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Under the effect of the Rosenblatt process, time-delay systems of nonlinear stochastic delay differential equations are considered. Utilizing the delayed matrix functions and exact solutions for these systems, the existence and Hyers–Ulam stability results are derived. First, depending on the fixed point theory, the existence and uniqueness of solutions are proven. Next, sufficient criteria for the Hyers–Ulam stability are established. Ultimately, to illustrate the importance of the results, an example is provided. [ABSTRACT FROM AUTHOR]

In this paper, by using the Riccati transformation and integral inequality technique, we establish several oscillation criteria for second-order Emden–Fowler neutral delay differential equations under the canonical case and non-canonical case, respectively. Compared with some recent results reported in the literature, we extend the range of the neutral coefficient. Therefore, our results generalize to some of the results presented in the literature. Furthermore, several examples are provided to illustrate our conclusions. [ABSTRACT FROM AUTHOR]

In this paper, a class of two-delay differential equations with coefficient-dependent delay is studied. The distribution of the roots of the eigenequation is discussed, and conditions for the stability of the internal equilibrium and the existence of Hopf bifurcation are obtained. Additionally, using the normal form method and the central manifold theory, the bifurcation direction and the stability for the periodic solution of Hopf bifurcation are calculated. Finally, the correctness of the theory is verified by numerical simulation. [ABSTRACT FROM AUTHOR]

The author considers a nonlinear Caputo fractional-order delay differential equation (CFrDDE) with multiple variable delays. First, we study the existence and uniqueness of the solutions of the CFrDDE with multiple variable delays. Second, we obtain two new results on the Ulam–Hyers–Mittag–Leffler (UHML) stability of the same equation in a closed interval using the Picard operator, Chebyshev norm, Bielecki norm and the Banach contraction principle. Finally, we present three examples to show the applications of our results. Although there is an extensive literature on the Lyapunov, Ulam and Mittag–Leffler stability of fractional differential equations (FrDEs) with and without delays, to the best of our knowledge, there are very few works on the UHML stability of FrDEs containing a delay. Thereby, considering a CFrDDE containing multiple variable delays and obtaining new results on the existence and uniqueness of the solutions and UHML stability of this kind of CFrDDE are the important aims of this work. [ABSTRACT FROM AUTHOR]

The pantograph equation is a basic model in the field of delay differential equations. This paper deals with an extended version of the pantograph delay equation by incorporating a variable coefficient of exponential order. At specific values of the involved parameters, the exact solution is obtained by applying the regular Maclaurin series expansion (MSE). A second approach is also applied on the current model based on a hybrid method combining the Laplace transform (LT) and the Adomian decomposition method (ADM) denoted as (LTADM). Although the MSE derives the exact solution in a straightforward manner, the LTADM determines the solution in a closed series form which is theoretically proved for convergence. Further, the accuracy of such a closed-form solution is examined through various comparisons with the exact solution. For validation, the residual errors are calculated and displayed in graphs. The results show that the solution obtained utilizing the LTADM is in full agreement with the exact solution using only a few terms of the closed-form series solution. Moreover, it is found that the residual errors tend to zero, which reflects the effectiveness of the LTADM. The present approach may merit further extension by including other types of linear delay differential equations with variable coefficients. [ABSTRACT FROM AUTHOR]

In this paper, a new effective technique for the investigation of the higher-order trinomial differential equations y (n) (t) + p (t) y (τ (t)) + q (t) y (σ (t)) = 0 is established. We offer new criteria for so-called property (A) and oscillation of the considered equation. Examples are provided to illustrate the importance of our results. [ABSTRACT FROM AUTHOR]

In this work, we consider the one-dimensional thermoelastic Bresse system by addressing the aspects of nonlinear damping and distributed delay term acting on the first and the second equations. We prove a stability result without the common assumption regarding wave speeds under Neumann boundary conditions. We discover a new relationship between the decay rate of the solution and the growth of ϖ at infinity. Our results were achieved using the multiplier method and the perturbed modified energy, named Lyapunov functions together with some properties of convex functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a faster iterative method for approximating the fixed points of generalized α -nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our findings, we present some nontrivial examples of the considered mappings. Furthermore, we show that the class of mappings considered is more general than some nonexpansive-type mappings. Also, we show numerically that the method studied in our article is more efficient than several existing methods. Lastly, we use our main results to approximate the solution of a delay fractional differential equation in the Caputo sense. Our results generalize and improve many well-known existing results. [ABSTRACT FROM AUTHOR]

