Search

In the case of oscillatory potentials, we give sufficient conditions for the oscillation of the forced nonlinear second order differential equations with delayed argument in the form x″(t)+q(t)x(τ(t))γsgnx(τ(t))=f(t) in the linear (γ=1) and the superlinear (γ>1) cases. [Copyright &y& Elsevier]

Copyright of HETS Online Journal is the property of Hispanic Educational Technology Services, Inc. (HETS) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We study locally maximal attractors of expanding dynamical systems. Our main result is a representation of these attractors with the help of topological Markov chains corresponding to the Markov partitions of these attractors, which allows us to describe the dynamics of system on them. Ya. G. Sinai was the first who constructed and used Markov partitions for Anosov's diffeomorphisms [Funk. Anal. Prilozh., 2, No 1, 64; No 3, 70 (1968); English translation:Funct. Anal. Appl., 2, No 1, 61; No 3, 245 (1968)]. Expanding endomorphisms regarded as the simplest representatives of endomorphisms were first studied by M. Shub [Amer. J. Math., 91, No 1, 175 (1969)]. To construct Markov partitions for expanding endomorphisms, we update Sinai's approach in the proper way. A more detailed historical overview can be found in the work by O. M. Sharkovsky [Ukr. Mat. Zh., 74, No. 12, 1709 (2023); English translation:Ukr. Math. J., 74, No. 12, 1950 (2023)]. In this work, Sharkovsky indicated that the methods used to prove the main results presented in [Dokl. Akad. Nauk SSSR, 170, No. 6, 1276 (1966); English translation:Sov. Math. Dokl., 7, No. 5, 1384 (1966)] were, in fact, published in the collection of papers "Dynamical systems and the problems of stability of solutions of differential equations" (1973) issued by the Institute of Mathematics of the Academy of Sciences of Ukraine. This collection is difficultly accessible and was never translated into English. Note that, in the indicated paper, these methods were applied to somewhat different objects. To the best of our knowledge, there is no information about publications of similar results. In view of the outlined history and importance of the described approach (based on Markov partitions and topological Markov chains) for the description of construction of the attractors, it seems reasonable to publish these results anew. [ABSTRACT FROM AUTHOR]

This paper is written to provide ideas and guide faculty who want to design a mathematical modeling course for undergraduate mathematics majors and minors. We discuss course goals, assignments, and projects that can be used to help students gain experience relevant for careers and mathematical modeling opportunities. The authors designed this course to build students' mathematical thought processes and toolbox, ability to analyze and evaluate mathematical models, mathematical modeling skills, and teamwork skills. The course described is intended as an upper division undergraduate course with a prerequisite of differential equations. [ABSTRACT FROM AUTHOR]

We derive Melnikov type conditions for the persistence of periodic solutions in perturbed slowly varying discontinuous differential equations. In contrast to Battelli and Fečkan (Mathematics 9(19):2449, 2021) we assume that the unperturbed (frozen) equation has a family of periodic solutions depending on some parameters. The result of this paper are motivated by and extend a result in Wiggins and Holmes (SIAM J Math Anal 18:592–611, 1987) where the authors considered a two dimensional hamiltonian family of smooth systems depending on a scalar variable which is the solution of a singularly perturbed equation. [ABSTRACT FROM AUTHOR]

In this paper, the existence of positive solutions of the following third-order three-point boundary value problem with p-Laplacian {(Φp(u"(t)))' + f(t, u(t)) = 0, t (0, 1), u(0) = au(η), u(1) = au(η), u"(0) = 0, is studied, where Φp(s) = |s|p-2s, p > 1. By using the fixed point index method, we establish sufficient conditions for the existence of at least one or at least two positive solutions for the above boundary value problem. The main result is demonstrated by providing an example as an application. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new technique to deduce oscillation of a second-order noncanonical advanced differential equation by using established criteria for second-order canonical advanced differential equations. We illustrate our results by presenting two examples. [ABSTRACT FROM AUTHOR]

We give counterexamples for some statements of Borwein and sufficient conditions for the validity of these. [ABSTRACT FROM AUTHOR]

