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21 pages, 357 KB

Open AccessArticle

A New Study on the Approximate Controllability of Sobolev-Type Stochastic ABC-Fractional Impulsive Differential Inclusions with Clarke Sub-Differential and Poisson Jumps

by Yousef Alnafisah, Hamdy M. Ahmed and A. M. Sayed Ahmed

Fractal Fract. 2025, 9(9), 605; https://doi.org/10.3390/fractalfract9090605 - 18 Sep 2025

Cited by 3 | Viewed by 809

Abstract

This paper undertakes a rigorous analytical exposition of the approximate controllability of a novel class of Sobolev-type stochastic impulsive differential inclusions, incorporating the Atangana–Baleanu fractional derivative in the Caputo configuration under the influence of Wiener process and Poissonian discontinuities. The system’s analytical landscape [...] Read more.

This paper undertakes a rigorous analytical exposition of the approximate controllability of a novel class of Sobolev-type stochastic impulsive differential inclusions, incorporating the Atangana–Baleanu fractional derivative in the Caputo configuration under the influence of Wiener process and Poissonian discontinuities. The system’s analytical landscape is further enriched by the incorporation of Clarke sub-differentials, facilitating the treatment of nonsmooth, nonconvex, and multivalued dynamics. The inherent complexity arising from the confluence of fractional memory, stochastic perturbations, and impulsive phenomena necessitates the deployment of a sophisticated apparatus from variational analysis, measurable selection theory, and multivalued fixed point frameworks within infinite-dimensional Banach spaces. This study delineates rigorous sufficient conditions, ensuring controllability under such hybrid influences, thereby generalizing classical paradigms to encompass nonlocal and discontinuous dynamical regimes. A precisely articulated exemplar is included to validate the theoretical constructs and demonstrate the operational efficacy of the proposed analytical methodology. Full article

(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)

22 pages, 352 KB

Open AccessArticle

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 1 | Viewed by 1924

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)

20 pages, 393 KB

Open AccessArticle

Trajectory Controllability of Clarke Subdifferential-Type Conformable Fractional Stochastic Differential Inclusions with Non-Instantaneous Impulsive Effects and Deviated Arguments

by Dimplekumar Chalishajar, Ramkumar Kasinathan, Ravikumar Kasinathan and Varshini Sandrasekaran

Fractal Fract. 2023, 7(7), 541; https://doi.org/10.3390/fractalfract7070541 - 13 Jul 2023

Cited by 7 | Viewed by 2061

Abstract

In this study, the multivalued fixed point theorem, Clarke subdifferential properties, fractional calculus, and stochastic analysis are used to arrive at the system’s mild solution (1). Furthermore, the mean square moment for the aforementioned system (1) confirms the conditions for trajectory (T-)controllability. The [...] Read more.

In this study, the multivalued fixed point theorem, Clarke subdifferential properties, fractional calculus, and stochastic analysis are used to arrive at the system’s mild solution (1). Furthermore, the mean square moment for the aforementioned system (1) confirms the conditions for trajectory (T-)controllability. The last part of the paper uses two numerical applications to explain the novel theoretical results that were reached. Full article

(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)

13 pages, 314 KB

Open AccessArticle

Time Optimal Feedback Control for 3D Navier–Stokes-Voigt Equations

by Yunxiang Li, Maojun Bin and Cuiyun Shi

Symmetry 2023, 15(5), 1127; https://doi.org/10.3390/sym15051127 - 22 May 2023

Cited by 1 | Viewed by 1933

Abstract

In this article, we discuss a time optimal feedback control for asymmetrical 3D Navier–Stokes–Voigt equations. Firstly, we consider the existence of the admissible trajectories for the asymmetrical 3D Navier–Stokes–Voigt equations by using the well-known Cesari property and the Fillippove’s theorem. Secondly, we study [...] Read more.