In this article, we consider a system of ordinary differential equations (ODEs) of second order with two variable time delays. We obtain new conditions for uniform ultimate bounded (UUB) solutions of the considered system. The technique of the proof is based on the Lyapunov–Krasovskii functional (LKF) method using a new LKF. The main result of this article extends and improves a recent result for ODEs of second order with a constant delay to a more general system of ODEs of second order with two variable time delays. In this particular case, we also give a numerical example to verify the application of the main result of this article. [ABSTRACT FROM AUTHOR]

We present a mathematical model that captures the dynamics of the immune system during allergic reactions. Using delay differential equations, we depict the evolution of T cells, APCs, and IL6, considering cell migration between various body compartments. The biological discussions and interpretations within the article revolve around drug desensitization, highlighting one potential application of the model. We conduct stability analysis on certain equilibrium points, demonstrating stability in some cases and only partial stability in others. Numerical simulations validate the theoretical findings. [ABSTRACT FROM AUTHOR]

In the sense of an arbitrary time scale, some new sufficient conditions on oscillation are presented in this paper for a class of nonlinear third-order delay dynamic equations involving a local fractional derivative with a super-linear neutral term. The established oscillation results include known Kamenev and Philos-type oscillation criteria and are new oscillation results so far in the literature. Some inequalities, the Riccati transformation, the integral technique, and the theory of time scale are used in the establishment of these oscillation criteria. The proposed results unify continuous and discrete analysis, and the process of deduction is further extended to another class of nonlinear third-order delay dynamic equations involving a local fractional derivative with a super-linear neutral term and a damping term. As applications for the established oscillation criteria, some examples are given. [ABSTRACT FROM AUTHOR]

This paper delves into the asymptotic and oscillatory behavior of all classes of solutions of fourth-order nonlinear neutral delay differential equations in the noncanonical form with damping terms. This research aims to improve the relationships between the solutions of these equations and their corresponding functions and derivatives. By refining these relationships, we unveil new insights into the asymptotic properties governing these solutions. These insights lead to the establishment of improved conditions that ensure the nonexistence of any positive solutions to the studied equation, thus obtaining improved oscillation criteria. In light of the broader context, our findings extend and build upon the existing literature in the field of neutral differential equations. To emphasize the importance of the results and their applicability, this paper concludes with some examples. [ABSTRACT FROM AUTHOR]

In this work, we delve into a nonlinear integral equation (IEq) with multiple variable time delays and a nonlinear integro-differential equation (IDEq) without delay. Global existence and uniqueness (GEU) of solutions of that IEq with multiple variable time delays and IDEq are investigated by the fixed point method using progressive contractions, which are due to T.A. Burton. We prove four new theorems including sufficient conditions with regard to GEU of solutions of the equations. The results generalize and improve some related published results of the relevant literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of random impulsive differential equations with infinite delay. When utilizing the nonlinear variation of Leray–Schauder's fixed-point principles together with a technique based on separable vector-valued metrics to establish sufficient conditions for the existence of solutions, under suitable assumptions on Y 1 , Y 2 and ϖ 1 , ϖ 2 , which greatly enriched the existence literature on this system, there is, however, no hope to discuss the uniqueness result in a convex case. In the present study, we analyzed the influence of the impulsive and infinite delay on the solutions to our system. In addition, to the best of our acknowledge, there is no result concerning coupled random system in the presence of impulsive and infinite delay. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper delves into the investigation of quasi-linear neutral differential equations in the third-order canonical case. In this study, we refine the relationship between the solution and its corresponding function, leading to improved preliminary results. These enhanced results play a crucial role in excluding the existence of positive solutions to the investigated equation. By building upon the improved preliminary results, we introduce novel criteria that shed light on the nature of these solutions. These criteria help to distinguish whether the solutions exhibit oscillatory behavior or tend toward zero. Moreover, we present oscillation criteria for all solutions. To demonstrate the relevance of our results, we present an illustrative example. This example validates the theoretical framework we have developed and offers practical insights into the behavior of solutions for quasi-linear third-order neutral differential equations. [ABSTRACT FROM AUTHOR]

