Mathematics

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24 pages, 353 KiB

Open AccessFeature PaperArticle

On the Generalized (p,q)-ϕ-Calculus with Respect to Another Function

by Sina Etemad, Ivanka Stamova, Sotiris K. Ntouyas and Jessada Tariboon

Mathematics 2024, 12(20), 3290; https://doi.org/10.3390/math12203290 - 20 Oct 2024

Viewed by 634

Abstract

In the present paper, we generalized some of the operators defined in

(p, q)

-calculus with respect to another function. More precisely, the generalized

(p, q)

-

ϕ

-derivatives and

(p, q)

-

ϕ

[...] Read more.

In the present paper, we generalized some of the operators defined in

(p, q)

-calculus with respect to another function. More precisely, the generalized

(p, q)

-

ϕ

-derivatives and

(p, q)

-

ϕ

-integrals were introduced with respect to the strictly increasing function

ϕ

with the help of different orders of the q-shifting, p-shifting, and

(q / p)

-shifting operators. Then, after proving some related properties, and as an application, we considered a generalized

(p, q)

-

ϕ

-difference problem and studied the existence property for its unique solutions with the help of the Banach contraction mapping principle. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications III)

15 pages, 291 KiB

Open AccessArticle

Orthogonal Stability and Solution of a Three-Variable Functional Equation in Extended Banach Spaces

by Jagjeet Jakhar, Shalu Sharma, Jyotsana Jakhar, Majeed A. Yousif, Pshtiwan Othman Mohammed, Alina Alb Lupas and Nejmeddine Chorfi

Mathematics 2024, 12(18), 2868; https://doi.org/10.3390/math12182868 - 14 Sep 2024

Viewed by 592

Abstract

This manuscript introduces a novel three-variable cubic functional equation and derives its general solution. Employing both the direct and fixed-point methods, we investigate the orthogonal stability of this equation within the frameworks of quasi-

β

-Banach spaces and multi-Banach spaces. Additionally, the study [...] Read more.

This manuscript introduces a novel three-variable cubic functional equation and derives its general solution. Employing both the direct and fixed-point methods, we investigate the orthogonal stability of this equation within the frameworks of quasi-

β

-Banach spaces and multi-Banach spaces. Additionally, the study explores the stability of the equation in various extended Banach spaces and provides a specific example illustrating the absence of stability in certain cases. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications III)

18 pages, 335 KiB

Open AccessArticle

Offset Linear Canonical Stockwell Transform for Boehmians

by Navneet Kaur, Bivek Gupta, Amit K. Verma and Ravi P. Agarwal

Mathematics 2024, 12(15), 2379; https://doi.org/10.3390/math12152379 - 31 Jul 2024

Viewed by 625

Abstract

In this article, we construct a Boehmian space using the convolution theorem that contains the offset linear canonical Stockwell transforms (OLCST) of all square-integrable Boehmians. It is also proven that the extended OLCST on square-integrable Boehmians is consistent with the traditional OLCST. Furthermore, [...] Read more.

In this article, we construct a Boehmian space using the convolution theorem that contains the offset linear canonical Stockwell transforms (OLCST) of all square-integrable Boehmians. It is also proven that the extended OLCST on square-integrable Boehmians is consistent with the traditional OLCST. Furthermore, it is one-to-one, linear, and continuous with respect to

Δ

-convergence as well as

Δ

-convergence. Subsequently, we introduce a discrete variant of the OLCST. Ultimately, we broaden the application of the presented work by examining the OLCST within the domain of almost periodic functions. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications III)

19 pages, 342 KiB

Open AccessArticle

Existence of Solutions for Planar Kirchhoff–Choquard Problems

by Rui Niu and Tianxing Wu

Mathematics 2023, 11(17), 3754; https://doi.org/10.3390/math11173754 - 31 Aug 2023

Cited by 1 | Viewed by 912

Abstract

In this article, we are interested in the study of the following Kirchhoff–Choquard equations: [...] Read more.

In this article, we are interested in the study of the following Kirchhoff–Choquard equations:

- (a + b \int_{R^{2}} {| \nabla u |}^{2} d x) Δ u + V (x) u = λ (ln | x | * u^{2}) u + f (u), x \in R^{2},

where

λ > 0, a > 0, b > 0

, V and f are continuous functions with some appropriate assumptions. We prove that when the parameter

λ

is sufficiently small, the above problem has a mountain pass solution, a least energy solution and a ground state solution by applying the variational methods and building some subtle inequalities. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications III)

14 pages, 292 KiB

Open AccessArticle

Bounded Solutions of Semi-Linear Parabolic Differential Equations with Unbounded Delay Terms

by Allaberen Ashyralyev and Sa’adu Bello Mu’azu

Mathematics 2023, 11(16), 3470; https://doi.org/10.3390/math11163470 - 10 Aug 2023

Viewed by 744

Abstract

In the present work, an initial boundary value problem (IBVP) for the semi-linear delay differential equation in a Banach space with unbounded positive operators is studied. The main theorem on the uniqueness and existence of a bounded solution (BS) of this problem is [...] Read more.

