Research

15 pages, 275 KiB

Open AccessArticle

Existence of Solutions: Investigating Fredholm Integral Equations via a Fixed-Point Theorem

by Faruk Özger, Merve Temizer Ersoy and Zeynep Ödemiş Özger

Axioms 2024, 13(4), 261; https://doi.org/10.3390/axioms13040261 - 14 Apr 2024

Viewed by 246

Integral equations, which are defined as “the equation containing an unknown function under the integral sign”, have many applications of real-world problems. The second type of Fredholm integral equations is generally used in radiation transfer theory, kinetic theory of gases, and neutron transfer [...] Read more.

Integral equations, which are defined as “the equation containing an unknown function under the integral sign”, have many applications of real-world problems. The second type of Fredholm integral equations is generally used in radiation transfer theory, kinetic theory of gases, and neutron transfer theory. A special case of these equations, known as the quadratic Chandrasekhar integral equation, given by

x (s) = 1 + λ x (s) \int_{0}^{1} \frac{s}{t + s} x (t) d t,

can be very often encountered in many applications, where x is the function to be determined,

λ

is a parameter, and

t, s \in [0, 1]

. In this paper, using a fixed-point theorem, the existence conditions for the solution of Fredholm integral equations of the form

χ (l) = ϱ (l) + χ (l) \int_{p}^{q} k (l, z) (V χ) (z) d z

are investigated in the space

C_{ω} [p, q],

where

χ

is the unknown function to be determined, V is a given operator, and

ϱ, k

are two given functions. Moreover, certain important applications demonstrating the applicability of the existence theorem presented in this paper are provided. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization)

14 pages, 297 KiB

Open AccessArticle

New Summation and Integral Representations for 2-Variable (p,q)-Hermite Polynomials

by Nusrat Raza, Mohammed Fadel and Wei-Shih Du

Axioms 2024, 13(3), 196; https://doi.org/10.3390/axioms13030196 - 15 Mar 2024

Cited by 1 | Viewed by 724

Abstract

In this paper, we introduce and study new features for 2-variable

(p, q)

-Hermite polynomials, such as the

(p, q)

-diffusion equation,

(p, q)

-differential formula and integral representations. In addition, we establish some [...] Read more.

In this paper, we introduce and study new features for 2-variable

(p, q)

-Hermite polynomials, such as the

(p, q)

-diffusion equation,

(p, q)

-differential formula and integral representations. In addition, we establish some summation models and their

(p, q)

-derivatives. Certain parting remarks and nontrivial examples are also provided. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization)

18 pages, 1090 KiB

Open AccessArticle

Finite-Time Stability of Impulsive Fractional Differential Equations with Pure Delays

by Tingting Xie and Mengmeng Li

Axioms 2023, 12(12), 1129; https://doi.org/10.3390/axioms12121129 - 15 Dec 2023

Viewed by 862

Abstract

This paper introduces a novel concept of the impulsive delayed Mittag–Leffler-type vector function, an extension of the Mittag–Leffler matrix function. It is essential to seek explicit formulas for the solutions to linear impulsive fractional differential delay equations. Based on explicit formulas of the [...] Read more.

This paper introduces a novel concept of the impulsive delayed Mittag–Leffler-type vector function, an extension of the Mittag–Leffler matrix function. It is essential to seek explicit formulas for the solutions to linear impulsive fractional differential delay equations. Based on explicit formulas of the solutions, the finite-time stability results of impulsive fractional differential delay equations are presented. Finally, we present four examples to illustrate the validity of our theoretical results. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization)

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15 pages, 336 KiB

Open AccessArticle

On Local Unique Solvability for a Class of Nonlinear Identification Problems

by Vladimir E. Fedorov, Marina V. Plekhanova and Daria V. Melekhina

Axioms 2023, 12(11), 1013; https://doi.org/10.3390/axioms12111013 - 27 Oct 2023

Viewed by 589

Abstract

Nonlinear identification problems for evolution differential equations, solved with respect to the highest-order Dzhrbashyan–Nersesyan fractional derivative, are studied. An equation of the considered class contains a linear unbounded operator, which generates analytic resolving families for the corresponding linear homogeneous equation, and a continuous [...] Read more.