In this paper, we establish a quadratic integral inequality involving the second order derivative of functions in the following form: for all f ∈ D , ∫ a b r | f ″ | 2 + p | f ′ | 2 + q | f | 2 ≥ μ 0 ∫ a b | f | 2. Here r , p , q are real- valued coefficient functions on the compact interval [ a , b ] with r (x) > 0 . D is a linear manifold in the Hilbert function space L 2 (a , b) such that all integrals of the above inequality are finite and μ 0 is a real number that can be determined in terms of the spectrum of a uniquely determined self adjoint differential operator in L 2 (a , b) . The inequality is the best possible, i.e., the number μ 0 cannot be increased. f is a complex-valued function in D. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and stability of solutions in connection to a non-local multiterm boundary value problem (BVP) with differential equations equipped with the Riemann-Liouville (RL) fractional derivative in the sense of Atangana-Baleanu of variable-order. The results about the existence property are investigated and proved via Krasnoselskii's fixed point theorem. Note that all theorems in the present research are studied based on piece-wise constant functions defined on generalized intervals. We shall convert our main BVP with the RL-fractional derivative of the Atangana-Baleanu type of variable-order to an equivalent BVP of constant order of the RL-Atangana-Baleanu derivative. In the next step, we examine the Ulam-Hyers stability for the supposed variable-order RL-Atangana-Baleanu BVP. Finally, we provide some examples to validate that our results are applicable. [ABSTRACT FROM AUTHOR]

Picard's theorem is among the most striking results in complex analysis and plays a decisive role in the development of the theory of entire and meromorphic functions. In this paper, we give an equivalence between Picard's theorem and entire solutions of (i) difference equation Δ η f + a (z) P (f) = 0 in C and (ii) differential equation D k f + h (z) g (f (z)) = 0 in terms of total derivatives in C n . We prove two Picard-type theorems that generalize the results of Lu (Kodai Math J 26:221–229, 2003). In addition, examples have been exhibited to show that certain assumptions in the main result cannot be improved. [ABSTRACT FROM AUTHOR]

In this article, the author presents his opinions about the article "Does demography need differential equations" by Thomas K. Burch. He agrees with Burch regarding the close association of demography to mathematical expression than other social scientific discipline. However, he warns the potential abuse of mathematics' use in population studies.

By introducing new variables in the systems of difference equations from the paper [B. Zlatanovska and D. Dimovski, Systems of difference equations approximating the Lorentz system of differential equations, Contrib. Sec. Math. Tech. Sci. 33 (2012) 75–96], new systems of difference equations are obtained. The solutions of these new systems of difference equations depend only on the initial values a 0 , b 0 , c 0 and the coefficients a i , i ∈ N. The power series whose coefficients are these solutions present the recurrent solutions for the Lorenz system of differential equations. A comparison between the behavior of the Lorenz system of differential equations and the local behavior of its recurrent solutions for small time steps is examined in examples by computer simulations, as in the papers [B. Zlatanovska and D. Dimovski, Systems of difference equations approximating the Lorentz system of differential equations, Contrib. Sec. Math. Tech. Sci. 33 (2012) 75–96; B. Zlatanovska and D. Dimovski, Systems of difference equations as a model for the Lorentz system, in Proc. Fifth Int. Scientific Conf. FMNS - 2013, Vol. I (Blagoevgrad, Bulgaria, 2013), pp. 102–107 and B. Zlatanovska, Approximation for the solutions of Lorenz system with systems of differential equations, Bull. Math. 41(1) (2017) 51–61]. [ABSTRACT FROM AUTHOR]

The analysis of real-world data in classical statistics is commenced by deriving a density function which requires a lengthy function selection and parameter estimation process. The process is supported by difficult integration to obtain the cumulative frequency function. In view of the difficulties in the existing methods, a super function called the Persian Probability Curve is proposed in this paper. This function is based on logical reasoning and mathematical concepts to overcome the shortcomings of the classical method. It is shown that the Persian probability is equivalent to that of the Burr in the governing differential equation. The validity of the work has been verified by comparing the obtained results with those of others. [ABSTRACT FROM AUTHOR]

This paper deals with the Hyers-Ulam-Rassias stability of nthorder linear partial differential equation. Result is obtained by using Laplace Transform. [ABSTRACT FROM AUTHOR]

Mathematics and its applications try to make available a simple and accurate approach to handle nonlinear equations. The present paper investigates for the first time the application of the linearizing method to determine both the approximate frequency and displacement amplitude for the third‐order differential equations. The main merit of the linearized method is a collection of simplicity and accuracy for the solution of high‐order nonlinear conservative or non‐conservative oscillators. The current work sheds light on the approximate proposition to the damped third‐order nonlinear differential equations. The fast estimated solution is simulated to the accurate solution gained through the numerical methods. It is significant to conclude that the non‐perturbative method considered here is simpler and easier to apply than the other perturbation methods. This paper enriches some existing literature. [ABSTRACT FROM AUTHOR]

The present paper dealt with the study of solitons of Lorentzian Kähler space-time manifolds. In this paper, we have discussed different conditions for solitons to be steady, expanding, or shrinking in terms of isotropic pressure, the cosmological constant, energy density, nonlinear equations, and gravitational constant in pseudo-quasi-conformally flat and pseudo-$ W_8 $ flat Lorentzian Kähler space-time manifolds.

