Variational Problems and Applications, 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Dynamical Systems".

Deadline for manuscript submissions: 31 July 2024 | Viewed by 10164

Special Issue Editor

Special Issue Information

Dear Colleagues,

In recent decades, there has been increased interest among researchers in solution procedures in variational (interval/fuzzy) analysis and robust control. To formulate the necessary and sufficient optimality/efficiency conditions and duality theorems for different classes of robust and interval-valued/fuzzy variational problems, various approaches have been proposed. The current Special Issue is focused on studies of uncertain variational problems and aims to develop the research in this field by formulating and demonstrating some characterization results of well-posedness and robust efficient solutions in new classes of (multiobjective) variational (control) problems governed by multiple and/or path-independent curvilinear integral cost functionals and robust mixed and/or isoperimetric constraints involving first- and second-order partial differential equations. Therefore, I cordially invite you to publish your results on related subjects (variational inequalities, evolutionary problems, and so on) in this Special Issue. 

Dr. Savin Treanta
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • optimization problems
  • optimal control
  • variational problems
  • well-posedness
  • partial differential equations
  • generalized convexity
  • dynamical systems

Published Papers (13 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

34 pages, 1306 KiB  
Article
Hyers–Ulam Stability of 2D-Convex Mappings and Some Related New Hermite–Hadamard, Pachpatte, and Fejér Type Integral Inequalities Using Novel Fractional Integral Operators via Totally Interval-Order Relations with Open Problem
by Waqar Afzal, Daniel Breaz, Mujahid Abbas, Luminiţa-Ioana Cotîrlă, Zareen A. Khan and Eleonora Rapeanu
Mathematics 2024, 12(8), 1238; https://doi.org/10.3390/math12081238 - 19 Apr 2024
Viewed by 481
Abstract
The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves [...] Read more.
The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves the development of Hermite–Hadamard inequality, including its weighted and product forms, by using a novel type of fractional operator having non-singular kernels. Moreover, we develop several nontrivial examples and remarks to demonstrate the validity of our main results. Finally, we examine approximate convex mappings and have left an open problem regarding the best optimal constants for two-dimensional approximate convexity. Full article
(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)
Show Figures

