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27 pages, 408 KiB

Open AccessArticle

Quadratic BSDEs with Singular Generators and Unbounded Terminal Conditions: Theory and Applications

by Wenbo Wang and Guangyan Jia

Mathematics 2025, 13(14), 2292; https://doi.org/10.3390/math13142292 - 17 Jul 2025

Viewed by 226

We investigate a class of quadratic backward stochastic differential equations (BSDEs) with generators that are singular in y. First, we establish the existence of solutions and a comparison theorem, thereby extending the existing results in the literature. Furthermore, we analyze the stability [...] Read more.

We investigate a class of quadratic backward stochastic differential equations (BSDEs) with generators that are singular in y. First, we establish the existence of solutions and a comparison theorem, thereby extending the existing results in the literature. Furthermore, we analyze the stability properties, derive the Feynman–Kac formula, and prove the uniqueness of viscosity solutions for the corresponding singular semi-linear partial differential equations (PDEs). Finally, we demonstrate applications in the context of robust control linked to stochastic differential utility and the certainty equivalent based on g-expectation. In these applications, the quadratic coefficients in the generators, respectively, quantify ambiguity aversion and absolute risk aversion. Full article

(This article belongs to the Special Issue Integration of Statistical Computing and Artificial Intelligence: Theoretical Methods and Empirical Application)

17 pages, 4274 KiB

Open AccessArticle

On the Study of Solutions for a Class of Third-Order Semilinear Nonhomogeneous Delay Differential Equations

by Wenjin Li, Jiaxuan Sun and Yanni Pang

Mathematics 2025, 13(12), 1926; https://doi.org/10.3390/math13121926 - 10 Jun 2025

Viewed by 266

Abstract

This paper mainly investigates a class of third-order semilinear delay differential equations with a nonhomogeneous term [...] Read more.

This paper mainly investigates a class of third-order semilinear delay differential equations with a nonhomogeneous term

{({[x^{″} (t)]}^{α})}^{'} + q (t) x^{α} (σ (t)) + f (t) = 0, t \geq t_{0} .

Under the oscillation criteria, we propose a sufficient condition to ensure that all solutions for the equation exhibit oscillatory behavior when

α

is the quotient of two positive odd integers, supported by concrete examples to verify the accuracy of these conditions. Furthermore, for the case

α = 1,

a sufficient condition is established to guarantee that the solutions either oscillate or asymptotically converge to zero. Moreover, under these criteria, we demonstrate that the global oscillatory behavior of solutions remains unaffected by time-delay functions, nonhomogeneous terms, or nonlinear perturbations when

α = 1 .

Finally, numerical simulations are provided to validate the effectiveness of the derived conclusions. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

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14 pages, 356 KiB

Open AccessArticle

Control of Semilinear Differential Equations with Moving Singularities

by Radu Precup, Andrei Stan and Wei-Shih Du

Fractal Fract. 2025, 9(4), 198; https://doi.org/10.3390/fractalfract9040198 - 24 Mar 2025

Viewed by 383

Abstract

In this paper, we present a control problem related to a semilinear differential equation with a moving singularity, i.e., the singular point depends on a parameter. The particularity of the controllability condition resides in the fact that it depends on the singular point, [...] Read more.

In this paper, we present a control problem related to a semilinear differential equation with a moving singularity, i.e., the singular point depends on a parameter. The particularity of the controllability condition resides in the fact that it depends on the singular point, which in turn depends on the control variable. We provide sufficient conditions to ensure that the functional determining the control is continuous over the entire domain of the parameter. Lower and upper solutions techniques combined with a bisection algorithm is used to prove the controllability of the equation and to approximate the control. An example is given together with some numerical simulations. The results naturally extend to fractional differential equations. Full article

(This article belongs to the Section General Mathematics, Analysis)

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14 pages, 294 KiB

Open AccessArticle

Existence, Uniqueness and Asymptotic Behavior of Solutions for Semilinear Elliptic Equations

by Lin-Lin Wang, Jing-Jing Liu and Yong-Hong Fan

Mathematics 2024, 12(22), 3624; https://doi.org/10.3390/math12223624 - 20 Nov 2024

Viewed by 803

Abstract

A class of semilinear elliptic differential equations was investigated in this study. By constructing the inverse function, using the method of upper and lower solutions and the principle of comparison, the existence of the maximum positive solution and the minimum positive solution was [...] Read more.

