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15 pages, 280 KiB

Open AccessFeature PaperArticle

Dirichlet μ-Parametric Differential Problem with Multivalued Reaction Term

by Mina Ghasemi, Calogero Vetro and Zhenfeng Zhang

Mathematics 2025, 13(8), 1295; https://doi.org/10.3390/math13081295 - 15 Apr 2025

Viewed by 269

We study a Dirichlet

μ

-parametric differential problem driven by a variable competing exponent operator, given by the sum of a negative p-Laplace differential operator and a positive q-Laplace differential operator, with a multivalued reaction term in the sense of a [...] Read more.

We study a Dirichlet

μ

-parametric differential problem driven by a variable competing exponent operator, given by the sum of a negative p-Laplace differential operator and a positive q-Laplace differential operator, with a multivalued reaction term in the sense of a Clarke subdifferential. The parameter

μ \in R

makes it possible to distinguish between the cases of an elliptic principal operator

(μ \leq 0

) and a non-elliptic principal operator (

μ > 0

). We focus on the well-posedness of the problem in variable exponent Sobolev spaces, starting with energy functional analysis. Using a Galerkin approach with a priori estimate and embedding results, we show that the functional associated with the problem is coercive; hence, we prove the existence of generalized and weak solutions. Full article

(This article belongs to the Section C: Mathematical Analysis)

34 pages, 414 KiB

Open AccessArticle

Existence Results and Gap Functions for Nonsmooth Weak Vector Variational-Hemivariational Inequality Problems on Hadamard Manifolds

by Balendu Bhooshan Upadhyay, Shivani Sain, Priyanka Mishra and Ioan Stancu-Minasian

Mathematics 2025, 13(6), 995; https://doi.org/10.3390/math13060995 - 18 Mar 2025

Viewed by 337

Abstract

In this paper, we consider a class of nonsmooth weak vector variational-hemivariational inequality problems (abbreviated as, WVVHVIP) in the framework of Hadamard manifolds. By employing an analogous to the KKM lemma, we establish the existence of the solutions for WVVHVIP without utilizing any [...] Read more.

In this paper, we consider a class of nonsmooth weak vector variational-hemivariational inequality problems (abbreviated as, WVVHVIP) in the framework of Hadamard manifolds. By employing an analogous to the KKM lemma, we establish the existence of the solutions for WVVHVIP without utilizing any monotonicity assumptions. Moreover, a uniqueness result for the solutions of WVVHVIP is established by using generalized geodesic strong monotonicity assumptions. We formulate Auslender, regularized, and Moreau-Yosida regularized type gap functions for WVVHVIP to establish necessary and sufficient conditions for the existence of the solutions to WVVHVIP. In addition to this, by employing the Auslender, regularized, and Moreau-Yosida regularized type gap functions, we derive the global error bounds for the solution of WVVHVIP under the generalized geodesic strong monotonicity assumptions. Several non-trivial examples are furnished in the Hadamard manifold setting to illustrate the significance of the established results. To the best of our knowledge, this is the first time that the existence results, gap functions, and global error bounds for WVVHVIP have been investigated in the framework of Hadamard manifolds via Clarke subdifferentials. Full article

45 pages, 512 KiB

Open AccessArticle

Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing Constraints

by Balendu Bhooshan Upadhyay, Shivani Sain and Ioan Stancu-Minasian

Axioms 2024, 13(9), 573; https://doi.org/10.3390/axioms13090573 - 23 Aug 2024

Viewed by 803

Abstract

This article deals with a class of nonsmooth interval-valued multiobjective semi-infinite programming problems with vanishing constraints (NIMSIPVC). We introduce the VC-Abadie constraint qualification (VC-ACQ) for NIMSIPVC and employ it to establish Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions for the considered problem. Regarding NIMSIPVC, we [...] Read more.

This article deals with a class of nonsmooth interval-valued multiobjective semi-infinite programming problems with vanishing constraints (NIMSIPVC). We introduce the VC-Abadie constraint qualification (VC-ACQ) for NIMSIPVC and employ it to establish Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions for the considered problem. Regarding NIMSIPVC, we formulate interval-valued weak vector, as well as interval-valued vector Lagrange-type dual and scalarized Lagrange-type dual problems. Subsequently, we establish the weak, strong, and converse duality results relating the primal problem NIMSIPVC and the corresponding dual problems. Moreover, we introduce the notion of saddle points for the interval-valued vector Lagrangian and scalarized Lagrangian of NIMSIPVC. Furthermore, we derive the saddle-point optimality criteria for NIMSIPVC by establishing the relationships between the solutions of NIMSIPVC and the saddle points of the corresponding Lagrangians of NIMSIPVC, under convexity assumptions. Non-trivial illustrative examples are provided to demonstrate the validity of the established results. The results presented in this paper extend the corresponding results derived in the existing literature from smooth to nonsmooth optimization problems, and we further generalize them for interval-valued multiobjective semi-infinite programming problems with vanishing constraints. Full article

(This article belongs to the Special Issue Optimization, Operations Research and Statistical Analysis)

18 pages, 427 KiB

Open AccessArticle

Efficient Automatic Subdifferentiation for Programs with Linear Branches

by Sejun Park

Mathematics 2023, 11(23), 4858; https://doi.org/10.3390/math11234858 - 3 Dec 2023

Viewed by 1182

Abstract

Computing an element of the Clarke subdifferential of a function represented by a program is an important problem in modern non-smooth optimization. Existing algorithms either are computationally inefficient in the sense that the computational cost depends on the input dimension or can only [...] Read more.

