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Symmetry
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Symmetry in Differential Equations and Integral Operators
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Symmetry in Differential Equations and Integral Operators

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 August 2024) | Viewed by 4424

Special Issue Editors

Dr. Reza Sadaati

E-Mail Website
Guest Editor
Mathematics Department, Iran University of Science and Technology, Tehran 13114-16846, Iran
Interests: mathematics and computer science
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Donal O’Regan

grade E-Mail Website
Guest Editor
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, University Road, H91 CF50 Galway, Ireland
Interests: integral equations in Banach spaces; groups and semigroups of linear operators; qualitative theory of discrete and continuous evolution equations in Banach spaces; Hyers–Ulam stability and its connections with exponential dichotomy; long time behavior for solutions of abstract Cauchy problems in Banach spaces; fixed point theory and its application
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Chenkuan Li

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada
Interests: distribution theory; Hankel transform; fractional calculus of generalized functions; integral equations; fractional differential equations with fixed point theories
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The symmetry of differential equation systems involves a transformation that maps any solution to another solution within the system. For a first-order ODE, the invariance of the ODE under a point symmetry is equivalent to the existence of a first integral for the ODE. However, in general, integral operators are arbitrary to some extent. Indeed, if we have any solution of a linear differential equation which depends on some parameters, then an integral of the solution multiplied by any function of the parameter represents an integral operator, in turn generating solutions of the equation. One could also consider operators which permit the development of a systematic and unified theory of solutions of partial differential equations on the basis of complex function theory. As a result, it seems that a certain type of integral operator is of particular interest for many situations. However, it is also important to study various other types of integral operators since many situations will generate equations where other types of integral operators are applicable and useful. In this SI, we hope to receive papers on the above topics.

[1] Agarwal, Ravi; Hristova, S.; O'Regan, D. Integral representations of scalar delay non-instantaneous impulsive Riemann-Liouville fractional differential equations. Appl. Anal. 101 (2022), no. 18, 6495--6513.

[2] Positive Solutions for a High-Order Riemann-Liouville Type Fractional Integral Boundary Value Problem Involving Fractional Derivatives. By Wuyang Wang, Jun Ye, Jiafa Xu and Donal O’Regan. Symmetry 2022, 14(11), 2320; https://doi.org/10.3390/sym14112320 - 04 Nov 2022.

[3] Jabeen, T.; Agarwal, R. P.; Lupulescu, V.; O'Regan, D. Existence of global solutions for some classes of integral equations; translated from Ukraïn. Mat. Zh. 70 (2018), no. 1, 130--148 Ukrainian Math. J. 70 (2018), no. 1, 142--163.

[4] Chaharpashlou, Reza; Atangana, Abdon; Saadati, Reza. On the fuzzy stability results for fractional stochastic volterra integral equation. Discrete Contin. Dyn. Syst. Ser. S 14 (2021), no. 10, 3529--3539.

[5] Chaharpashlou, R.; O'Regan, Donal; Park, Choonkil; Saadati, Reza. $C^*$-algebra valued fuzzy normed spaces with application of Hyers-Ulam stability of a random integral equation. Adv. Difference Equ. 2020, Paper No. 326, 9 pp.

[6] Murza, Adrian C. Heteroclinic cycles in ODEs with the symmetry of the quaternion $\bold Q_8$ group. Math. Rep. (Bucur.) 22(72) (2020), no. 2, 87--98.

[7] Positive Solutions for a System of Riemann–Liouville Type Fractional-Order Integral Boundary Value Problems. By Keyu Zhang, Fehaid Salem Alshammari, Jiafa Xu and Donal O’Regan. Fractal Fract. 2022, 6(9), 480; https://doi.org/10.3390/fractalfract6090480 - 29 Aug 2022.

[8] Ndogmo, Jean-Claude. Some variational principles associated with ODEs of maximal symmetry. Part 2: the general case. J. Appl. Anal. 24 (2018), no. 2, 175--183.

[9] New Stability Results of an ABC Fractional Differential Equation in the Symmetric Matrix-Valued FBS by Zahra Eidinejad,Reza Saadati,Radko Mesiar and Chenkuan Li. Symmetry 2022, 14(12), 2667; https://doi.org/10.3390/sym14122667 - 16 Dec 2022.

Dr. Reza Sadaati
Prof. Dr. Donal O'Regan
Prof. Dr. Chenkuan Li
Guest Editors

Manuscript Submission Information

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Published Papers (4 papers)

