Axioms

Research

16 pages, 293 KiB

Open AccessArticle

On Ulam Stability of the Davison Functional Equation in m-Banach Spaces

by El-sayed El-hady and Janusz Brzdęk

Axioms 2025, 14(2), 107; https://doi.org/10.3390/axioms14020107 - 30 Jan 2025

Viewed by 307

Abstract

We prove new Ulam stability results for the Davison functional equation,

h (s v) + h (s + v) = h (s v + s) + h (v),

in the class of mappings h [...] Read more.

We prove new Ulam stability results for the Davison functional equation,

h (s v) + h (s + v) = h (s v + s) + h (v),

in the class of mappings h from a ring F into an m-Banach space. In this way, we complement several earlier outcomes, by extending them to the case of m-normed spaces. Our proofs are based on an earlier Ulam stability result obtained for some functional equation in a single variable. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

28 pages, 375 KiB

Open AccessArticle

Functional Differential Equations with an Advanced Neutral Term: New Monotonic Properties of Recursive Nature to Optimize Oscillation Criteria

by Amany Nabih, Wedad Albalawi, Mohammad S. Jazmati, Ali Elrashidi, Hegagi M. Ali and Osama Moaaz

Axioms 2024, 13(12), 847; https://doi.org/10.3390/axioms13120847 - 2 Dec 2024

Viewed by 634

Abstract

The goal of this study is to derive new conditions that improve the testing of the oscillatory and asymptotic features of fourth-order differential equations with an advanced neutral term. By using Riccati techniques and comparison with lower-order equations, we establish new criteria that [...] Read more.

The goal of this study is to derive new conditions that improve the testing of the oscillatory and asymptotic features of fourth-order differential equations with an advanced neutral term. By using Riccati techniques and comparison with lower-order equations, we establish new criteria that verify the absence of positive solutions and, consequently, the oscillation of all solutions to the investigated equation. Using our results to analyze a few specific instances of the examined equation, we can ultimately clarify the significance of the new inequalities. Our results are an extension of previous results that considered equations with a neutral delay term and also an improvement of previous results that considered only equations with an advanced neutral term. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

10 pages, 251 KiB

Open AccessArticle

Spectral Properties of the Laplace Operator with Variable Dependent Boundary Conditions in a Disk

by Aishabibi Dukenbayeva and Makhmud Sadybekov

Axioms 2024, 13(11), 794; https://doi.org/10.3390/axioms13110794 - 16 Nov 2024

Viewed by 651

Abstract

In this work, we study the spectral properties of the Laplace operator with variable dependent boundary conditions in a disk. The boundary conditions include periodic and antiperiodic boundary conditions as well as the generalized Samarskii–Ionkin-type boundary conditions. We show eigenfunctions and eigenvalues of [...] Read more.

In this work, we study the spectral properties of the Laplace operator with variable dependent boundary conditions in a disk. The boundary conditions include periodic and antiperiodic boundary conditions as well as the generalized Samarskii–Ionkin-type boundary conditions. We show eigenfunctions and eigenvalues of these problems in an explicit form. Moreover, the completeness of their eigenfunctions is investigated. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

20 pages, 330 KiB

Open AccessArticle

Third-Order Neutral Differential Equations with Non-Canonical Forms: Novel Oscillation Theorems

by Barakah Almarri, Belal Batiha, Omar Bazighifan and Fahd Masood

Axioms 2024, 13(11), 755; https://doi.org/10.3390/axioms13110755 - 31 Oct 2024

Cited by 1 | Viewed by 671

Abstract

This paper explores the asymptotic and oscillatory properties of a class of third-order neutral differential equations with multiple delays in a non-canonical form. The main objective is to simplify the non-canonical form by converting it to a canonical form, which reduces the complexity [...] Read more.

This paper explores the asymptotic and oscillatory properties of a class of third-order neutral differential equations with multiple delays in a non-canonical form. The main objective is to simplify the non-canonical form by converting it to a canonical form, which reduces the complexity of the possible cases of positive solutions and their derivatives from four cases in the non-canonical form to only two cases in the canonical form, which facilitates the process of inference and development of results. New criteria are provided that exclude the existence of positive solutions or Kneser-type solutions for this class of equations. New criteria that guarantee the oscillatory behavior of all solutions that satisfy the conditions imposed on the studied equation are also derived. This work makes a qualitative contribution to the development of previous studies in the field of neutral differential equations, as it provides new insights into the oscillatory behavior of neutral equations with multiple delays. To confirm the strength and effectiveness of the results, three examples are included that highlight the accuracy of the derived criteria and their practical applicability, which enhances the value of this research and expands the scope of its use in the field. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