Nonlinear identification problems for evolution differential equations, solved with respect to the highest-order Dzhrbashyan–Nersesyan fractional derivative, are studied. An equation of the considered class contains a linear unbounded operator, which generates analytic resolving families for the corresponding linear homogeneous equation, and a continuous nonlinear operator, which depends on lower-order Dzhrbashyan–Nersesyan derivatives and a depending on time unknown element. The identification problem consists of the equation, Dzhrbashyan–Nersesyan initial value conditions and an abstract overdetermination condition, which is defined by a linear continuous operator. Using the contraction mappings theorem, we prove the unique local solvability of the identification problem. The cases of mild and classical solutions are studied. The obtained abstract results are applied to an investigation of a nonlinear identification problem to a linearized phase field system with time dependent unknown coefficients at Dzhrbashyan–Nersesyan time-derivatives of lower orders. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization)

15 pages, 338 KiB

Open AccessArticle

Common Fixed Point of (ψ, β, L)-Generalized Contractive Mapping in Partially Ordered b-Metric Spaces

by Binghua Jiang, Huaping Huang and Stojan Radenović

Axioms 2023, 12(11), 1008; https://doi.org/10.3390/axioms12111008 - 26 Oct 2023

Viewed by 802

Abstract

The purpose of this paper is to attain the existence of coincidences and common fixed points in four mappings satisfying

(ψ, β, L)

-generalized contractive conditions in the framework of partially ordered b-metric spaces. The main results presented [...] Read more.

The purpose of this paper is to attain the existence of coincidences and common fixed points in four mappings satisfying

(ψ, β, L)

-generalized contractive conditions in the framework of partially ordered b-metric spaces. The main results presented in this paper generalize some recent results in the existing literature. Furthermore, a nontrivial example is presented to support the obtained results. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization)

25 pages, 362 KiB

Open AccessArticle

On the Study of Pseudo 𝒮-Asymptotically Periodic Mild Solutions for a Class of Neutral Fractional Delayed Evolution Equations

by Naceur Chegloufa, Belkacem Chaouchi, Marko Kostić and Wei-Shih Du

Axioms 2023, 12(8), 800; https://doi.org/10.3390/axioms12080800 - 20 Aug 2023

Viewed by 548

Abstract

The goal of this paper is to investigate the existence and uniqueness of pseudo

S

-asymptotically periodic mild solutions for a class of neutral fractional evolution equations with finite delay. We essentially use the fractional powers of closed linear operators, the semigroup theory [...] Read more.

The goal of this paper is to investigate the existence and uniqueness of pseudo

S

-asymptotically periodic mild solutions for a class of neutral fractional evolution equations with finite delay. We essentially use the fractional powers of closed linear operators, the semigroup theory and some classical fixed point theorems. Furthermore, we provide an example to illustrate the applications of our abstract results. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization)

15 pages, 510 KiB

Open AccessArticle

Pricing of Credit Risk Derivatives with Stochastic Interest Rate

by Wujun Lv and Linlin Tian

Axioms 2023, 12(8), 782; https://doi.org/10.3390/axioms12080782 - 12 Aug 2023

Cited by 1 | Viewed by 751

Abstract

This paper deals with a credit derivative pricing problem using the martingale approach. We generalize the conventional reduced-form credit risk model for a credit default swap market, assuming that the firms’ default intensities depend on the default states of counterparty firms and that [...] Read more.