This paper studies the stochastic pantograph model, which is considered a subcategory of stochastic delay differential equations. A more general jump process, which is called the Lévy process, is added to the model for better performance and modeling situations, having sudden changes and extreme events such as market crashes in finance. By utilizing the truncation technique, we propose the diffused split-step truncated Euler–Maruyama method, which is considered as an explicit scheme, and apply it to the addressed model. By applying the Khasminskii-type condition, the convergence rate of the proposed scheme is attained in L p (p ≥ 2) sense where the non-jump coefficients grow super-linearly while the jump coefficient acts linearly. Also, the rate of convergence of the proposed scheme in L p (0 < p < 2) sense is addressed where all the three coefficients grow beyond linearly. Finally, theoretical findings are manifested via some numerical examples. [ABSTRACT FROM AUTHOR]

In this work, new oscillation criteria are established for a second-order differential equation with several sublinear neutral terms and in the canonical case. To determine the oscillation conditions, we followed the Riccati approach and also compared the studied equation with a first-order delay equation. Obtaining the oscillation conditions required deducing some new relationships linking the solution to the corresponding function as well as its derivatives. The paper addresses some interesting analytical points in the study of the oscillation of equations with several sublinear neutral terms. These new findings complement some well-known findings in the literature. Furthermore, an example is provided to show the importance of the results. [ABSTRACT FROM AUTHOR]

The asymptotical properties of a special dynamic two-person game are examined under best-response dynamics in both discrete and continuos time scales. The direction of strategy changes by the players depend on the best responses to the strategies of the competitors and on their own strategies. Conditions are given first for the local asymptotical stability of the equilibrium if instantaneous data are available to the players concerning all current strategies. Next, it is assumed that only delayed information is available about one or more strategies. In the discrete case, the presence of delays has an effect on only the order of the governing difference equations. Under continuous scales, several possibilities are considered: each player has a delay in the strategy of its competitor; player 1 has identical delays in both strategies; the players have identical delays in their own strategies; player 1 has different delays in both strategies; and the players have different delays in their own strategies. In all cases, it is assumed that the equilibrium is asymptotically stable without delays, and we examine how delays can make the equilibrium unstable. For small delays, the stability is preserved. In the cases of one-delay models, the critical value of the delay is determined when stability changes to instability. In the cases of two and three delays, the stability-switching curves are determined in the two-dimensional space of the delays, where stability becomes lost if the delay pair crosses this curve. The methodology is different for the one-, two-, and three-delay cases outlined in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a fixed stepsize Euler method for a class of linear impulsive neutral delay differential equations. By taking the partition nodes for the Euler scheme and employing the linear interpolation, we strictly prove the method is convergent of order one. Two examples illustrating the efficiency results are also presented. [ABSTRACT FROM AUTHOR]

We present a new method for obtaining stability conditions for certain classes of delay differential equations. The method is based on the transition from an individual equation to a family of equations, and next the selection of a representative of this family, the test equation, asymptotic properties of which determine those of all equations in the family. This approach allows us to obtain the conditions that are the criteria for the stability of all equations of a given family. These conditions are formulated in terms of the parameters of the class of equations being studied, and are effectively verifiable. The main difference of the proposed method from the known general methods (using Lyapunov–Krasovsky functionals, Razumikhin functions, and Azbelev W-substitution) is the emphasis on the exactness of the result; the difference from the known exact methods is a significant expansion of the range of applicability. The method provides an algorithm for checking stability conditions, which is carried out in a finite number of operations and allows the use of numerical methods. [ABSTRACT FROM AUTHOR]

This paper is focused on energy decay rates for the viscoelastic wave equation that includes nonlinear time-varying delay, nonlinear damping at the boundary, and acoustic boundary conditions. We derive general decay rate results without requiring the condition a 2 > 0 and without imposing any restrictive growth assumption on the damping term f 1 , using the multiplier method and some properties of the convex functions. Here we investigate the relaxation function ψ , namely ψ ′ (t) ≤ − μ (t) G (ψ (t)) , where G is a convex and increasing function near the origin, and μ is a positive nonincreasing function. Moreover, the energy decay rates depend on the functions μ and G, as well as the function F defined by f 0 , which characterizes the growth behavior of f 1 at the origin. [ABSTRACT FROM AUTHOR]

