Mathematics

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33 pages, 449 KiB

Open AccessArticle

Bounds of Different Integral Operators in Tensorial Hilbert and Variable Exponent Function Spaces

by Waqar Afzal, Mujahid Abbas and Omar Mutab Alsalami

Mathematics 2024, 12(16), 2464; https://doi.org/10.3390/math12162464 - 9 Aug 2024

Viewed by 332

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of [...] Read more.

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space (

l^{q (\cdot)} (L^{p (\cdot)})

). Moreover, it was developed using classical Lebesgue space (

L_{p}

) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not. Full article

(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)

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24 pages, 609 KiB

Open AccessArticle

A Discrete Hamilton–Jacobi Theory for Contact Hamiltonian Dynamics

by Oğul Esen, Cristina Sardón and Marcin Zajac

Mathematics 2024, 12(15), 2342; https://doi.org/10.3390/math12152342 - 26 Jul 2024

Viewed by 274

Abstract

In this paper, we propose a discrete Hamilton–Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton–Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, [...] Read more.

In this paper, we propose a discrete Hamilton–Jacobi theory for (discrete) Hamiltonian dynamics defined on a (discrete) contact manifold. To this end, we first provide a novel geometric Hamilton–Jacobi theory for continuous contact Hamiltonian dynamics. Then, rooting on the discrete contact Lagrangian formulation, we obtain the discrete equations for Hamiltonian dynamics by the discrete Legendre transformation. Based on the discrete contact Hamilton equation, we construct a discrete Hamilton–Jacobi equation for contact Hamiltonian dynamics. We show how the discrete Hamilton–Jacobi equation is related to the continuous Hamilton–Jacobi theory presented in this work. Then, we propose geometric foundations of the discrete Hamilton–Jacobi equations on contact manifolds in terms of discrete contact flows. At the end of the paper, we provide a numerical example to test the theory. Full article

(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)

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34 pages, 1306 KiB

Open AccessArticle

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

Cited by 1 | Viewed by 832

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)

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19 pages, 312 KiB

Open AccessArticle

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 699

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

Open AccessFeature PaperArticle

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 527

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

Open AccessArticle

Weighted Fejér, Hermite–Hadamard, and Trapezium-Type Inequalities for

(h_{1}, h_{2})

–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 812

Abstract

This note introduces a new class of preinvexity called

(h_{1}, h_{2})

-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

(h_{1}, h_{2})

-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

Open AccessArticle

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 696

Abstract

The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder

Ω \subset R^{3}

with the axis of symmetry.

S_{1}

is the boundary of the cylinder parallel to the axis of symmetry, and

S_{2}

is perpendicular to [...] Read more.

The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder

Ω \subset R^{3}

with the axis of symmetry.

S_{1}

is the boundary of the cylinder parallel to the axis of symmetry, and

S_{2}

is perpendicular to it. We have two parts of

S_{2}

. On

S_{1}

and

S_{2}

, 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

S_{1}

and the normal derivative of the angular component of velocity vanishes on

S_{2}

. 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

S_{2}

. 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

Open AccessArticle

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 709

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

H^{s}

-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

Open AccessArticle

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 609

Abstract

The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder

Ω \subset R^{3}

with the axis of symmetry.

S_{1}

is the boundary of the cylinder parallel to the axis of symmetry and

S_{2}

is perpendicular to [...] Read more.

The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder

Ω \subset R^{3}

with the axis of symmetry.

S_{1}

is the boundary of the cylinder parallel to the axis of symmetry and

S_{2}

is perpendicular to it. We have two parts of

S_{2}

. For simplicity, we assume the periodic boundary conditions on

S_{2}

. On

S_{1}

, 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

S_{2}

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

S_{2}

. 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

Open AccessArticle

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 794

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

Open AccessArticle

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 818

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 (

H - H

) 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)

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14 pages, 714 KiB

Open AccessArticle

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 834

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

δ^{- 1}

. 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)

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16 pages, 4470 KiB

Open AccessArticle

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 1035

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

Open AccessArticle

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 956

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

Open AccessArticle

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 14 | Viewed by 1663

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

(h_{1}, h_{2})

-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)

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