This paper deals with a credit derivative pricing problem using the martingale approach. We generalize the conventional reduced-form credit risk model for a credit default swap market, assuming that the firms’ default intensities depend on the default states of counterparty firms and that the stochastic interest rate follows a jump-diffusion Cox–Ingersoll–Ross process. First, we derive the joint Laplace transform of the distribution of the vector process

(r_{t}, R_{t})

by applying piecewise deterministic Markov process theory and martingale theory. Then, using the joint Laplace transform, we obtain the explicit pricing of defaultable bonds and a credit default swap. Lastly, numerical examples are presented to illustrate the dynamic relationships between defaultable securities (defaultable bonds, credit default swap) and the maturity date. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization)

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14 pages, 291 KiB

Open AccessArticle

Determinantal Expressions, Identities, Concavity, Maclaurin Power Series Expansions for van der Pol Numbers, Bernoulli Numbers, and Cotangent

by Zhen-Ying Sun, Bai-Ni Guo and Feng Qi

Axioms 2023, 12(7), 665; https://doi.org/10.3390/axioms12070665 - 05 Jul 2023

Cited by 2 | Viewed by 948

Abstract

In this paper, basing on the generating function for the van der Pol numbers, utilizing the Maclaurin power series expansion and two power series expressions of a function involving the cotangent function, and by virtue of the Wronski formula and a derivative formula [...] Read more.

In this paper, basing on the generating function for the van der Pol numbers, utilizing the Maclaurin power series expansion and two power series expressions of a function involving the cotangent function, and by virtue of the Wronski formula and a derivative formula for the ratio of two differentiable functions, the authors derive four determinantal expressions for the van der Pol numbers, discover two identities for the Bernoulli numbers and the van der Pol numbers, prove the increasing property and concavity of a function involving the cotangent function, and establish two alternative Maclaurin power series expansions of a function involving the cotangent function. The coefficients of the Maclaurin power series expansions are expressed in terms of specific Hessenberg determinants whose elements contain the Bernoulli numbers and binomial coefficients. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization)

16 pages, 350 KiB

Open AccessArticle

Fixed Point Results in Generalized Menger Probabilistic Metric Spaces with Applications to Decomposable Measures

by Huaping Huang, Tatjana Došenović, Dušan Rakić and Stojan Radenović

Axioms 2023, 12(7), 660; https://doi.org/10.3390/axioms12070660 - 03 Jul 2023

Viewed by 709

Abstract

The aim of this paper is to give some fixed point results in generalized Menger probabilistic metric spaces. Moreover, some nontrivial examples are presented to illustrate the superiority of the obtained results. In addition, several interesting applications are given to show that our [...] Read more.

The aim of this paper is to give some fixed point results in generalized Menger probabilistic metric spaces. Moreover, some nontrivial examples are presented to illustrate the superiority of the obtained results. In addition, several interesting applications are given to show that our results are meaningful and valuable. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization)

14 pages, 356 KiB

Open AccessArticle

Optimality Conditions of the Approximate Efficiency for Nonsmooth Robust Multiobjective Fractional Semi-Infinite Optimization Problems

by Liu Gao, Guolin Yu and Wenyan Han

Axioms 2023, 12(7), 635; https://doi.org/10.3390/axioms12070635 - 27 Jun 2023

Cited by 1 | Viewed by 610

Abstract

This paper is devoted to the investigation of optimality conditions and saddle point theorems for robust approximate quasi-weak efficient solutions for a nonsmooth uncertain multiobjective fractional semi-infinite optimization problem (NUMFP). Firstly, a necessary optimality condition is established by using the properties of the [...] Read more.

This paper is devoted to the investigation of optimality conditions and saddle point theorems for robust approximate quasi-weak efficient solutions for a nonsmooth uncertain multiobjective fractional semi-infinite optimization problem (NUMFP). Firstly, a necessary optimality condition is established by using the properties of the Gerstewitz’s function. Furthermore, a kind of approximate pseudo/quasi-convex function is defined for the problem (NUMFP), and under its assumption, a sufficient optimality condition is obtained. Finally, we introduce the notion of a robust approximate quasi-weak saddle point to the problem (NUMFP) and prove corresponding saddle point theorems. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization)

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