This paper delves into the analysis of oscillation characteristics within third-order quasilinear delay equations, focusing on the canonical case. Novel sufficient conditions are introduced, aimed at discerning the nature of solutions—whether they exhibit oscillatory behavior or converge to zero. By expanding the literature, this study enriches the existing knowledge landscape within this field. One of the foundations on which we rely in proving the results is the symmetry between the positive and negative solutions, so that we can, using this feature, obtain criteria that guarantee the oscillation of all solutions. The paper enhances comprehension through the provision of illustrative examples that effectively showcase the outcomes and implications of the established findings. [ABSTRACT FROM AUTHOR]

We present a systematic introduction to a class of functions that provide fundamental solutions for autonomous linear integer-order and fractional-order delay differential equations. These functions, referred to as delay functions, are defined through power series or fractional power series, with delays incorporated into their series representations. Using this approach, we have defined delay exponential functions, delay trigonometric functions and delay fractional Mittag-Leffler functions, among others. We obtained Laplace transforms of the delay functions and demonstrated how they can be employed in finding solutions to delay differential equations. Our results, which extend and unify previous work, offer a consistent framework for defining and using delay functions. [ABSTRACT FROM AUTHOR]

This research paper focuses on investigating the solvability of a constrained problem involving a nonlinear delay functional equation subject to a quadratic functional integral constraint, in two significant cases: firstly, the existence of nondecreasing solutions in a bounded interval L 1 [ 0 , T ] and, secondly, the existence of nonincreasing solutions in unbounded interval L 1 (R +) . Moreover, the paper explores various qualitative properties associated with these solutions for the given problem. To establish the validity of our claims, we employ the De Blasi measure of noncompactness (MNC) technique as a basic tool for our proofs. In the first case, we provide sufficient conditions for the uniqueness of the solution ψ ∈ L 1 [ 0 , T ] and rigorously demonstrate its continuous dependence on some parameters. Additionally, we establish the equivalence between the constrained problem and an implicit hybrid functional integral equation (IHFIE). Furthermore, we delve into the study of Hyers–Ulam stability. In the second case, we examine both the asymptotic stability and continuous dependence of the solution ψ ∈ L 1 (R +) on some parameters. Finally, some examples are provided to verify our investigation. [ABSTRACT FROM AUTHOR]

In this paper, the global asymptotic stability and global Mittag–Leffler stability of a class of fractional-order fuzzy bidirectional associative memory (BAM) neural networks with distributed delays is investigated. Necessary conditions are obtained by means of the Lyapunov functional method and inequality techniques. The hybrid feedback controllers are then developed to ensure the global asymptotic synchronization of these neural networks, resulting in two additional synchronization criteria. The derived conditions are applied to check the fractional-order fuzzy BAM neural network's Mittag–Leffler stability and synchronization. Three examples are given to demonstrate the effectiveness of the achieved results. [ABSTRACT FROM AUTHOR]

In this paper, the idea of two transformations is first proposed and applied. Some new different sufficient conditions for the asymptotical stability of the exact solutions of nonlinear impulsive neutral delay differential equations (INDDEs) are obtained. A new numerical scheme for INDDEs is also constructed based on the idea. The numerical methods that can preserve the stability and asymptotical stability of the exact solutions are provided. Two numerical examples are provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

We study several stability properties on a finite or infinite interval of inhomogeneous linear neutral fractional systems with distributed delays and Caputo-type derivatives. First, a continuous dependence of the solutions of the corresponding initial problem on the initial functions is established. Then, with the obtained result, we apply our approach based on the integral representation of the solutions instead on some fixed-point theorems and derive sufficient conditions for Hyers–Ulam and Hyers–Ulam–Rassias stability of the investigated systems. A number of connections between each of the Hyers–Ulam, Hyers–Ulam–Rassias, and finite-time Lyapunov stability and the continuous dependence of the solutions on the initial functions are established. Some results for stability of the corresponding nonlinear perturbed homogeneous fractional linear neutral systems are obtained, too. [ABSTRACT FROM AUTHOR]

