Editorial

5 pages, 273 KiB

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Special Issue of Symmetry: “Symmetry in Mathematical Analysis and Functional Analysis”

by Octav Olteanu

Symmetry 2022, 14(12), 2665; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14122665 - 16 Dec 2022

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This Special Issue consists of 11 papers recently published in MDPI’s journal Symmetry under the general thematic title “Symmetry in Mathematical Analysis and Functional Analysis” (see [...] Full article

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23 pages, 377 KiB

Open AccessArticle

Relation-Theoretic Coincidence and Common Fixed Point Results in Extended Rectangular b-Metric Spaces with Applications

by Yan Sun and Xiaolan Liu

Symmetry 2022, 14(8), 1588; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081588 - 02 Aug 2022

Cited by 2 | Viewed by 1085

Abstract

The objective of this paper is to obtain new relation-theoretic coincidence and common fixed point results for some mappings F and g via hybrid contractions and auxiliary functions in extended rectangular b-metric spaces, which improve the existing results and give some relevant [...] Read more.

The objective of this paper is to obtain new relation-theoretic coincidence and common fixed point results for some mappings F and g via hybrid contractions and auxiliary functions in extended rectangular b-metric spaces, which improve the existing results and give some relevant results. Finally, some nontrivial examples and applications to justify the main results. Full article

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20 pages, 326 KiB

Open AccessArticle

Sufficiency for Weak Minima in Optimal Control Subject to Mixed Constraints

by Gerardo Sánchez Licea

Symmetry 2022, 14(8), 1520; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081520 - 25 Jul 2022

Cited by 1 | Viewed by 880

Abstract

For optimal control problems of Bolza involving time-state-control mixed constraints, containing inequalities and equalities, fixed initial end-point, variable final end-point, and nonlinear dynamics, sufficient conditions for weak minima are derived. The proposed algorithm allows us to avoid hypotheses such as the continuity of [...] Read more.

For optimal control problems of Bolza involving time-state-control mixed constraints, containing inequalities and equalities, fixed initial end-point, variable final end-point, and nonlinear dynamics, sufficient conditions for weak minima are derived. The proposed algorithm allows us to avoid hypotheses such as the continuity of the second derivatives of the functions delimiting the problems, the continuity of the optimal controls or the parametrization of the final variable end-point. We also present a relaxation relative to some similar works, in the sense that we arrive essentially to the same conclusions but making weaker assumptions. Full article

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15 pages, 299 KiB

Open AccessArticle

Abstraction of Interpolative Reich-Rus-Ćirić-Type Contractions and Simplest Proof Technique

by Monairah Alansari and Muhammad Usman Ali

Symmetry 2022, 14(8), 1504; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081504 - 22 Jul 2022

Cited by 1 | Viewed by 863

Abstract

The concept of symmetry is a very vast topic that is involved in the studies of several phenomena. This concept enables us to discuss the phenomenon in some systematic pattern depending upon the type of phenomenon. Each phenomenon has its own type of [...] Read more.

The concept of symmetry is a very vast topic that is involved in the studies of several phenomena. This concept enables us to discuss the phenomenon in some systematic pattern depending upon the type of phenomenon. Each phenomenon has its own type of symmetry. The phenomenon that is used in the discussion of this article is a symmetric distance-measuring function. This article presents the notions of abstract interpolative Reich-Rus-Ćirić-type contractions with a shrink map and examines the existence of

ϕ

-fixed points for such maps in complete metric space. These notions are defined through special types of simulation functions. The proof technique of the results presented in this article is easy to understand compared with the existing literature on interpolative Reich-Rus-Ćirić-type contractions. Full article

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14 pages, 289 KiB

Open AccessArticle

Solving a System of Differential Equations with Infinite Delay by Using Tripled Fixed Point Techniques on Graphs

by Hasanen A. Hammad and Mohra Zayed

Symmetry 2022, 14(7), 1388; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14071388 - 06 Jul 2022

Cited by 23 | Viewed by 1285

Abstract

In this manuscript, some similar tripled fixed point results under certain restrictions on a

b -

metric space endowed with graphs are established. Furthermore, an example is provided to support our results. The obtained results extend, generalize, and unify several similar significant contributions [...] Read more.