22 pages, 341 KiB

Open AccessArticle

Unified Framework for Continuous and Discrete Relations of Gehring and Muckenhoupt Weights on Time Scales

by Samir H. Saker, Naglaa Mohammed, Haytham M. Rezk, Ahmed I. Saied, Khaled Aldwoah and Ayman Alahmade

Axioms 2024, 13(11), 754; https://doi.org/10.3390/axioms13110754 - 31 Oct 2024

Viewed by 576

Abstract

This article contains some relations, which include some embedding and transition properties, between the Muckenhoupt classes

M_{γ};

γ > 1

and the Gehring classes

G_{δ};

δ > 1

of bi-Sobolev weights on a time scale

T

. In addition, [...] Read more.

This article contains some relations, which include some embedding and transition properties, between the Muckenhoupt classes

M_{γ};

γ > 1

and the Gehring classes

G_{δ};

δ > 1

of bi-Sobolev weights on a time scale

T

. In addition, we establish the relations between Muckenhoupt and Gehring classes, where we define a new time scale

\tilde{T} = v (T)

, to indicate that if the

\tilde{Δ}

derivative of the inverse of a bi-Sobolev weight belongs to the Gehring class, then the

Δ

derivative of a bi-Sobolev weight on a time scale

T

belongs to the Muckenhoupt class. Furthermore, our results, which will be established by a newly developed technique, show that the study of the properties in the continuous and discrete classes of weights can be unified. As special cases of our results, when

T = R

, one can obtain classical continuous results, and when

T = N

, one can obtain discrete results which are new and interesting for the reader. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

15 pages, 299 KiB

Open AccessArticle

Investigation of the Oscillatory Properties of Fourth-Order Delay Differential Equations Using a Comparison Approach with First- and Second-Order Equations

by Osama Moaaz, Shaimaa Elsaeed, Asma Al-Jaser, Samia Ibrahim and Amira Essam

Axioms 2024, 13(9), 652; https://doi.org/10.3390/axioms13090652 - 23 Sep 2024

Cited by 1 | Viewed by 842

Abstract

This paper investigates the oscillatory behavior of solutions to fourth-order functional differential equations (FDEs) with multiple delays and a middle term. By employing a different comparison method approach with lower-order equations, the study introduces enhanced oscillation criteria. A key strength of the proposed [...] Read more.

This paper investigates the oscillatory behavior of solutions to fourth-order functional differential equations (FDEs) with multiple delays and a middle term. By employing a different comparison method approach with lower-order equations, the study introduces enhanced oscillation criteria. A key strength of the proposed method is its ability to reduce the complexity of the fourth-order equation by converting it into first- and second-order forms, allowing for the application of well-established oscillation theories. This approach not only extends existing criteria to higher-order FDEs but also offers more efficient and broadly applicable results. Detailed comparisons with previous research confirm the method’s effectiveness and broader relevance while demonstrating the feasibility and significance of our results as an expansion and improvement of previous results. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

12 pages, 397 KiB

Open AccessArticle

Nonlocal Extensions of First Order Initial Value Problems

by Ravi Shankar

Axioms 2024, 13(8), 567; https://doi.org/10.3390/axioms13080567 - 21 Aug 2024

Viewed by 591

Abstract

We study certain Volterra integral equations that extend and recover first order ordinary differential equations (ODEs). We formulate the former equations from the latter by replacing classical derivatives with nonlocal integral operators with anti-symmetric kernels. Replacements of spatial derivatives have seen success in [...] Read more.

We study certain Volterra integral equations that extend and recover first order ordinary differential equations (ODEs). We formulate the former equations from the latter by replacing classical derivatives with nonlocal integral operators with anti-symmetric kernels. Replacements of spatial derivatives have seen success in fracture mechanics, diffusion, and image processing. In this paper, we consider nonlocal replacements of time derivatives which contain future data. To account for the nonlocal nature of the operators, we formulate initial “volume” problems (IVPs) for these integral equations; the initial data is prescribed on a time interval rather than at a single point. As a nonlocality parameter vanishes, we show that the solutions to these equations recover those of classical ODEs. We demonstrate this convergence with exact solutions of some simple IVPs. However, we find that the solutions of these nonlocal models exhibit several properties distinct from their classical counterparts. For example, the solutions exhibit discontinuities at periodic intervals. In addition, for some IVPs, a continuous initial profile develops a measure-valued singularity in finite time. At subsequent periodic intervals, these solutions develop increasingly higher order distributional singularities. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

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13 pages, 275 KiB

Open AccessArticle

Linearized Stability Analysis of Nonlinear Delay Differential Equations with Impulses

by Mostafa Bachar

Axioms 2024, 13(8), 524; https://doi.org/10.3390/axioms13080524 - 2 Aug 2024

Viewed by 806

Abstract

This paper explores the linearized stability of nonlinear delay differential equations (DDEs) with impulses. The classical results on the existence of periodic solutions are extended from ordinary differential equations (ODEs) to DDEs with impulses. Furthermore, the classical results of linearized stability for nonlinear [...] Read more.