In this study, we present a mild solution to the Hilfer fractional differential equations with infinite delay. Firstly, we establish the results on an infinite interval; to achieve this, we use the generalized Ascoli–Arzelà theorem and Mönch's fixed point theorem via a measure of noncompactness. Secondly, we consider the existence of a mild solution when the semigroup is compact, and the Schauder fixed-point theorem is used. The outcome is demonstrated using an infinitesimal operator, fractional calculus, semigroup theory, and abstract space. Finally, we present an example to support the results. [ABSTRACT FROM AUTHOR]

This paper delves into the enhancement of asymptotic and oscillatory behaviors in solutions to even-order neutral differential equations with multiple delays. The main objective is to establish improved inequalities to advance the understanding of oscillation theory for these equations. The paper's approach is centered on improving the understanding of the intricate relationship between solutions and their corresponding functions. This is achieved by harnessing the modified monotonic properties of positive solutions, which provide valuable insights into oscillation behavior. Furthermore, leveraging the symmetry between positive and negative solutions, we derived criteria that ensure oscillation for all solutions, with a specific emphasis on excluding only positive solutions. To illustrate the significance of our findings, we provide an illustrative example. [ABSTRACT FROM AUTHOR]

This study aims to examine the oscillatory behavior of third-order differential equations involving various delays within the context of functional differential equations of the neutral type. The oscillation criteria for the solutions of our equation have been obtained in this study to extend and supplement existing findings in the literature. In this study, a technique that relies on repeatedly improving monotonic properties was used in order to exclude positive solutions to the studied equation. Negative solutions are excluded based on the symmetry between the positive and negative solutions. Our results are important because they become sharper when applied to a Euler-type equation as compared to previous studies of the same equation. The significance of the findings was illustrated through the application of these findings to specific cases of the investigated equation. [ABSTRACT FROM AUTHOR]

The method of studying the stability in the probability for nonlinear systems of stochastic difference equations is demonstrated on two systems with exponential and fractional nonlinearities. The proposed method can be applied to nonlinear systems of higher dimensions and with other types of nonlinearity, both for difference equations and for differential equations with delay. [ABSTRACT FROM AUTHOR]

This study investigates the oscillatory properties of a fourth-order delay functional differential equation. This study's methodology is built around two key tenets. First, we propose optimized relationships between the solution and its derivatives by making use of some improved monotonic features. By using a comparison technique to connect the oscillation of the studied equation with some second-order equations, the second aspect takes advantage of the significant progress made in the study of the oscillation of second-order equations. Numerous applications of functional differential equations of the neutral type served as the inspiration for the study of a subclass of these equations. [ABSTRACT FROM AUTHOR]

In the present paper, a transmission problem of the Timoshenko beam in the presence of distributed delay is considered. Under appropriate assumptions, we prove the well-posedness by using the semi-group theory. Furthermore, we study the asymptotic behavior of solutions using the multiplier method. We investigate the techniques and ideas used by the second author to extend the recent results. [ABSTRACT FROM AUTHOR]

The fractional Legendre polynomials (FLPs) that we present as an effective method for solving fractional delay differential equations (FDDEs) are used in this work. The Liouville–Caputo sense is used to characterize fractional derivatives. This method uses the spectral collocation technique based on FLPs. The proposed method converts FDDEs into a set of algebraic equations. We lay out a study of the convergence analysis and figure out the upper bound on error for the approximate solution. Examples are provided to demonstrate the precision of the suggested approach. [ABSTRACT FROM AUTHOR]

In this article, by using the differential Caputo–Fabrizio operator, we suggest a novel family of piecewise differential equations (DEs). The issue under study contains a mixed delay period under the criteria of anti-periodic boundaries. It is possible to utilize the piecewise derivative to describe a variety of complex, multi-step, real-world situations that arise from nature. Using fixed point (FP) techniques, like Banach's FP theorem, Schauder's FP theorem, and Arzelá Ascoli's FP theorem, the Hyer–Ulam (HU) stability and the existence theorem conclusions are investigated for the considered problem. Eventually, a supportive example is given to demonstrate the applicability and efficacy of the applied concept. [ABSTRACT FROM AUTHOR]