Computing an element of the Clarke subdifferential of a function represented by a program is an important problem in modern non-smooth optimization. Existing algorithms either are computationally inefficient in the sense that the computational cost depends on the input dimension or can only cover simple programs such as polynomial functions with branches. In this work, we show that a generalization of the latter algorithm can efficiently compute an element of the Clarke subdifferential for programs consisting of analytic functions and linear branches, which can represent various non-smooth functions such as max, absolute values, and piecewise analytic functions with linear boundaries, as well as any program consisting of these functions such as neural networks with non-smooth activation functions. Our algorithm first finds a sequence of branches used for computing the function value at a random perturbation of the input; then, it returns an element of the Clarke subdifferential by running the backward pass of the reverse-mode automatic differentiation following those branches. The computational cost of our algorithm is at most that of the function evaluation multiplied by some constant independent of the input dimension n, if a program consists of piecewise analytic functions defined by linear branches, whose arities and maximum depths of branches are independent of n. Full article

(This article belongs to the Special Issue High-Speed Computing and Parallel Algorithms)

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20 pages, 393 KiB

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 4 | Viewed by 1468

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 KiB

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

Viewed by 1585

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)

12 pages, 307 KiB

Open AccessArticle

Fractional Stochastic Evolution Inclusions with Control on the Boundary

by Hamdy M. Ahmed, Mahmoud M. El-Borai, Wagdy G. El-Sayed and Alaa Y. Elbadrawi

Symmetry 2023, 15(4), 928; https://doi.org/10.3390/sym15040928 - 17 Apr 2023

Cited by 7 | Viewed by 1601

Abstract

Symmetry in systems arises as a result of natural design and provides a pivotal mechanism for crucial system properties. In the field of control theory, scattered research has been carried out concerning the control of group-theoretic symmetric systems. In this manuscript, the principles [...] Read more.

Symmetry in systems arises as a result of natural design and provides a pivotal mechanism for crucial system properties. In the field of control theory, scattered research has been carried out concerning the control of group-theoretic symmetric systems. In this manuscript, the principles of stochastic analysis, the fixed-point theorem, fractional calculus, and multivalued map theory are implemented to investigate the null boundary controllability (NBC) of stochastic evolution inclusion (SEI) with the Hilfer fractional derivative (HFD) and the Clarke subdifferential. Moreover, an example is depicted to show the effect of the obtained results. Full article

(This article belongs to the Special Issue Stochastic Analysis with Applications and Symmetry)

22 pages, 436 KiB

Open AccessArticle

Existence of Sobolev-Type Hilfer Fractional Neutral Stochastic Evolution Hemivariational Inequalities and Optimal Controls

by Sivajiganesan Sivasankar, Ramalingam Udhayakumar, Venkatesan Muthukumaran, Saradha Madhrubootham, Ghada AlNemer and Ahmed M. Elshenhab

Fractal Fract. 2023, 7(4), 303; https://doi.org/10.3390/fractalfract7040303 - 30 Mar 2023

Cited by 7 | Viewed by 2110

Abstract

This article concentrates on a control system with a nonlocal condition that is driven by neutral stochastic evolution hemivariational inequalities (HVIs) of Sobolev-type Hilfer fractional (HF). In order to illustrate the necessary requirements for the existence of mild solutions to the required control [...] Read more.

This article concentrates on a control system with a nonlocal condition that is driven by neutral stochastic evolution hemivariational inequalities (HVIs) of Sobolev-type Hilfer fractional (HF). In order to illustrate the necessary requirements for the existence of mild solutions to the required control system, we first use the characteristics of the modified Clarke sub-differential and a fixed point approach for multivalued functions. Then, we show that there are optimal state-control sets that are driven by Sobolev-type HF neutral stochastic evolution HVIs utilizing constrained Lagrange optimal systems. The optimal control (OC) results are created without taking the uniqueness of the control system solutions into account. Finally, the main finding is shown by an example. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

18 pages, 374 KiB

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 14 | Viewed by 1571

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)

10 pages, 307 KiB

Open AccessArticle

Null Controllability of Hilfer Fractional Stochastic Differential Inclusions

by Hamdy M. Ahmed, Mahmoud M. El-Borai, Wagdy El-Sayed and Alaa Elbadrawi

Fractal Fract. 2022, 6(12), 721; https://doi.org/10.3390/fractalfract6120721 - 5 Dec 2022

Cited by 6 | Viewed by 1637

Abstract

This paper gives the null controllability for nonlocal stochastic differential inclusion with the Hilfer fractional derivative and Clarke subdifferential. Sufficient conditions for null controllability of nonlocal Hilfer fractional stochastic differential inclusion are established by using the fixed-point approach with the proof that the [...] Read more.