The basic reproduction, or reproductive number, is a useful index that indicates whether or not there will be an epidemic. However, it is also very important to determine whether an epidemic will eventually decrease and disappear or persist as an endemic. Different infectious diseases have different behaviors and mathematical models used to simulated them should capture the most important processes; however, the models also involve simplifications. Influenza epidemics are usually short-lived and can be modeled with ordinary differential equations without considering demographics. Delays such as the infection time can change the behavior of the solutions. The same is true if there is permanent or temporary immunity, or complete or partial immunity. Vaccination, isolation and the use of antivirals can also change the outcome. In this paper, we introduce several new models and use them to find the effects of all the above factors paying special attention to whether the model can represent an infectious process that eventually disappears. We determine the equilibrium solutions and establish the stability of the disease-free equilibrium using various methods. We also show that many models of influenza or other epidemics with a short duration do not have solutions with a disappearing epidemic. The main objective of the paper is to introduce different ways of modeling immunity in epidemic models. Several scenarios with different immunities are studied since a person may not be re-infected because he/she has total or partial immunity or because there were no close contacts. We show that some relatively small changes, such as in the vaccination rate, can significantly change the dynamics; for example, the existence and number of the disease-free equilibria. We also illustrate that while introducing delays makes the models more realistic, the dynamics have the same qualitative behavior.

The nodes and their connection relationships are the two main bodies for dynamic complex networks. In existing theoretical researches, the phenomena of stabilization and synchronization for complex dynamical networks are generally regarded as the dynamic characteristic behaviors of the nodes, which are mainly caused by coupling effect of connection relationships between nodes. However, the connection relationships between nodes are also one main body of a time-varying dynamic complex network, and thus they may evolve with time and maybe show certain characteristic phenomena. For example, the structural balance in the social networks and the synaptic facilitation in the biological neural networks. Therefore, it is important to investigate theoretically the reasons in dynamics for the occurrence. Especially, from the angle of large-scale systems, how the dynamic behaviors of nodes (such as the individuals, neurons) contribute to the connection relationships is one of worthy research directions. In this paper, according to the structural balance theory of triad proposed by F. Heider, we mainly focus on the connection relationships body, which is regarded as one of the two subsystems (another is the nodes body), and try to find the dynamic mechanism of the structural balance with the internal state behaviors of the nodes. By using the Riccati linear matrix differential equation as the dynamic model of connection relationships subsystem, it is proved under some mathematic conditions that the connection relationships subsystem is asymptotical structural balance via the effects of the coupling roles with the internal state of nodes. Finally, the simulation example is given to show the validity of the method in this paper. [ABSTRACT FROM AUTHOR]

In this paper, new analytical solutions for a class of conformable fractional differential equations (CFDEs) and some more results about Laplace transform introduced by Abdeljawad are investigated. The Laplace transform method is developed to get the exact solution of CFDEs. The aim of this paper is to convert the CFDEs into ordinary differential equations (ODEs), this is done by using the fractional Laplace transform of (α + β) order. [ABSTRACT FROM AUTHOR]

The Bessel process in low dimension (0 ≤ δ ≤ 1) is not an Itô process and it is a semimartingale only in the cases δ = 1 and δ = 0. In this paper we first characterize it as the unique solution of an SDE with distributional drift or more precisely its related martingale problem. In a second part, we introduce a suitable notion of path-dependent Bessel processes and we characterize them as solutions of path-dependent SDEs with distributional drift. [ABSTRACT FROM AUTHOR]