We investigate the oscillation of the fourth-order differential equation for a class of functional differential equations of the neutral type. We obtain a new single-oscillation criterion for the oscillation of all the solutions of our equation. We establish new monotonic properties for some cases of positive solutions of the studied equation. Moreover, we improve these properties by using an iterative method. This development of monotonic properties contributes to obtaining new and more efficient criteria for verifying the oscillation of the equation. The results obtained extend and improve previous findings in the literature by using an Euler-type equation as an example. The importance of the results was clarified by applying them to some special cases of the studied equation. The fourth-order delay differential equations have great practical importance due to their wide applications in civil, mechanical, and aeronautical engineering. Research on this type of equation is still ongoing due to its remarkable importance in many fields. [ABSTRACT FROM AUTHOR]

This paper is devoted to a model of epidemic progression, taking into account vaccination and immunity waning. The model consists of a system of delay differential equations with time delays determined by the disease duration and immunity loss. Periodic epidemic outbreaks emerge as a result of the instability of a positive stationary solution if the basic reproduction number exceeds some critical value. Vaccination can change epidemic dynamics, resulting in more complex aperiodic oscillations confirmed by some data on Influenza A in Norway. Furthermore, the measures of social distancing during the COVID-19 pandemic weakened seasonal influenza in 2021, but increased it during the next year. Optimal control allows for the minimization of epidemic cost by vaccination. [ABSTRACT FROM AUTHOR]

This manuscript aims to investigate the existence and uniqueness of fuzzy mild solutions for non-local impulsive neutral functional differential equations of both first and second order, incorporating finite delay. Furthermore, the study explores the properties of fuzzy set-valued mappings of a real variable, where these mappings exhibit characteristics such as normality, convexity, upper semi-continuity, and compact support. The application of the Banach fixed-point theorem is employed to derive the results. The research extensively employs fundamental concepts from fuzzy set theory, functional analysis, and the Hausdorff metric. Additionally, an illustrative example is provided to exemplify the practical implementation of the proposed concept. [ABSTRACT FROM AUTHOR]

Forest pests and diseases have been seriously threatening ecological security. Effective prevention and control of such threats can extend the growth cycle of forest trees and increase the amount of forest carbon sink, which makes a contribution to achieving China's goal of "emission peak and carbon neutrality". In this paper, based on the insect-vector populations (this refers to Monochamus alternatus, which is the main vector in Asia) in pine wilt disease, we establish a two-dimensional delay differential equation model to investigate disease control and the impact of time delay on the effectiveness of it. Then, we analyze the existence and stability of the equilibrium of the system and the existence of Hopf bifurcation, derive the normal form of Hopf bifurcation by using a multiple time scales method, and conduct numerical simulations with realistic parameters to verify the correctness of the theoretical analysis. Eventually, according to theoretical analysis and numerical simulations, some specific suggestions are put forward for prevention and control of pine wilt disease. [ABSTRACT FROM AUTHOR]

This research delves into the field of fractional differential equations with both non-instantaneous impulses and delay within the framework of Banach spaces. Our objective is to establish adequate conditions that ensure the existence, uniqueness, and Ulam–Hyers–Rassias stability results for our problems. The studied problems encompass abstract impulsive fractional differential problems with finite delay, infinite delay, state-dependent finite delay, and state-dependent infinite delay. To provide clarity and depth, we augment our theoretical results with illustrative examples, illustrating the practical implications of our work. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we construct a more realistic mathematical model to study toxoplasmosis dynamics. The model considers two discrete time delays. The first delay is related to the latent phase, which is the time lag between when a susceptible cat has effective contact with an oocyst and when it begins to produce oocysts. The second discrete time delay is the time that elapses from when the oocysts become present in the environment to when they are able to infect. The main aim in this paper is to find the conditions under which the toxoplasmosis can disappear from the cat population and to study whether the time delays can affect the qualitative properties of the model. Thus, we investigate the impact of the combination of two discrete time delays on the toxoplasmosis dynamics. Using dynamical systems theory, we are able to find the basic reproduction number R 0 d that determines the global long-term dynamics of the toxoplasmosis. We prove that, if R 0 d < 1 , the toxoplasmosis will be eradicated and that the toxoplasmosis-free equilibrium is globally stable. We design a Lyapunov function in order to prove the global stability of the toxoplasmosis-free equilibrium. We also prove that, if the threshold parameter R 0 d is greater than one, then there is only one toxoplasmosis-endemic equilibrium point, but the stability of this point is not theoretically proven. However, we obtained partial theoretical results and performed numerical simulations that suggest that, if R 0 d > 1 , then the toxoplasmosis-endemic equilibrium point is globally stable. In addition, other numerical simulations were performed in order to help to support the theoretical stability results. [ABSTRACT FROM AUTHOR]