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Research

34 pages, 434 KiB  
Article
Existence Results for Tempered-Hilfer Fractional Differential Problems on Hölder Spaces
by Hussein A. H. Salem, Mieczysław Cichoń and Wafa Shammakh
Symmetry 2024, 16(6), 700; https://doi.org/10.3390/sym16060700 - 5 Jun 2024
Cited by 1 | Viewed by 600
Abstract
This paper considers a nonlinear fractional-order boundary value problem [...] Read more.
This paper considers a nonlinear fractional-order boundary value problem  HDa,gα1,β,μx(t)+f(t,x(t),HDa,gα2,β,μx(t))=0, for t∈[a,b], α1∈(1,2], α2∈(0,1], β∈[0,1] with appropriate integral boundary conditions on the Hölder spaces. Here, f is a real-valued function that satisfies the Hölder condition, and  HDa,gα,β,μ represents the tempered-Hilfer fractional derivative of order α>0 with parameter μ∈R+ and type β∈[0,1]. The corresponding integral problem is introduced in the study of this issue. This paper addresses a fundamental issue in the field, namely the circumstances under which differential and integral problems are equivalent. This approach enables the study of differential problems using integral operators. In order to achieve this, tempered fractional calculus and the equivalence problem of the studied problems are introduced and studied. The selection of an appropriate function space is of fundamental importance. This paper investigates the applicability of these operators on Hölder spaces and provides a comprehensive rationale for this choice. Full article
(This article belongs to the Special Issue Symmetry in Differential Equations and Integral Operators)
40 pages, 9936 KiB  
Article
Stability Results and Parametric Delayed Mittag–Leffler Matrices in Symmetric Fuzzy–Random Spaces with Application
by Donal O’Regan, Safoura Rezaei Aderyani, Reza Saadati and Chenkuan Li
Symmetry 2023, 15(10), 1880; https://doi.org/10.3390/sym15101880 - 6 Oct 2023
Cited by 1 | Viewed by 854
Abstract
We introduce a matrix-valued fractional delay differential system in diverse cases and present Fox type stability results with applications of aggregated special functions. In addition we present an example showing the numerical solutions based on the second type Kudryashov method. Finally, via the [...] Read more.
We introduce a matrix-valued fractional delay differential system in diverse cases and present Fox type stability results with applications of aggregated special functions. In addition we present an example showing the numerical solutions based on the second type Kudryashov method. Finally, via the method of variation of constants, and some properties of the parametric Mittag–Leffler matrices, we obtain both symmetric random and symmetric fuzzy finite-time stability results for the governing fractional delay model. A numerical example is considered to illustrate applicability of the study. Full article
(This article belongs to the Special Issue Symmetry in Differential Equations and Integral Operators)
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10 pages, 298 KiB  
Article
Uniqueness Results and Asymptotic Behaviour of Nonlinear Schrödinger–Kirchhoff Equations
by Dongdong Sun
Symmetry 2023, 15(10), 1856; https://doi.org/10.3390/sym15101856 - 2 Oct 2023
Viewed by 849
Abstract
In this paper, we first study the uniqueness and symmetry of solution of nonlinear Schrödinger–Kirchhoff equations with constant coefficients. Then, we show the uniqueness of the solution of nonlinear Schrödinger–Kirchhoff equations with the polynomial potential. In the end, we investigate the asymptotic behaviour [...] Read more.
In this paper, we first study the uniqueness and symmetry of solution of nonlinear Schrödinger–Kirchhoff equations with constant coefficients. Then, we show the uniqueness of the solution of nonlinear Schrödinger–Kirchhoff equations with the polynomial potential. In the end, we investigate the asymptotic behaviour of the positive least energy solutions to nonlinear Schrödinger–Kirchhoff equations with vanishing potentials. The vanishing potential means that the zero set of the potential is non-empty. The uniqueness results of Schrödinger equations and the scaling technique are used in our proof. The elliptic estimates and energy analysis are applied in the proof of the asymptotic behaviour of the above Schrödinger–Kirchhoff-type equations. Full article
(This article belongs to the Special Issue Symmetry in Differential Equations and Integral Operators)
22 pages, 1410 KiB  
Article
Symmetrical Impulsive Inertial Neural Networks with Unpredictable and Poisson-Stable Oscillations
by Marat Akhmet, Madina Tleubergenova, Roza Seilova and Zakhira Nugayeva
Symmetry 2023, 15(10), 1812; https://doi.org/10.3390/sym15101812 - 22 Sep 2023
Cited by 2 | Viewed by 1269
Abstract
This paper explores the novel concept of discontinuous unpredictable and Poisson-stable motions within impulsive inertial neural networks. The primary focus is on a specific neural network architecture where impulses mimic the structure of the original model, that is, continuous and discrete parts are [...] Read more.
This paper explores the novel concept of discontinuous unpredictable and Poisson-stable motions within impulsive inertial neural networks. The primary focus is on a specific neural network architecture where impulses mimic the structure of the original model, that is, continuous and discrete parts are symmetrical. This unique modeling decision aligns with the real-world behavior of systems, where voltage typically remains smooth and continuous but may exhibit sudden changes due to various factors such as switches, sudden loads, or faults. The paper introduces the representation of these abrupt voltage transitions as discontinuous derivatives, providing a more accurate depiction of real-world scenarios. Thus, the focus of the research is a model, exceptional in its generality. To study Poisson stability, the method of included intervals is extended for discontinuous functions and B-topology. The theoretical findings are substantiated with numerical examples, demonstrating the practical feasibility of the proposed model. Full article
(This article belongs to the Special Issue Symmetry in Differential Equations and Integral Operators)
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