A class of semilinear elliptic differential equations was investigated in this study. By constructing the inverse function, using the method of upper and lower solutions and the principle of comparison, the existence of the maximum positive solution and the minimum positive solution was explored. Furthermore, the uniqueness of the positive solution and its asymptotic estimation at the origin were evaluated. The results show that the asymptotic estimation is similar to that of the corresponding boundary blowup problems. Compared with the conclusions of Wei’s work in 2017, the asymptotic behavior of the solution only depends on the asymptotic behavior of

b (x)

at the origin and the asymptotic behavior of g at infinity. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

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19 pages, 299 KiB

Open AccessArticle

A Study of Positive Solutions for Semilinear Fractional Measure Driven Functional Differential Equations in Banach Spaces

by Jing Zhang and Haide Gou

Mathematics 2024, 12(17), 2696; https://doi.org/10.3390/math12172696 - 29 Aug 2024

Viewed by 837

Abstract

In this paper, we deal with the delayed measure differential equations with nonlocal conditions via measure of noncompactness in ordered Banach spaces. Combining

(β, γ_{k})

-resolvent family, regulated functions and fixed point theorem with respect to convex-power condensing operator [...] Read more.

In this paper, we deal with the delayed measure differential equations with nonlocal conditions via measure of noncompactness in ordered Banach spaces. Combining

(β, γ_{k})

-resolvent family, regulated functions and fixed point theorem with respect to convex-power condensing operator and measure of noncompactness, we investigate the existence of positive mild solutions for the mentioned system under the situation that the nonlinear function satisfies measure conditions and order conditions. In addition, we provide an example to verify the rationality of our conclusion. Full article

17 pages, 916 KiB

Open AccessArticle

Positive Fitted Finite Volume Method for Semilinear Parabolic Systems on Unbounded Domain

by Miglena N. Koleva and Lubin G. Vulkov

Axioms 2024, 13(8), 507; https://doi.org/10.3390/axioms13080507 - 27 Jul 2024

Viewed by 804

Abstract

This work deals with a semilinear system of parabolic partial differential equations (PDEs) on an unbounded domain, related to environmental pollution modeling. Although we study a one-dimensional sub-model of a vertical advection–diffusion, the results can be extended in each direction for any number [...] Read more.

This work deals with a semilinear system of parabolic partial differential equations (PDEs) on an unbounded domain, related to environmental pollution modeling. Although we study a one-dimensional sub-model of a vertical advection–diffusion, the results can be extended in each direction for any number of spatial dimensions and different boundary conditions. The transformation of the independent variable is applied to convert the nonlinear problem into a finite interval, which can be selected in advance. We investigate the positivity of the solution of the new, degenerated parabolic system with a non-standard nonlinear right-hand side. Then, we design a fitted finite volume difference discretization in space and prove the non-negativity of the solution. The full discretization is obtained by implicit–explicit time stepping, taking into account the sign of the coefficients in the nonlinear term so as to preserve the non-negativity of the numerical solution and to avoid the iteration process. The method is realized on adaptive graded spatial meshes to attain second-order of accuracy in space. Some results from computations are presented. Full article

(This article belongs to the Special Issue Advances in Numerical Analysis and Meshless Methods)

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24 pages, 378 KiB

Open AccessArticle

Optimal Solutions for a Class of Impulsive Differential Problems with Feedback Controls and Volterra-Type Distributed Delay: A Topological Approach

by Paola Rubbioni

Mathematics 2024, 12(14), 2293; https://doi.org/10.3390/math12142293 - 22 Jul 2024

Viewed by 1005

Abstract

In this paper, the existence of optimal solutions for problems governed by differential equations involving feedback controls is established for when the problem must account for a Volterra-type distributed delay and is subject to the action of impulsive external forces. The problem is [...] Read more.

In this paper, the existence of optimal solutions for problems governed by differential equations involving feedback controls is established for when the problem must account for a Volterra-type distributed delay and is subject to the action of impulsive external forces. The problem is reformulated within the class of impulsive semilinear integro-differential inclusions in Banach spaces and is studied by using topological methods and multivalued analysis. The paper concludes with an application to a population dynamics model. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

14 pages, 280 KiB

Open AccessArticle

Topological Degree via a Degree of Nondensifiability and Applications

by Noureddine Ouahab, Juan J. Nieto and Abdelghani Ouahab

Axioms 2024, 13(7), 482; https://doi.org/10.3390/axioms13070482 - 18 Jul 2024

Cited by 1 | Viewed by 956

Abstract

The goal of this work is to introduce the notion of topological degree via the principle of the degree of nondensifiability (DND for short). We establish some new fixed point theorems, concerning, Schaefer’s fixed point theorem and the nonlinear alternative of Leray–Schauder type. [...] Read more.