This paper explores the linearized stability of nonlinear delay differential equations (DDEs) with impulses. The classical results on the existence of periodic solutions are extended from ordinary differential equations (ODEs) to DDEs with impulses. Furthermore, the classical results of linearized stability for nonlinear semigroups are generalized to periodic DDEs with impulses. A significant challenge arises from the need for a discontinuous initial function to obtain periodic solutions. To address this, first-kind discontinuous spaces

R ([a, b], R^{n})

are introduced for defining DDEs with impulses, providing key existence and uniqueness results. This study also establishes linear stability results by linearizing the Poincaré operator for DDEs with impulses. Additionally, the stability properties of equilibrium solutions for these equations are analyzed, highlighting their importance due to the wide range of applications in various scientific fields. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

12 pages, 260 KiB

Open AccessArticle

Stability Results for Some Classes of Cubic Functional Equations

by El-sayed El-hady, Yamin Sayyari, Mehdi Dehghanian and Ymnah Alruwaily

Axioms 2024, 13(7), 480; https://doi.org/10.3390/axioms13070480 - 18 Jul 2024

Cited by 1 | Viewed by 807

Abstract

Applications involving functional equations (FUEQs) are commonplace. They are essential to various applications, such as fog computing. Ulam’s notion of stability is highly helpful since it provides a range of estimates between exact and approximate solutions. Using Brzdȩk’s fixed point technique (FPT), we [...] Read more.

Applications involving functional equations (FUEQs) are commonplace. They are essential to various applications, such as fog computing. Ulam’s notion of stability is highly helpful since it provides a range of estimates between exact and approximate solutions. Using Brzdȩk’s fixed point technique (FPT), we establish the stability of the following cubic type functional equations (CFUEQs):

F (\sqrt[3]{ξ_{1}^{3} + ξ_{2}^{3}}) + F (\sqrt[3]{ξ_{1}^{3} - ξ_{2}^{3}}) = 2 F (ξ_{1}) + 2 F (ξ_{2}),

2 F (\sqrt[3]{\frac{ξ_{1}^{3} + ξ_{2}^{3}}{2}}) = F (ξ_{1}) + F (ξ_{2})

for all

ξ_{1}, ξ_{2} \in R

. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

14 pages, 293 KiB

Open AccessArticle

Some Results of Stochastic Differential Equations

by Shuai Guo, Wei Li and Guangying Lv

Axioms 2024, 13(6), 405; https://doi.org/10.3390/axioms13060405 - 16 Jun 2024

Viewed by 712

Abstract

In this paper, there are two aims: one is Schauder and Sobolev estimates for the one-dimensional heat equation; the other is the stabilization of differential equations by stochastic feedback control based on discrete-time state observations. The nonhomogeneous Poisson stochastic process is used to [...] Read more.

In this paper, there are two aims: one is Schauder and Sobolev estimates for the one-dimensional heat equation; the other is the stabilization of differential equations by stochastic feedback control based on discrete-time state observations. The nonhomogeneous Poisson stochastic process is used to show how knowing Schauder and Sobolev estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs. The properties of a jump process is used. The stabilization of differential equations by stochastic feedback control is based on discrete-time state observations. Firstly, the stability results of the auxiliary system is established. Secondly, by comparing it with the auxiliary system and using the continuity method, the stabilization of the original system is obtained. Both parts focus on the impact of probability theory. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

12 pages, 713 KiB

Open AccessArticle

Analytical and Numerical Investigation for the Inhomogeneous Pantograph Equation

by Faten Aldosari and Abdelhalim Ebaid

Axioms 2024, 13(6), 377; https://doi.org/10.3390/axioms13060377 - 4 Jun 2024

Cited by 1 | Viewed by 660

Abstract

This paper investigates the inhomogeneous version of the pantograph equation. The current model includes the exponential function as the inhomogeneous part of the pantograph equation. The Maclaurin series expansion (MSE) is a well-known standard method for solving initial value problems; it may be [...] Read more.