For a quasi-periodically forced differential equation, response solutions are quasi-periodic ones whose frequency vector coincides with that of the forcing function and they are known to play a fundamental role in the harmonic and synchronizing behaviors of quasi-periodically forced oscillators. These solutions are well-understood in quasi-periodically perturbed nonlinear oscillators either in the presence of large damping or in the non-degenerate cases with small or free damping. In this paper, we consider the existence of response solutions in quasi-periodically perturbed, second order differential equations, including nonlinear oscillators, of the form x ¨ + λ x l = ϵ f (ω t , x , x ˙) , x ∈ R , where λ is a constant, 0 < ϵ ≪ 1 is a small parameter, l > 1 is an integer, ω ∈ R d is a frequency vector, and f : T d × R 2 → R 1 is real analytic and non-degenerate in x up to a given order p ≥ 0 , i.e., [ f (· , 0 , 0) ] = [ ∂ f (· , 0 , 0) ∂ x ] = [ ∂ 2 f (· , 0 , 0) ∂ x 2 ] = ⋯ = [ ∂ p - 1 f (· , 0 , 0) ∂ x p - 1 ] = 0 and [ ∂ p f (· , 0 , 0) ∂ x p ] ≠ 0 , where [ ] denotes the average value of a continuous function on T d . In the case that λ = 0 and f is independent of x ˙ , the existence of response solutions was first shown by Gentile (Ergod Theory Dyn Syst 27:427–457, 2007) when p = 1 . This result was later generalized by Corsi and Gentile (Commun Math Phys 316:489–529, 2012; Ergod Theory Dyn Syst 35:1079–1140, 2015; Nonlinear Differ Equ Appl 24(1):article 3, 2017) to the case that p > 1 is odd. In the case λ ≠ 0 , the existence of response solutions is studied by the authors Si and Yi (Nonlinearity 33(11):6072–6099, 2020) when p = 0 . The present paper is devoted to the study of response solutions of the above quasi-periodically perturbed differential equations for the case λ ≠ 0 by allowing p > 0 . Under the conditions that 0 ≤ p < l / 2 and λ [ ∂ p f (· , 0 , 0) ∂ x p ] > 0 when l - p is even, we obtain a general result which particularly implies the following: (1) If either l is odd and λ < 0 or l is even and [ ∂ p f (· , 0 , 0) ∂ x p ] > 0 , then as ϵ sufficiently small response solutions exist for each ω satisfying a Brjuno-like non-resonant condition; (2) If either l is odd and λ > 0 or l is even and [ ∂ p f (· , 0 , 0) ∂ x p ] < 0 , then there exists an ϵ ∗ > 0 sufficiently small and a Cantor set E ∈ (0 , ϵ ∗) with almost full Lebesgue measure such that response solutions exist for each ϵ ∈ E and ω satisfying a Diophantine condition. Similar results are also obtained in the case λ = ± ϵ which particularly concern the existence of large amplitude response solutions. [ABSTRACT FROM AUTHOR]

This paper proposes a modified D-iteration to approximate the solutions of three quasinonexpansive multivalued mappings in a real Hilbert space. Due to the incorporation of an inertial step in the iteration, the sequence generated by the modified method converges faster to the common fixed point of the mappings. Furthermore, the generated sequence strongly converges to the required solution using a shrinking technique. Numerical results obtained indicate that the proposed iteration is computationally effcient and outperforms the standard forward-backward with inertial step. [ABSTRACT FROM AUTHOR]

In this paper, we take advantage of implicit relationships to come up with a new concept called "Aϑ-α-contraction mapping". We utilized our new notion to formulate and prove some common fixed point theorems for two and four self-mappings over complete extended quasi b-metric spaces under a set of conditions. Our main results widen and improve many existing results in the literature. To support our research, we present some examples as applications to our main findings. [ABSTRACT FROM AUTHOR]

This paper studies the existence of solutions for Caputo-Hadamard fractional nonlinear differential equations of variable order (CHFDEVO). We obtain some needed conditions for this purpose by providing an auxiliary constant order system of the given CHFDEVO. In other words, with the help of piece-wise constant order functions on some continuous subintervals of a partition, we convert the main variable order initial value problem (IVP) to a constant order IVP of the Caputo-Hadamard differential equations. By calculating and obtaining equivalent solutions in the form of a Hadamard integral equation, our results are established with the help of the upper-lower-solutions method. Finally, a numerical example is presented to express the validity of our results. [ABSTRACT FROM AUTHOR]

In this paper, we aim to solving the open question left in [Nie, Yuan: Nonlinear Anal 196 (2020); J. Math. Anal. Appl 505 (2022) and Xiao, Fei: J. Math. Anal. Appl 514 (2022)]. We prove that a multidimensional chemotaxis system is ill-posedness in B ˙ 2 d , r - 3 2 × ( B ˙ 2 d , r - 1 2 ) d when 1 ≤ r < d due to the lack of continuity of the solution. [ABSTRACT FROM AUTHOR]

Existence of symmetric solutions to the Gaussian Minkowski problem was established by Huang et al. In this paper, we show the existence of non-symmetric solutions to this problem by studying the related Monge–Ampère type equation on the sphere. [ABSTRACT FROM AUTHOR]

Separation of variables is a common method for producing an analytical based solution to partial differential equations. Despite the wide application of this method, often the physical phenomena described by the differential equations are not adequately involved in the discourse over the appropriate methods to solve a given problem, particularly in mathematics curricula. However, as mathematics is the tool to better understanding of the physical world, the meaning of the differential equation, boundary conditions, and initial conditions cannot be detached from the methods used to solve the differential equations. Failure to recognize the physical conditions being studied can lead to solution methods that are invalid or unphysical. This paper demonstrates how awareness of the physical nature of the system being investigated and its relationship to the mathematics can guide the selection of the relevant solution methods. To illustrate the importance of the comprehension of the physical meaning behind the mathematical equations and representations and the need to avoid rote application a solution technique, the logic behind the selection of the appropriate solution techniques for the one-dimensional transient heat conduction equation is considered under different imposed conditions which lead to different trends in system operation. [ABSTRACT FROM AUTHOR]