This paper gives the null controllability for nonlocal stochastic differential inclusion with the Hilfer fractional derivative and Clarke subdifferential. Sufficient conditions for null controllability of nonlocal Hilfer fractional stochastic differential inclusion are established by using the fixed-point approach with the proof that the corresponding linear system is controllable. Finally, the theoretical results are verified with an example. Full article

(This article belongs to the Section General Mathematics, Analysis)

18 pages, 322 KiB

Open AccessArticle

Stationary Condition for Borwein Proper Efficient Solutions of Nonsmooth Multiobjective Problems with Vanishing Constraints

by Hui Huang and Haole Zhu

Mathematics 2022, 10(23), 4569; https://doi.org/10.3390/math10234569 - 2 Dec 2022

Cited by 2 | Viewed by 1269

Abstract

This paper discusses optimality conditions for Borwein proper efficient solutions of nonsmooth multiobjective optimization problems with vanishing constraints. A new notion in terms of contingent cone and upper directional derivative is introduced, and a necessary condition for the Borwein proper efficient solution of [...] Read more.

This paper discusses optimality conditions for Borwein proper efficient solutions of nonsmooth multiobjective optimization problems with vanishing constraints. A new notion in terms of contingent cone and upper directional derivative is introduced, and a necessary condition for the Borwein proper efficient solution of the considered problem is derived. The concept of

ε

proper Abadie data qualification is also introduced, and a necessary condition which is called a strictly strong stationary condition for Borwein proper efficient solutions is obtained. In view of the strictly strong stationary condition, convexity of the objective functions, and quasi-convexity of constrained functions, sufficient conditions for the Borwein proper efficient solutions are presented. Some examples are given to illustrate the reasonability of the obtained results. Full article

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

17 pages, 312 KiB

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 1837

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)

18 pages, 340 KiB

Open AccessFeature PaperArticle

Monotonicity Arguments for Variational–Hemivariational Inequalities in Hilbert Spaces

by Mircea Sofonea

Axioms 2022, 11(3), 136; https://doi.org/10.3390/axioms11030136 - 16 Mar 2022

Cited by 1 | Viewed by 2155

Abstract

We consider a variational–hemivariational inequality in a real Hilbert space, which depends on two parameters. We prove that the inequality is governed by a maximal monotone operator, then we deduce various existence, uniqueness and equivalence results. The proofs are based on the theory [...] Read more.

We consider a variational–hemivariational inequality in a real Hilbert space, which depends on two parameters. We prove that the inequality is governed by a maximal monotone operator, then we deduce various existence, uniqueness and equivalence results. The proofs are based on the theory of maximal monotone operators, fixed point arguments and the properties of the subdifferential, both in the sense of Clarke and in the sense of convex analysis. These results lay the background in the study of various classes of inequalities. We use them to prove existence, uniqueness and continuous dependence results for the solution of elliptic and history-dependent variational–hemivariational inequalities. We also present some iterative methods in solving these inequalities, together with various convergence results. Full article

(This article belongs to the Special Issue Advances in General Topology and Its Application)

15 pages, 314 KiB

Open AccessArticle

Minty Variational Principle for Nonsmooth Interval-Valued Vector Optimization Problems on Hadamard Manifolds

by Savin Treanţă, Priyanka Mishra and Balendu Bhooshan Upadhyay

Mathematics 2022, 10(3), 523; https://doi.org/10.3390/math10030523 - 7 Feb 2022

Cited by 23 | Viewed by 2391

Abstract

This article deals with the classes of approximate Minty- and Stampacchia-type vector variational inequalities on Hadamard manifolds and a class of nonsmooth interval-valued vector optimization problems. By using the Clarke subdifferentials, we define a new class of functions on Hadamard manifolds, namely, the [...] Read more.

This article deals with the classes of approximate Minty- and Stampacchia-type vector variational inequalities on Hadamard manifolds and a class of nonsmooth interval-valued vector optimization problems. By using the Clarke subdifferentials, we define a new class of functions on Hadamard manifolds, namely, the geodesic

L U

-approximately convex functions. Under geodesic

L U

-approximate convexity hypothesis, we derive the relationship between the solutions of these approximate vector variational inequalities and nonsmooth interval-valued vector optimization problems. This paper extends and generalizes some existing results in the literature. Full article

21 pages, 342 KiB

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 15 | Viewed by 2094

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)

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