The goal of this work is to introduce the notion of topological degree via the principle of the degree of nondensifiability (DND for short). We establish some new fixed point theorems, concerning, Schaefer’s fixed point theorem and the nonlinear alternative of Leray–Schauder type. As applications, we study the existence of mild solution of functional semilinear integro-differential equations. Full article

(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)

14 pages, 303 KiB

Open AccessArticle

(ω,c)-Periodic Solution to Semilinear Integro-Differential Equations with Hadamard Derivatives

by Ahmad Al-Omari, Hanan Al-Saadi and Fawaz Alharbi

Fractal Fract. 2024, 8(2), 86; https://doi.org/10.3390/fractalfract8020086 - 28 Jan 2024

Viewed by 1551

Abstract

This study aims to prove the existence and uniqueness of the

(ω, c)

-periodic solution as a specific solution to Hadamard impulsive boundary value integro-differential equations with fixed lower limits. The results are proven using the Banach contraction, Schaefer’s fixed [...] Read more.

This study aims to prove the existence and uniqueness of the

(ω, c)

-periodic solution as a specific solution to Hadamard impulsive boundary value integro-differential equations with fixed lower limits. The results are proven using the Banach contraction, Schaefer’s fixed point theorem, and the Arzelà–Ascoli theorem. Furthermore, we establish the necessary conditions for a set of solutions to the explored boundary values with impulsive fractional differentials. Finally, we present two examples as applications for our results. Full article

(This article belongs to the Special Issue Advances in Fractional Integral and Derivative Operators with Applications)

29 pages, 408 KiB

Open AccessArticle

Optimal Weak Order and Approximation of the Invariant Measure with a Fully-Discrete Euler Scheme for Semilinear Stochastic Parabolic Equations with Additive Noise

by Qiu Lin and Ruisheng Qi

Mathematics 2024, 12(1), 112; https://doi.org/10.3390/math12010112 - 28 Dec 2023

Cited by 1 | Viewed by 1141

Abstract

In this paper, we consider the ergodic semilinear stochastic partial differential equation driven by additive noise and the long-time behavior of its full discretization realized by a spectral Galerkin method in spatial direction and an Euler scheme in the temporal direction, which admits [...] Read more.

In this paper, we consider the ergodic semilinear stochastic partial differential equation driven by additive noise and the long-time behavior of its full discretization realized by a spectral Galerkin method in spatial direction and an Euler scheme in the temporal direction, which admits a unique invariant probability measure. Under the condition that the nonlinearity is once differentiable, the optimal convergence orders of the numerical invariant measures are obtained based on the time-independent weak error, but not relying on the associated Kolmogorov equation. More precisely, the obtained convergence orders are

O (λ_{N}^{- γ})

in space and

O (τ^{γ})

in time, where

γ \in (0, 1]

from the assumption

∥ A^{\frac{γ - 1}{2}} Q^{\frac{1}{2}} ∥_{L_{2}}

is used to characterize the spatial correlation of the noise process. Finally, numerical examples confirm the theoretical findings. Full article

(This article belongs to the Section E: Applied Mathematics)

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23 pages, 326 KiB

Open AccessArticle

A System of Coupled Impulsive Neutral Functional Differential Equations: New Existence Results Driven by Fractional Brownian Motion and the Wiener Process

by Abdelkader Moumen, Mohamed Ferhat, Amin Benaissa Cherif, Mohamed Bouye and Mohamad Biomy

Mathematics 2023, 11(24), 4949; https://doi.org/10.3390/math11244949 - 13 Dec 2023

Viewed by 1153

Abstract

Conditions for the existence and uniqueness of mild solutions for a system of semilinear impulsive differential equations with infinite fractional Brownian movements and the Wiener process are established. Our approach is based on a novel application of Burton and Kirk’s fixed point theorem [...] Read more.