In this article, we discuss a time optimal feedback control for asymmetrical 3D Navier–Stokes–Voigt equations. Firstly, we consider the existence of the admissible trajectories for the asymmetrical 3D Navier–Stokes–Voigt equations by using the well-known Cesari property and the Fillippove’s theorem. Secondly, we study the existence result of a time optimal control for the feedback control systems. Lastly, asymmetrical Clarke’s subdifferential inclusions and asymmetrical 3D Navier–Stokes–Voigt differential variational inequalities are given to explain our main results. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

18 pages, 374 KB

Open AccessArticle

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 1888

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, 312 KB

Open AccessArticle

Characterizations of Well-Posedness for Generalized Hemivariational Inequalities Systems with Derived Inclusion Problems Systems in Banach Spaces

by Lu-Chuan Ceng, Jian-Ye Li, Cong-Shan Wang, Fang-Fei Zhang, Hui-Ying Hu, Yun-Ling Cui and Long He

Symmetry 2022, 14(7), 1341; https://doi.org/10.3390/sym14071341 - 29 Jun 2022

Viewed by 2113

Abstract

In real Banach spaces, the concept of

α

-well-posedness is extended to the class of generalized hemivariational inequalities systems consisting of two parts which are of symmetric structure mutually. First, certain concepts of

α

-well-posedness for generalized hemivariational inequalities systems are put forward. [...] Read more.

In real Banach spaces, the concept of

α

-well-posedness is extended to the class of generalized hemivariational inequalities systems consisting of two parts which are of symmetric structure mutually. First, certain concepts of

α

-well-posedness for generalized hemivariational inequalities systems are put forward. Second, certain metric characterizations of

α

-well-posedness for generalized hemivariational inequalities systems are presented. Lastly, certain equivalence results between strong

α

-well-posedness of both the system of generalized hemivariational inequalities and its system of derived inclusion problems are established. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory II)

21 pages, 342 KB

Open AccessArticle

A General Class of Differential Hemivariational Inequalities Systems in Reflexive Banach Spaces

by Lu-Chuan Ceng, Ching-Feng Wen, Yeong-Cheng Liou and Jen-Chih Yao

Mathematics 2021, 9(24), 3173; https://doi.org/10.3390/math9243173 - 9 Dec 2021

Cited by 19 | Viewed by 2430

Abstract

We consider an abstract system consisting of the parabolic-type system of hemivariational inequalities (SHVI) along with the nonlinear system of evolution equations in the frame of the evolution triple of product spaces, which is called a system of differential hemivariational inequalities (SDHVI). A [...] Read more.

We consider an abstract system consisting of the parabolic-type system of hemivariational inequalities (SHVI) along with the nonlinear system of evolution equations in the frame of the evolution triple of product spaces, which is called a system of differential hemivariational inequalities (SDHVI). A hybrid iterative system is proposed via the temporality semidiscrete technique on the basis of the Rothe rule and feedback iteration approach. Using the surjective theorem for pseudomonotonicity mappings and properties of the partial Clarke’s generalized subgradient mappings, we establish the existence and priori estimations for solutions to the approximate problem. Whenever studying the parabolic-type SHVI, the surjective theorem for pseudomonotonicity mappings, instead of the KKM theorems exploited by other authors in recent literature for a SHVI, guarantees the successful continuation of our demonstration. This overcomes the drawback of the KKM-based approach. Finally, via the limitation process for solutions to the hybrid iterative system, we derive the solvability of the SDHVI with no convexity of functions

u \mapsto f_{l} (t, x, u), l = 1, 2

and no compact property of

C_{0}

-semigroups

e^{A_{l} (t)}, l = 1, 2

. Full article

(This article belongs to the Special Issue New Advances in Functional Analysis)

10 pages, 249 KB

Open AccessArticle

S-Subgradient Projection Methods with S-Subdifferential Functions for Nonconvex Split Feasibility Problems

by Jinzuo Chen, Mihai Postolache and Yonghong Yao

Symmetry 2019, 11(12), 1517; https://doi.org/10.3390/sym11121517 - 14 Dec 2019

Cited by 2 | Viewed by 2820

Abstract

In this paper, the original

C Q

algorithm, the relaxed

C Q

algorithm, the gradient projection method (

G P M

) algorithm, and the subgradient projection method (

S P M

) algorithm for the convex split feasibility problem are reviewed, and [...] Read more.

In this paper, the original

C Q

algorithm, the relaxed

C Q

algorithm, the gradient projection method (

G P M

) algorithm, and the subgradient projection method (

S P M

) algorithm for the convex split feasibility problem are reviewed, and a renewed

S P M

algorithm with S-subdifferential functions to solve nonconvex split feasibility problems in finite dimensional spaces is suggested. The weak convergence theorem is established. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

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