Figure 1

19 pages, 312 KiB  
Article
On Semi-Infinite Optimization Problems with Vanishing Constraints Involving Interval-Valued Functions
by Bhuwan Chandra Joshi, Murari Kumar Roy and Abdelouahed Hamdi
Mathematics 2024, 12(7), 1008; https://doi.org/10.3390/math12071008 - 28 Mar 2024
Viewed by 507
Abstract
In this paper, we examine a semi-infinite interval-valued optimization problem with vanishing constraints (SIVOPVC) that lacks differentiability and involves constraints that tend to vanish. We give definitions of generalized convex functions along with supportive examples. We investigate duality theorems for the SIVOPVC problem. [...] Read more.
In this paper, we examine a semi-infinite interval-valued optimization problem with vanishing constraints (SIVOPVC) that lacks differentiability and involves constraints that tend to vanish. We give definitions of generalized convex functions along with supportive examples. We investigate duality theorems for the SIVOPVC problem. We establish these theorems by creating duality models, which establish a relationship between SIVOPVC and its corresponding dual models, assuming generalized convexity conditions. Some examples are also given to illustrate the results. Full article
(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)
21 pages, 351 KiB  
Article
Abnormality and Strict-Sense Minimizers That Are Not Extended Minimizers
by Giovanni Fusco and Monica Motta
Mathematics 2024, 12(7), 943; https://doi.org/10.3390/math12070943 - 22 Mar 2024
Viewed by 395
Abstract
We consider a constrained optimal control problem and an extension of it, in which the set of strict-sense trajectories is enlarged. Extension is a common procedure in optimal control used to derive necessary and sufficient optimality conditions for the original problem from the [...] Read more.
We consider a constrained optimal control problem and an extension of it, in which the set of strict-sense trajectories is enlarged. Extension is a common procedure in optimal control used to derive necessary and sufficient optimality conditions for the original problem from the extended one, which usually admits a minimizer and has a more regular structure. However, this procedure fails if the two problems have different infima. Therefore, it is relevant to identify such situations. Following on from earlier work by Warga but adopting perturbation techniques developed in nonsmooth analysis, we investigate the relation between the occurrence of an infimum gap and the abnormality of necessary conditions. For the notion of a local minimizer based on control distance and an extension, including the impulsive one, we prove that (i) a local extended minimizer that is not a local minimizer of the original problem, and (ii) a local strict-sense minimizer that is not a local minimizer of the extended problem both satisfy the extended maximum principle in abnormal form. The main novelty is result (ii), as until now, it has only been shown that a strict-sense minimizer that is not an extended minimizer is abnormal for an ‘averaged version’ of the maximum principle. Full article
(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)
28 pages, 431 KiB  
Article
Weighted Fejér, Hermite–Hadamard, and Trapezium-Type Inequalities for (h1,h2)–Godunova–Levin Preinvex Function with Applications and Two Open Problems
by Abdullah Ali H. Ahmadini, Waqar Afzal, Mujahid Abbas and Elkhateeb S. Aly
Mathematics 2024, 12(3), 382; https://doi.org/10.3390/math12030382 - 24 Jan 2024
Cited by 4 | Viewed by 667
Abstract
This note introduces a new class of preinvexity called (h1,h2)-Godunova-Levin preinvex functions that generalize earlier findings. Based on these notions, we developed Hermite-Hadamard, weighted Fejér, and trapezium type inequalities. Furthermore, we constructed some non-trivial examples in [...] Read more.
This note introduces a new class of preinvexity called (h1,h2)-Godunova-Levin preinvex functions that generalize earlier findings. Based on these notions, we developed Hermite-Hadamard, weighted Fejér, and trapezium type inequalities. Furthermore, we constructed some non-trivial examples in order to verify all the developed results. In addition, we discussed some applications related to the trapezoidal formula, probability density functions, special functions and special means. Lastly, we discussed the importance of order relations and left two open problems for future research. As an additional benefit, we believe that the present work can provide a strong catalyst for enhancing similar existing literature. Full article
(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)
50 pages, 536 KiB  
Article
Global Regular Axially Symmetric Solutions to the Navier–Stokes Equations: Part 2
by Wojciech M. Zajączkowski
Mathematics 2024, 12(2), 263; https://doi.org/10.3390/math12020263 - 12 Jan 2024
Viewed by 587
Abstract
The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder ΩR3 with the axis of symmetry. S1 is the boundary of the cylinder parallel to the axis of symmetry, and S2 is perpendicular to [...] Read more.
The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder ΩR3 with the axis of symmetry. S1 is the boundary of the cylinder parallel to the axis of symmetry, and S2 is perpendicular to it. We have two parts of S2. On S1 and S2, we impose vanishing of the normal component of velocity and the angular component of vorticity. Moreover, we assume that the angular component of velocity vanishes on S1 and the normal derivative of the angular component of velocity vanishes on S2. We prove the existence of global regular solutions. To prove this, the coordinate of velocity along the axis of symmetry must vanish on it. We have to emphasize that the technique of weighted spaces applied to the stream function plays a crucial role in the proof of global regular axially symmetric solutions. The paper is a generalization of Part 1, where the periodic boundary conditions are prescribed on S2. The transformation is not trivial because it needs to examine many additional boundary terms and derive new estimates. Full article
(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)
21 pages, 370 KiB  
Article
On a Generalized Gagliardo–Nirenberg Inequality with Radial Symmetry and Decaying Potentials
by Mirko Tarulli and George Venkov
Mathematics 2024, 12(1), 8; https://doi.org/10.3390/math12010008 - 19 Dec 2023
Viewed by 582
Abstract
We present a generalized version of a Gagliardo–Nirenberg inequality characterized by radial symmetry and involving potentials exhibiting pure power polynomial behavior. As an application of our result, we investigate the existence of extremals for this inequality, which also correspond to stationary solutions for [...] Read more.
We present a generalized version of a Gagliardo–Nirenberg inequality characterized by radial symmetry and involving potentials exhibiting pure power polynomial behavior. As an application of our result, we investigate the existence of extremals for this inequality, which also correspond to stationary solutions for the nonlinear Schrödinger equation with inhomogeneous nonlinearity, competing with Hs-subcritical nonlinearities, either of a local or nonlocal nature. Full article
(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)
46 pages, 509 KiB  
Article
Global Regular Axially Symmetric Solutions to the Navier–Stokes Equations: Part 1
by Wojciech M. Zaja̧czkowski
Mathematics 2023, 11(23), 4731; https://doi.org/10.3390/math11234731 - 22 Nov 2023
Cited by 2 | Viewed by 496
Abstract
The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder ΩR3 with the axis of symmetry. S1 is the boundary of the cylinder parallel to the axis of symmetry and S2 is perpendicular to [...] Read more.
The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder ΩR3 with the axis of symmetry. S1 is the boundary of the cylinder parallel to the axis of symmetry and S2 is perpendicular to it. We have two parts of S2. For simplicity, we assume the periodic boundary conditions on S2. On S1, we impose the vanishing of the normal component of velocity, the angular component of velocity, and the angular component of vorticity. We prove the existence of global regular solutions. To prove this, it is necessary that the coordinate of velocity along the axis of symmetry vanishes on it. We have to emphasize that the technique of weighted spaces applied to the stream function plays a crucial role in the proof of global regular axially symmetric solutions. The weighted spaces used are such that the stream function divided by the radius must vanish on the axis of symmetry. Currently, we do not know how to relax this restriction. In part 2 of this topic, the periodic boundary conditions on S2 are replaced by the conditions that both the normal component of velocity and the angular component of vorticity must vanish. Moreover, it is assumed that the normal derivative of the angular component of velocity also vanishes on S2. A transformation from part 1 to part 2 is not trivial because it needs new boundary value problems, so new estimates must be derived. Full article
(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)
12 pages, 236 KiB  
Article
Superiorization with a Projected Subgradient Algorithm on the Solution Sets of Common Fixed Point Problems
by Alexander J. Zaslavski
Mathematics 2023, 11(21), 4536; https://doi.org/10.3390/math11214536 - 3 Nov 2023
Viewed by 658
Abstract
In this work, we investigate a minimization problem with a convex objective function on a domain, which is the solution set of a common fixed point problem with a finite family of nonexpansive mappings. Our algorithm is a combination of a projected subgradient [...] Read more.
In this work, we investigate a minimization problem with a convex objective function on a domain, which is the solution set of a common fixed point problem with a finite family of nonexpansive mappings. Our algorithm is a combination of a projected subgradient algorithm and string-averaging projection method with variable strings and variable weights. This algorithm generates a sequence of iterates which are approximate solutions of the corresponding fixed point problem. Additionally, either this sequence also has a minimizing subsequence for our optimization problem or the sequence is strictly Fejer monotone regarding the approximate solution set of the common fixed point problem. Full article
(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)
21 pages, 420 KiB  
Article
Some New Estimates of Hermite–Hadamard Inequalities for Harmonical cr-h-Convex Functions via Generalized Fractional Integral Operator on Set-Valued Mappings
by Yahya Almalki and Waqar Afzal
Mathematics 2023, 11(19), 4041; https://doi.org/10.3390/math11194041 - 23 Sep 2023
Cited by 5 | Viewed by 645
Abstract
The application of fractional calculus to interval analysis is vital for the precise derivation of integral inequalities on set-valued mappings. The objective of this article is to reformulated the well-known Hermite–Hadamard inequality into various new variants via fractional integral operator (Riemann–Liouville) and generalize [...] Read more.
The application of fractional calculus to interval analysis is vital for the precise derivation of integral inequalities on set-valued mappings. The objective of this article is to reformulated the well-known Hermite–Hadamard inequality into various new variants via fractional integral operator (Riemann–Liouville) and generalize the various previously published results on set-valued mappings via center and radius order relations using harmonical h-convex functions. First, using these notions, we developed the Hermite–Hadamard (HH) inequality, and then constructed some product form of these inequalities for harmonically convex functions. Moreover, to demonstrate the correctness of these results, we constructed some interesting non-trivial examples. Full article
(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)
Show Figures