Conditions for the existence and uniqueness of mild solutions for a system of semilinear impulsive differential equations with infinite fractional Brownian movements and the Wiener process are established. Our approach is based on a novel application of Burton and Kirk’s fixed point theorem in extended Banach spaces. This paper aims to extend current results to a differential-inclusions scenario. The motivation of this paper for impulsive neutral differential equations is to investigate the existence of solutions for impulsive neutral differential equations with fractional Brownian motion and a Wiener process (topics that have not been considered and are the main focus of this paper). Full article

(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)

17 pages, 302 KiB

Open AccessArticle

Existence and Uniqueness Results of Fractional Differential Inclusions and Equations in Sobolev Fractional Spaces

by Safia Meftah, Elhabib Hadjadj, Mohamad Biomy and Fares Yazid

Axioms 2023, 12(11), 1063; https://doi.org/10.3390/axioms12111063 - 20 Nov 2023

Cited by 2 | Viewed by 1581

Abstract

In this work, by using the iterative method, we discuss the existence and uniqueness of solutions for multiterm fractional boundary value problems. Next, we examine some existence and uniqueness returns for semilinear fractional differential inclusions and equations for multiterm problems by using some [...] Read more.

In this work, by using the iterative method, we discuss the existence and uniqueness of solutions for multiterm fractional boundary value problems. Next, we examine some existence and uniqueness returns for semilinear fractional differential inclusions and equations for multiterm problems by using some notions and properties on set-valued maps and give some examples to explain our main results. We explore and use in this paper the fundamental properties of set-valued maps, which are needed for the study of differential inclusions. It began only in the mid-1900s, when mathematicians realized that their uses go far beyond a mere generalization of single-valued maps. Full article

(This article belongs to the Special Issue Differential and Dynamic Equations on Time Scales and Their Applications)

16 pages, 326 KiB

Open AccessFeature PaperArticle

Linearization of Second-Order Non-Linear Ordinary Differential Equations: A Geometric Approach

by Michael Tsamparlis

Symmetry 2023, 15(11), 2082; https://doi.org/10.3390/sym15112082 - 18 Nov 2023

Cited by 2 | Viewed by 1674

Abstract

Using the coefficients of a system semilinear cubic in the first derivative second order differential equations one defines a connection in the space of the independent and dependent variables, which is specified modulo two free parameters. In this way, to any such equation [...] Read more.

Using the coefficients of a system semilinear cubic in the first derivative second order differential equations one defines a connection in the space of the independent and dependent variables, which is specified modulo two free parameters. In this way, to any such equation one associates an affine space which is not necessarily Riemannian, that is, a metric is not required. If such a metric exists, then under the Cartan parametrization the geodesic equations of the metric coincide with the system of the considered semilinear equations. In the present work, we consider semilinear cubic in the first derivative second order differential equations whose Lie symmetry algebra is the

s l (3, R)

. The covariant condition for these equations is the vanishing of the curvature tensor. We demonstrate the method in the solution of the Painlevé-Ince equation and in a system of two equations. Because the approach is geometric, the number of equations in the system is not important besides the complication in the calculations. It is shown that it is possible to linearize an equation in this form using a different covariant condition, for example, assuming the space to be of constant non-vanishing curvature. Finally, it is shown that one computes the associated metric to a semilinear cubic in the first derivatives differential equation using the inverse transformation derived from the transformation of the connection. Full article

(This article belongs to the Special Issue Symmetry in Mathematical Physics: History, Advances and Applications)

19 pages, 334 KiB

Open AccessFeature PaperArticle

Approximate Controllability for a Class of Semi-Linear Fractional Integro-Differential Impulsive Evolution Equations of Order 1 < α < 2 with Delay

by Daliang Zhao

Mathematics 2023, 11(19), 4069; https://doi.org/10.3390/math11194069 - 25 Sep 2023

Cited by 1 | Viewed by 1402

Abstract

This article is mainly concerned with the approximate controllability for some semi-linear fractional integro-differential impulsive evolution equations of order

1 < α < 2

with delay in Banach spaces. Firstly, we study the existence of the

P C

-mild solution for our objective [...] Read more.

This article is mainly concerned with the approximate controllability for some semi-linear fractional integro-differential impulsive evolution equations of order

1 < α < 2

with delay in Banach spaces. Firstly, we study the existence of the

P C

-mild solution for our objective system via some characteristic solution operators related to the Mainardi’s Wright function. Secondly, by using the spatial decomposition techniques and the range condition of control operator B, some new results of approximate controllability for the fractional delay system with impulsive effects are obtained. The results cover and extend some relevant outcomes in many related papers. The main tools utilized in this paper are the theory of cosine families, fixed-point strategy, and the Grönwall-Bellman inequality. At last, an example is given to demonstrate the effectiveness of our research results. Full article

(This article belongs to the Special Issue Fractional Dynamical Systems and Its Applications in Science and Engineering)

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 956

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)

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