Hilfer fractional stochastic differential equations with delay are discussed in this paper. Firstly, the solutions to the corresponding equations are given using the Laplace transformation and its inverse. Afterwards, the Picard iteration technique and the contradiction method are brought up to demonstrate the existence and uniqueness of understanding, respectively. Further, finite-time stability is obtained using the generalized Grönwall–Bellman inequality. As verification, an example is provided to support the theoretical results. [ABSTRACT FROM AUTHOR]

Iterative differential equations provide a new idea to study functional differential equations. The study of iterative equations can provide new methods for the study of differential equations with state-dependent delays. In this paper, we are concerned with proving the existence of infinitely many positive solutions to nonlinear first-order iterative systems of singular BVPs on time scales by using Krasnoselskii's cone fixed point theorem in a Banach space. It is worth pointing out that in this paper, we can use the symmetry of the iterative process and Green's function to transform the considered differential equation into an equivalent integral equation, which plays a key role in the proof of the theorem in this paper. [ABSTRACT FROM AUTHOR]

This paper is concerned with collocation methods for one class of impulsive delay differential equations (IDDEs). Some results for the convergence, global superconvergence and local superconvergence of collocation methods are given. We choose a suitable piecewise continuous collocation space to obtain high-order numerical methods. Some illustrative examples are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The purpose of this research is to investigate the asymptotic and oscillatory characteristics of odd-order neutral differential equation solutions with multiple delays. The relationship between the solution and its derivatives of different orders, as well as their related functions, must be understood in order to determine the oscillation terms of the studied equation. In order to contribute to this subject, we create new and significant relationships and inequalities. We use these relationships to create conditions in which positive and N-Kneser solutions of the considered equation are excluded. To obtain these terms, we employ the comparison method and the Riccati technique. Furthermore, we use the relationships obtained to create new criteria, so expanding the existing literature on the field. Finally, we provide an example from the general case to demonstrate the results' significance. The findings given in this work provide light on the behavior of odd-order neutral differential equation solutions with multiple delays. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to study the asymptotic properties of the solutions of linear fractional system with Riemann–Liouville-type derivatives and distributed delays. We prove under natural assumptions (similar to those used in the case when the derivatives are first (integer) order) the existence and uniqueness of the solutions in the initial problem for these systems with discontinuous initial functions. As a consequence, we also prove the existence of a unique fundamental matrix for the homogeneous system, which allows us to establish an integral representation of the solutions to the initial problem for the corresponding inhomogeneous system. Then, we introduce for the studied systems a concept for Hyers–Ulam in time stability and Hyers–Ulam–Rassias in time stability. As an application of the obtained results, we propose a new approach (instead of the standard fixed point approach) based on the obtained integral representation and establish sufficient conditions, which guarantee Hyers–Ulam-type stability in time. Finally, it is proved that the Hyers–Ulam-type stability in time leads to Lyapunov stability in time for the investigated homogeneous systems. [ABSTRACT FROM AUTHOR]

In this paper, we explain the approximate controllability of Ψ -Hilfer fractional neutral differential equations with infinite delay. The outcome is demonstrated using the infinitesimal operator, fractional calculus, semigroup theory, and the Krasnoselskii's fixed point theorem. To begin, we emphasise the presence of the mild solution and show that the Ψ -Hilfer fractional system is approximately controllable. Additionally, we present theoretical and practical examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the averaging principle for ψ -Capuo fractional stochastic delay differential equations (FSDDEs) with Poisson jumps. Based on fractional calculus, Burkholder-Davis-Gundy's inequality, Doob's martingale inequality, and the H o ¨ lder inequality, we prove that the solution of the averaged FSDDEs converges to that of the standard FSDDEs in the sense of L p . Our result extends some known results in the literature. Finally, an example and simulation is performed to show the effectiveness of our result. [ABSTRACT FROM AUTHOR]