In this manuscript, some similar tripled fixed point results under certain restrictions on a

b -

metric space endowed with graphs are established. Furthermore, an example is provided to support our results. The obtained results extend, generalize, and unify several similar significant contributions in the literature. Finally, to further extend our results, the existence of a solution to a system of ordinary differential equations with infinite delay is derived. Full article

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9 pages, 248 KiB

Open AccessArticle

Sharp Bounds for Trigonometric and Hyperbolic Functions with Application to Fractional Calculus

by Vuk Stojiljković, Slobodan Radojević, Eyüp Çetin, Vesna Šešum Čavić and Stojan Radenović

Symmetry 2022, 14(6), 1260; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14061260 - 18 Jun 2022

Cited by 5 | Viewed by 1636

Abstract

Sharp bounds for

\frac{cosh (x)}{x}, \frac{sinh (x)}{x}

, and

\frac{sin (x)}{x}

were obtained, as well as one new bound for

\frac{e^{x} + arctan (x)}{\sqrt{x}}

. A new situation to [...] Read more.

Sharp bounds for

\frac{cosh (x)}{x}, \frac{sinh (x)}{x}

, and

\frac{sin (x)}{x}

were obtained, as well as one new bound for

\frac{e^{x} + arctan (x)}{\sqrt{x}}

. A new situation to note about the obtained boundaries is the symmetry in the upper and lower boundary, where the upper boundary differs by a constant from the lower boundary. New consequences of the inequalities were obtained in terms of the Riemann–Liovuille fractional integral and in terms of the standard integral. Full article

(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Functional Analysis)

18 pages, 809 KiB

Open AccessArticle

Hermite–Hadamard Type Inclusions for Interval-Valued Coordinated Preinvex Functions

by Kin Keung Lai, Shashi Kant Mishra, Jaya Bisht and Mohd Hassan

Symmetry 2022, 14(4), 771; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14040771 - 08 Apr 2022

Cited by 6 | Viewed by 1156

Abstract

The connection between generalized convexity and symmetry has been studied by many authors in recent years. Due to this strong connection, generalized convexity and symmetry have arisen as a new topic in the subject of inequalities. In this paper, we introduce the concept [...] Read more.

The connection between generalized convexity and symmetry has been studied by many authors in recent years. Due to this strong connection, generalized convexity and symmetry have arisen as a new topic in the subject of inequalities. In this paper, we introduce the concept of interval-valued preinvex functions on the coordinates in a rectangle from the plane and prove Hermite–Hadamard type inclusions for interval-valued preinvex functions on coordinates. Further, we establish Hermite–Hadamard type inclusions for the product of two interval-valued coordinated preinvex functions. These results are motivated by the symmetric results obtained in the recent article by Kara et al. in 2021 on weighted Hermite–Hadamard type inclusions for products of coordinated convex interval-valued functions. Our established results generalize and extend some recent results obtained in the existing literature. Moreover, we provide suitable examples in the support of our theoretical results. Full article

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15 pages, 357 KiB

Open AccessArticle

Fractional Calculus for Convex Functions in Interval-Valued Settings and Inequalities

by Muhammad Bilal Khan, Hatim Ghazi Zaini, Savin Treanțǎ, Gustavo Santos-García, Jorge E. Macías-Díaz and Mohamed S. Soliman

Symmetry 2022, 14(2), 341; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020341 - 07 Feb 2022

Cited by 11 | Viewed by 1841

Abstract

In this paper, we discuss the Riemann–Liouville fractional integral operator for left and right convex interval-valued functions (left and right convex I∙V-F), as well as various related notions and concepts. First, the authors used the Riemann–Liouville fractional integral to prove [...] Read more.

In this paper, we discuss the Riemann–Liouville fractional integral operator for left and right convex interval-valued functions (left and right convex I∙V-F), as well as various related notions and concepts. First, the authors used the Riemann–Liouville fractional integral to prove Hermite–Hadamard type (𝓗–𝓗 type) inequality. Furthermore, 𝓗–𝓗 type inequalities for the product of two left and right convex I∙V-Fs have been established. Finally, for left and right convex I∙V-Fs, we found the Riemann–Liouville fractional integral Hermite–Hadamard type inequality (𝓗–𝓗 Fejér type inequality). The findings of this research show that this methodology may be applied directly and is computationally simple and precise. Full article

(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Functional Analysis)

22 pages, 370 KiB

Open AccessArticle

Some Fuzzy Riemann–Liouville Fractional Integral Inequalities for Preinvex Fuzzy Interval-Valued Functions

by Muhammad Bilal Khan, Hatim Ghazi Zaini, Jorge E. Macías-Díaz, Savin Treanțǎ and Mohamed S. Soliman

Symmetry 2022, 14(2), 313; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020313 - 03 Feb 2022

Cited by 8 | Viewed by 1282

Abstract

The main objective of this study is to introduce new versions of fractional integral inequalities in fuzzy fractional calculus utilizing the introduced preinvexity. Due to the behavior of its definition, the idea of preinvexity plays a significant role in the subject of inequalities. [...] Read more.