This paper investigates the inhomogeneous version of the pantograph equation. The current model includes the exponential function as the inhomogeneous part of the pantograph equation. The Maclaurin series expansion (MSE) is a well-known standard method for solving initial value problems; it may be easier than any other approaches. Moreover, the MSE can be used in a straightforward manner in contrast to the other analytical methods. Thus, the MSE is extended in this paper to treat the inhomogeneous pantograph equation. The solution is obtained in a closed series form with an explicit formula for the series coefficients and the convergence of the series is proved. Also, the analytic solutions of some models in the literature are recovered as special cases of the present work. The accuracy of the results is examined through several comparisons with the available exact solutions of some classes in the relevant literature. Finally, the residuals are calculated and then used to validate the accuracy of the present approximations for some classes which have no exact solutions. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

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21 pages, 354 KiB

Open AccessArticle

Exponential Stability of the Numerical Solution of a Hyperbolic System with Nonlocal Characteristic Velocities

by Rakhmatillo Djuraevich Aloev, Abdumauvlen Suleimanovich Berdyshev, Vasila Alimova and Kymbat Slamovna Bekenayeva

Axioms 2024, 13(5), 334; https://doi.org/10.3390/axioms13050334 - 17 May 2024

Cited by 1 | Viewed by 857

Abstract

In this paper, we investigate the problem of the exponential stability of a stationary solution for a hyperbolic system with nonlocal characteristic velocities and measurement error. The formulation of the initial boundary value problem of boundary control for the specified hyperbolic system is [...] Read more.

In this paper, we investigate the problem of the exponential stability of a stationary solution for a hyperbolic system with nonlocal characteristic velocities and measurement error. The formulation of the initial boundary value problem of boundary control for the specified hyperbolic system is given. A difference scheme is constructed for the numerical solution of the considered initial boundary value problem. The definition of the exponential stability of the numerical solution in

ℓ^{2}

-norm with respect to a discrete perturbation of the equilibrium state of the initial boundary value difference problem is given. A discrete Lyapunov function for a numerical solution is constructed, and a theorem on the exponential stability of a stationary solution of the initial boundary value difference problem in

ℓ^{2}

-norm with respect to a discrete perturbation is proved. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

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17 pages, 1459 KiB

Open AccessArticle

On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach

by Nada A. M. Alshomrani, Abdelhalim Ebaid, Faten Aldosari and Mona D. Aljoufi

Axioms 2024, 13(2), 129; https://doi.org/10.3390/axioms13020129 - 19 Feb 2024

Cited by 1 | Viewed by 1444

Abstract

The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind [...] Read more.

The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind of first-order scalar differential equation. The suggested approach transforms the given first-order scalar differential equation to an equivalent second-order ordinary differential equation (ODE) without the advance parameter. Using this method, we are able to construct the exact solution of both the transformed model and the given original model. The exact solution is obtained in a wave form with specified amplitude and phase. Furthermore, several special cases are investigated at certain values/relationships of the involved parameters. It is shown that the exact solution in the absence of the advance parameter reduces to the corresponding solution in the literature. In addition, it is declared that the current model enjoys various kinds of solutions, such as constant solutions, polynomial solutions, and periodic solutions under certain constraints of the included parameters. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

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15 pages, 330 KiB

Open AccessFeature PaperArticle

A Strong Maximum Principle for Nonlinear Nonlocal Diffusion Equations

by Tucker Hartland and Ravi Shankar

Axioms 2023, 12(11), 1059; https://doi.org/10.3390/axioms12111059 - 18 Nov 2023

Viewed by 1313

Abstract

We consider a class of nonlinear integro-differential equations that model degenerate nonlocal diffusion. We investigate whether the strong maximum principle is valid for this nonlocal equation. For degenerate parabolic PDEs, the strong maximum principle is not valid. In contrast, for nonlocal diffusion, we [...] Read more.

We consider a class of nonlinear integro-differential equations that model degenerate nonlocal diffusion. We investigate whether the strong maximum principle is valid for this nonlocal equation. For degenerate parabolic PDEs, the strong maximum principle is not valid. In contrast, for nonlocal diffusion, we can formulate a strong maximum principle for nonlinearities satisfying a geometric condition related to the flux operator of the equation. In our formulation of the strong maximum principle, we find a physical re-interpretation and generalization of the standard PDE conclusion of the principle: we replace constant solutions with solutions of zero flux. We also consider nonlinearities outside the scope of our principle. For highly degenerate conductivities, we demonstrate the invalidity of the strong maximum principle. We also consider intermediate, inconclusive examples, and provide numerical evidence that the strong maximum principle is valid. This suggests that our geometric condition is sharp. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

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