Figure 1

14 pages, 714 KiB  
Article
Variational Approach to Modeling of Curvilinear Thin Inclusions with Rough Boundaries in Elastic Bodies: Case of a Rod-Type Inclusion
by Evgeny Rudoy and Sergey Sazhenkov
Mathematics 2023, 11(16), 3447; https://doi.org/10.3390/math11163447 - 8 Aug 2023
Viewed by 703
Abstract
In the framework of 2D-elasticity, an equilibrium problem for an inhomogeneous body with a curvilinear inclusion located strictly inside the body is considered. The elastic properties of the inclusion are assumed to depend on a small positive parameter δ characterizing its width and [...] Read more.
In the framework of 2D-elasticity, an equilibrium problem for an inhomogeneous body with a curvilinear inclusion located strictly inside the body is considered. The elastic properties of the inclusion are assumed to depend on a small positive parameter δ characterizing its width and are assumed to be proportional to δ1. Moreover, it is supposed that the inclusion has a curvilinear rough boundary. Relying on the variational formulation of the equilibrium problem, we perform the asymptotic analysis, as δ tends to zero. As a result, a variational model of an elastic body containing a thin curvilinear rod is constructed. Numerical calculations give a relative error between the initial and limit problems depending on δ. Full article
(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)
Show Figures