In the present work, an initial boundary value problem (IBVP) for the semi-linear delay differential equation in a Banach space with unbounded positive operators is studied. The main theorem on the uniqueness and existence of a bounded solution (BS) of this problem is established. The application of the main theorem to four different semi-linear delay parabolic differential equations is presented. The first- and second-order accuracy difference schemes (FSADSs) for the solution of a one-dimensional semi-linear time-delay parabolic equation are considered. The new desired numerical results of this paper and their discussion are presented. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications III)

26 pages, 407 KiB

Open AccessArticle

Towards a Proof of Bahri–Coron’s Type Theorem for Mixed Boundary Value Problems

by Azeb Alghanemi, Slim Chaabane, Hichem Chtioui and Abdellahi Soumaré

Mathematics 2023, 11(8), 1955; https://doi.org/10.3390/math11081955 - 20 Apr 2023

Viewed by 1155

Abstract

We consider a nonlinear variational elliptic problem with critical nonlinearity on a bounded domain of

R^{n}, n \geq 3

and mixed Dirichlet–Neumann boundary conditions. We study the effect of the domain’s topology on the existence of solutions as Bahri–Coron did in [...] Read more.

We consider a nonlinear variational elliptic problem with critical nonlinearity on a bounded domain of

R^{n}, n \geq 3

and mixed Dirichlet–Neumann boundary conditions. We study the effect of the domain’s topology on the existence of solutions as Bahri–Coron did in their famous work on the homogeneous Dirichlet problem. However, due to the influence of the part of the boundary where the Neumann condition is prescribed, the blow-up picture in the present setting is more complicated and makes the mixed boundary problems different with respect to the homogeneous ones. Such complexity imposes modification of the argument of Bahri–Coron and demands new constructions and extra ideas. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications III)

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12 pages, 274 KiB

Open AccessFeature PaperArticle

Qualitative Properties of Solutions to a Class of Sixth-Order Equations

by Cristian-Paul Danet

Mathematics 2023, 11(6), 1280; https://doi.org/10.3390/math11061280 - 7 Mar 2023

Viewed by 1444

Abstract

In this paper, we present a detailed study of a class of sixth-order semilinear PDEs: existence, regularity and uniqueness. The uniqueness results are a consequence of a maximum principle called the

P

-function method. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications III)

11 pages, 265 KiB

Open AccessArticle

On Some Error Bounds for Milne’s Formula in Fractional Calculus

by Muhammad Aamir Ali, Zhiyue Zhang and Michal Fečkan

Mathematics 2023, 11(1), 146; https://doi.org/10.3390/math11010146 - 28 Dec 2022

Cited by 10 | Viewed by 1878

Abstract

In this paper, we found the error bounds for one of the open Newton–Cotes formulas, namely Milne’s formula for differentiable convex functions in the framework of fractional and classical calculus. We also give some mathematical examples to show that the newly established bounds [...] Read more.

In this paper, we found the error bounds for one of the open Newton–Cotes formulas, namely Milne’s formula for differentiable convex functions in the framework of fractional and classical calculus. We also give some mathematical examples to show that the newly established bounds are valid for Milne’s formula. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications III)

11 pages, 993 KiB

Open AccessArticle

Solitary Wave Solutions for the Stochastic Fractional-Space KdV in the Sense of the M-Truncated Derivative

by Wael W. Mohammed, Clemente Cesarano, Farah M. Al-Askar and Mahmoud El-Morshedy

Mathematics 2022, 10(24), 4792; https://doi.org/10.3390/math10244792 - 16 Dec 2022

Cited by 10 | Viewed by 1518

Abstract

The stochastic fractional-space Korteweg–de Vries equation (SFSKdVE) in the sense of the M-truncated derivative is examined in this article. In the Itô sense, the SFSKdVE is forced by multiplicative white noise. To produce new trigonometric, hyperbolic, rational, and elliptic stochastic fractional solutions, the [...] Read more.

The stochastic fractional-space Korteweg–de Vries equation (SFSKdVE) in the sense of the M-truncated derivative is examined in this article. In the Itô sense, the SFSKdVE is forced by multiplicative white noise. To produce new trigonometric, hyperbolic, rational, and elliptic stochastic fractional solutions, the tanh–coth and Jacobi elliptic function methods are used. The obtained solutions are useful in interpreting certain fascinating physical phenomena because the KdV equation is essential for understanding the behavior of waves in shallow water. To demonstrate how the multiplicative noise and the M-truncated derivative impact the precise solutions of the SFSKdVE, different 3D and 2D graphical representations are plotted. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications III)

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14 pages, 317 KiB

Open AccessArticle

On Solvability of Fractional (p,q)-Difference Equations with (p,q)-Difference Anti-Periodic Boundary Conditions

by Ravi P. Agarwal, Hana Al-Hutami and Bashir Ahmad

Mathematics 2022, 10(23), 4419; https://doi.org/10.3390/math10234419 - 23 Nov 2022

Cited by 3 | Viewed by 1148

Abstract

We discuss the solvability of a

(p, q)

-difference equation of fractional order

α \in (1, 2]

, equipped with anti-periodic boundary conditions involving the first-order

(p, q)

-difference operator. The desired results are [...] Read more.

We discuss the solvability of a

(p, q)

-difference equation of fractional order

α \in (1, 2]

, equipped with anti-periodic boundary conditions involving the first-order

(p, q)

-difference operator. The desired results are accomplished with the aid of standard fixed point theorems. Examples are presented for illustrating the obtained results. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications III)

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