansive mappings in Hilbert spaces, we consider a modified Krasnosel’skiǐ–Mann algorithm for which we prove a strong convergence theorem. We also empirically compare the rate of convergence of the modified Krasnosel’skiǐ–Mann algorithm and of the simple Krasnosel’skiǐ fixed point algorithm. Based on the numerical experiments reported in the paper we conclude that, for the class of enriched nonexpansive mappings, it is more convenient to work with the simple Krasnosel’skiǐ fixed point algorithm than with the modified Krasnosel’skiǐ–Mann algorithm. Full article

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

11 pages, 230 KiB

Open AccessArticle

Turnpike Properties for Dynamical Systems Determined by Differential Inclusions

by Alexander J. Zaslavski

Symmetry 2021, 13(12), 2326; https://doi.org/10.3390/sym13122326 - 04 Dec 2021

Cited by 1 | Viewed by 1215

Abstract

In this paper, we study the turnpike phenomenon for trajectories of continuous-time dynamical systems generated by differential inclusions, which have a prototype in mathematical economics. In particular, we show that, if the differential inclusion has a certain symmetric property, the turnpike possesses the [...] Read more.

In this paper, we study the turnpike phenomenon for trajectories of continuous-time dynamical systems generated by differential inclusions, which have a prototype in mathematical economics. In particular, we show that, if the differential inclusion has a certain symmetric property, the turnpike possesses the corresponding symmetric property. If we know a finite number of approximate trajectories of our system, then we know the turnpike and this information can be useful if we need to find new trajectories of our system or their approximations. Full article

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

17 pages, 2246 KiB

Open AccessArticle

A Modified Tseng’s Method for Solving the Modified Variational Inclusion Problems and Its Applications

by Thidaporn Seangwattana, Kamonrat Sombut, Areerat Arunchai and Kanokwan Sitthithakerngkiet

Symmetry 2021, 13(12), 2250; https://doi.org/10.3390/sym13122250 - 25 Nov 2021

Cited by 7 | Viewed by 1693

Abstract

The goal of this study was to show how a modified variational inclusion problem can be solved based on Tseng’s method. In this study, we propose a modified Tseng’s method and increase the reliability of the proposed method. This method is to modify [...] Read more.

The goal of this study was to show how a modified variational inclusion problem can be solved based on Tseng’s method. In this study, we propose a modified Tseng’s method and increase the reliability of the proposed method. This method is to modify the relaxed inertial Tseng’s method by using certain conditions and the parallel technique. We also prove a weak convergence theorem under appropriate assumptions and some symmetry properties and then provide numerical experiments to demonstrate the convergence behavior of the proposed method. Moreover, the proposed method is used for image restoration technology, which takes a corrupt/noisy image and estimates the clean, original image. Finally, we show the signal-to-noise ratio (SNR) to guarantee image quality. Full article

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

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14 pages, 323 KiB

Open AccessArticle

Existence and Convergence Results for Generalized Mixed Quasi-Variational Hemivariational Inequality Problem

by Shih-Sen Chang, Salahuddin, Lin Wang, Gang Wang and Yunhe Zhao

Symmetry 2021, 13(10), 1882; https://doi.org/10.3390/sym13101882 - 05 Oct 2021

Cited by 4 | Viewed by 1302

Abstract

The main purpose of this paper is threefold. One is to study the existence and convergence problem of solutions for a class of generalized mixed quasi-variational hemivariational inequalities. The second one is to study the existence of optimal control for such kind of [...] Read more.

The main purpose of this paper is threefold. One is to study the existence and convergence problem of solutions for a class of generalized mixed quasi-variational hemivariational inequalities. The second one is to study the existence of optimal control for such kind of generalized mixed quasi-variational hemivariational inequalities under given control

u \in U

. The third one is to study the relationship between the optimal control and the data for the underlying generalized mixed quasi-variational inequality problems and a class of minimization problem. As an application, we utilize our results to study the elastic frictional problem in a class of Hilbert spaces. The results presented in the paper extend and improve upon some recent results. Full article

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

14 pages, 288 KiB

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 7 | Viewed by 1450

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 [...] Read more.

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)

23 pages, 359 KiB

Open AccessFeature PaperArticle

A Multiplicity Theorem for Superlinear Double Phase Problems

by Beata Derȩgowska, Leszek Gasiński and Nikolaos S. Papageorgiou

Symmetry 2021, 13(9), 1556; https://doi.org/10.3390/sym13091556 - 24 Aug 2021

Cited by 2 | Viewed by 1563

Abstract

We consider a nonlinear Dirichlet problem driven by the double phase differential operator and with a superlinear reaction which need not satisfy the Ambrosetti–Rabinowitz condition. Using the Nehari manifold, we show that the problem has at least three nontrivial bounded solutions: nodal, positive [...] Read more.

We consider a nonlinear Dirichlet problem driven by the double phase differential operator and with a superlinear reaction which need not satisfy the Ambrosetti–Rabinowitz condition. Using the Nehari manifold, we show that the problem has at least three nontrivial bounded solutions: nodal, positive and by the symmetry of the behaviour at

+ \infty

and

- \infty

also negative. Full article

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

15 pages, 43747 KiB

Open AccessArticle

Control Theory Application for Swing Up and Stabilisation of Rotating Inverted Pendulum

by Xhevahir Bajrami, Arbnor Pajaziti, Ramë Likaj, Ahmet Shala, Rinor Berisha and Mirlind Bruqi

Symmetry 2021, 13(8), 1491; https://doi.org/10.3390/sym13081491 - 13 Aug 2021

Cited by 9 | Viewed by 4592

Abstract

This paper introduces a new scheme for sliding mode control using symmetry principles for a rotating inverted pendulum, with the possibility of extension of this control scheme to other dynamic systems. This was proven for swing up and stabilisation control problems via the [...] Read more.

This paper introduces a new scheme for sliding mode control using symmetry principles for a rotating inverted pendulum, with the possibility of extension of this control scheme to other dynamic systems. This was proven for swing up and stabilisation control problems via the new sliding mode control scheme using both simulations and experiments of rotary inverted pendulum (RIP) underactuated systems. According to the Lyapunov theory, a section of the pendulum was compensated with a scale error in the upright position, as the desired trajectory was followed by the pendulum arm section. As the RIP’s dynamic equations were nonlinearly complex and coupled, the complex internal dynamics made the task of controller design difficult. The system control for the pathway of the reference model of the rotational actuator with the application of the sliding mode technique for moving back and forth up the inverted pendulum’s structure, till the arm to reach the linear range round the vertical upright position, was created and tested in an existent device. The stabilisation scheme was switched on in the sliding mode as soon as the arm reached the linear range. A comparison of the stabilisation performance for the same rotating inverted pendulum as discussed by other authors revealed that the proposed controller was more flexible and reliable in terms of the swing up and stabilisation time. Full article

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

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