Figure 1

16 pages, 4470 KiB  
Article
Study of Log Convex Mappings in Fuzzy Aunnam Calculus via Fuzzy Inclusion Relation over Fuzzy-Number Space
by Tareq Saeed, Muhammad Bilal Khan, Savin Treanță, Hamed H. Alsulami and Mohammed Sh. Alhodaly
Mathematics 2023, 11(9), 2043; https://doi.org/10.3390/math11092043 - 25 Apr 2023
Cited by 1 | Viewed by 901
Abstract
In this paper, with the use of newly defined class up and down log–convex fuzzy-number valued mappings, we offer a few new and original mappings defined by applying some mild restrictions over the definition of up and down log–convex fuzzy-number valued mapping. With [...] Read more.
In this paper, with the use of newly defined class up and down log–convex fuzzy-number valued mappings, we offer a few new and original mappings defined by applying some mild restrictions over the definition of up and down log–convex fuzzy-number valued mapping. With the use of these mappings, we are able to develop partners of Fejér-type inequalities for up and down log–convexity, which improve upon certain previously established findings. The discussion also includes these mappings’ characteristics. Moreover, some nontrivial examples are also provided to prove the validation of our main results. Full article
(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)
17 pages, 321 KiB  
Article
Duality Results for a Class of Constrained Robust Nonlinear Optimization Problems
by Savin Treanţă and Tareq Saeed
Mathematics 2023, 11(1), 192; https://doi.org/10.3390/math11010192 - 29 Dec 2022
Cited by 1 | Viewed by 818
Abstract
In this paper, we establish various results of duality for a new class of constrained robust nonlinear optimization problems. For this new class of problems, involving functionals of (path-independent) curvilinear integral type and mixed constraints governed by partial derivatives of second order and [...] Read more.
In this paper, we establish various results of duality for a new class of constrained robust nonlinear optimization problems. For this new class of problems, involving functionals of (path-independent) curvilinear integral type and mixed constraints governed by partial derivatives of second order and uncertain data, we formulate and study Wolfe, Mond-Weir and mixed type robust dual optimization problems. In this regard, by considering the concept of convex curvilinear integral vector functional, determined by controlled second-order Lagrangians including uncertain data, and the notion of robust weak efficient solution associated with the considered problem, we create a new mathematical context to state and prove the duality theorems. Furthermore, an illustrative application is presented. Full article
(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)
16 pages, 362 KiB  
Article
Some New Generalizations of Integral Inequalities for Harmonical cr-(h1,h2)-Godunova–Levin Functions and Applications
by Tareq Saeed, Waqar Afzal, Mujahid Abbas, Savin Treanţă and Manuel De la Sen
Mathematics 2022, 10(23), 4540; https://doi.org/10.3390/math10234540 - 1 Dec 2022
Cited by 13 | Viewed by 1491
Abstract
The interval analysis is famous for its ability to deal with uncertain data. This method is useful for addressing models with data that contain inaccuracies. Different concepts are used to handle data uncertainty in an interval analysis, including a pseudo-order relation, inclusion relation, [...] Read more.
The interval analysis is famous for its ability to deal with uncertain data. This method is useful for addressing models with data that contain inaccuracies. Different concepts are used to handle data uncertainty in an interval analysis, including a pseudo-order relation, inclusion relation, and center–radius (cr)-order relation. This study aims to establish a connection between inequalities and a cr-order relation. In this article, we developed the Hermite–Hadamard (H.H) and Jensen-type inequalities using the notion of harmonical (h1,h2)-Godunova–Levin (GL) functions via a cr-order relation which is very novel in the literature. These new definitions have allowed us to identify many classical and novel special cases that illustrate our main findings. It is possible to unify a large number of well-known convex functions using the principle of this type of convexity. Furthermore, for the sake of checking the validity of our main findings, some nontrivial examples are given. Full article
(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)
Show Figures

Figure 1

Back to TopTop