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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 4 | Viewed by 851

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)

15 pages, 280 KB

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

Cited by 5 | Viewed by 691

Abstract

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 KB

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 1112

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

18 pages, 427 KB

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 1697

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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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 1906

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 2136

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)

15 pages, 314 KB

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 33 | Viewed by 2805

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 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 20 | Viewed by 2451

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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