The main objective of this study is to introduce new versions of fractional integral inequalities in fuzzy fractional calculus utilizing the introduced preinvexity. Due to the behavior of its definition, the idea of preinvexity plays a significant role in the subject of inequalities. The concepts of preinvexity and symmetry have a tight connection thanks to the significant correlation that has developed between both in recent years. In this study, we attain the Hermite-Hadamard (H·H) and Hermite-Hadamard-Fejér (H·H Fejér) type inequalities for preinvex fuzzy-interval-valued functions (preinvex F·I·V·Fs) via Condition C and fuzzy Riemann–Liouville fractional integrals. Furthermore, we establish some refinements of fuzzy fractional H·H type inequality. There are also some specific examples of the reported results for various preinvex functions deduced. To support the newly introduced ideal, we have provided some nontrivial and logical examples. The results presented in this research are a significant improvement over earlier results. This paper’s awe-inspiring notions and formidable tools may energize and revitalize future research on this worthwhile and fascinating topic. Full article

(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Functional Analysis)

10 pages, 779 KiB

Open AccessArticle

Multiobjective Convex Optimization in Real Banach Space

by Kin Keung Lai, Mohd Hassan, Jitendra Kumar Maurya, Sanjeev Kumar Singh and Shashi Kant Mishra

Symmetry 2021, 13(11), 2148; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112148 - 10 Nov 2021

Cited by 2 | Viewed by 1345

Abstract

In this paper, we consider convex multiobjective optimization problems with equality and inequality constraints in real Banach space. We establish saddle point necessary and sufficient Pareto optimality conditions for considered problems under some constraint qualifications. These results are motivated by the symmetric results [...] Read more.

In this paper, we consider convex multiobjective optimization problems with equality and inequality constraints in real Banach space. We establish saddle point necessary and sufficient Pareto optimality conditions for considered problems under some constraint qualifications. These results are motivated by the symmetric results obtained in the recent article by Cobos Sánchez et al. in 2021 on Pareto optimality for multiobjective optimization problems of continuous linear operators. The discussions in this paper are also related to second order symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones due to Mishra and Wang in 2005. Further, we establish Karush–Kuhn–Tucker optimality conditions using saddle point optimality conditions for the differentiable cases and present some examples to illustrate our results. The study in this article can also be seen and extended as symmetric results of necessary and sufficient optimality conditions for vector equilibrium problems on Hadamard manifolds by Ruiz-Garzón et al. in 2019. Full article

(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Functional Analysis)

14 pages, 324 KiB

Open AccessArticle

The Well Posedness for Nonhomogeneous Boussinesq Equations

by Yan Liu and Baiping Ouyang

Symmetry 2021, 13(11), 2110; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112110 - 06 Nov 2021

Cited by 1 | Viewed by 991

Abstract

This paper is devoted to studying the Cauchy problem for non-homogeneous Boussinesq equations. We built the results on the critical Besov spaces [...] Read more.

This paper is devoted to studying the Cauchy problem for non-homogeneous Boussinesq equations. We built the results on the critical Besov spaces

(θ, u) \in L_{T}^{\infty} ({\dot{B}}_{p, 1}^{N / p}) \times L_{T}^{\infty} ({\dot{B}}_{p, 1}^{N / p - 1}) ⋂ L_{T}^{1} ({\dot{B}}_{p, 1}^{N / p + 1})

with

1 < p < 2 N

. We proved the global existence of the solution when the initial velocity is small with respect to the viscosity, as well as the initial temperature approaches a positive constant. Furthermore, we proved the uniqueness for

1 < p \leq N

. Our results can been seen as a version of symmetry in Besov space for the Boussinesq equations. Full article

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12 pages, 318 KiB

Open AccessReview

On Special Properties for Continuous Convex Operators and Related Linear Operators

by Octav Olteanu

Symmetry 2022, 14(7), 1390; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14071390 - 06 Jul 2022

Cited by 2 | Viewed by 1155

Abstract

This paper provides a uniform boundedness theorem for a class of convex operators, such as Banach–Steinhaus theorem for families of continuous linear operators. The case of continuous symmetric sublinear operators is outlined. Second, a general theorem characterizing the existence of the solution of [...] Read more.

This paper provides a uniform boundedness theorem for a class of convex operators, such as Banach–Steinhaus theorem for families of continuous linear operators. The case of continuous symmetric sublinear operators is outlined. Second, a general theorem characterizing the existence of the solution of the Markov moment problem is reviewed, and a related minimization problem is solved. Convexity is the common point of the two aims of the paper mentioned above. Full article

(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Functional Analysis)

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