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In this paper, we have formulated and analyzed a mathematical model with multiple time delays of the genetic regulatory network of the microRNA‐17‐92 (miR‐17‐92) cluster. We have focused on the effect of the associated transcriptional and translational time delays of both positive and negative feedback loops on the miR‐17‐92 regulatory network. To gain better insight, we have explored the role of individual time delay in controlling the transition of the G1/S cell cycle process. We pay particular attention to the concentration of E2F and Myc, which trigger the protein ON and OFF switches. To analyze the situation mathematically, we employ delay‐dependent stability analysis and bifurcation analysis using delay differential equations and a biological study of the time‐delayed random transitions between the ON and OFF states for protein synthesis. Based on our study, we can conclude that the translational time delay (τ2)$$ \left({\tau}_2\right) $$ can halt the healthy transition of cells in the G1/S cell cycle phase as the small increase in (τ2)$$ \left({\tau}_2\right) $$ can destroy bistability of the system. Simulations are carried out to numerically show the solution trajectories under the combined effect of delays. Finally, we summarize the simulation results and the impact of delays on the dual behavior of miR‐17‐92 as it can act as an oncogene or as a tumor suppressor gene. [ABSTRACT FROM AUTHOR]

This article is devoted towards some necessary conditions for the existence of the mild solution of 'second‐order semilinear stochastic delay differential equations' having a variable delay in the state with the help of Banach fixed point theorem and theory of cosine family which is strongly continuous. In the last section, we have discussed an example to understand the theory of provided results in a better manner. [ABSTRACT FROM AUTHOR]

In this article, we initially provided the relationship between the RL fractional integral and the Caputo fractional derivative of different orders. Additionally, it is clear from the literature that studies into boundary value problems involving multi‐term operators have been conducted recently, and the aforementioned idea is used in the formulation of several novel models. We offer a unique coupled system of fractional delay differential equations with proper respect for the role that multi‐term operators play in the research of fractional differential equations, taking into account the newly established solution for fractional integral and derivative. We also made the assumptions that connected integral boundary conditions would be added on top of n$$ n $$‐fractional differential derivatives. The requirements for the existence and uniqueness of solutions are also developed using fixed‐point theorems. While analyzing various sorts of Ulam's stability results, the qualitative elements of the underlying model will also be examined. In the paper's final section, an example is given for purposes of demonstration. [ABSTRACT FROM AUTHOR]

This paper deals with the free boundary problem for a double‐layered tumor filled with quiescent cells and proliferating cells, where time delay τ>0$$ \tau >0 $$ in cell proliferation is taken into account. These two types of living cells exhibit different metabolic responses and consume nutrients σ$$ \sigma $$ at different rates λ1$$ {\lambda}_1 $$ and λ2$$ {\lambda}_2 $$ ( λ1⩽λ2$$ {\lambda}_1\leqslant {\lambda}_2 $$). Time delay happens between the time at which a cell commences mitosis and the time at which the daughter cells are produced. The problem is reduced to a delay differential equation on the tumor radius R(t)$$ R(t) $$ over time, and the difficulty arises from the jump discontinuity of the consumption rate function. We give rigorous analysis on this new model and study the dynamical behavior of the global solutions for any initial φ(t)$$ \varphi (t) $$. [ABSTRACT FROM AUTHOR]

This work introduces a new stochastic volatility model with delay parameters in the volatility process, extending the Barndorff–Nielsen and Shephard model. It establishes an analytical expression for the log price characteristic function, which can be applied to price European options. Empirical analysis on S&P500 European call options shows that adding delay parameters reduces mean squared error. This is the first instance of providing an analytical formula for the log price characteristic function in a stochastic volatility model with multiple delay parameters. We also provide a Monte Carlo scheme that can be used to simulate the model. [ABSTRACT FROM AUTHOR]

This paper trains a multilayered neural network (MLNN) for solving fractional delay differential equations (FDDEs), including nonlinear and singular types. The proposed methodology involves replacing the unknown functions in the equations with a truncated power series expansion. Subsequently, a collection of algebraic equations is solved utilizing an iterative minimization technique that leverages the capabilities of the MLNN architecture. The outcomes demonstrate that the MLNN architecture provides the required accuracy and strong stability compared to several numerical methods. [ABSTRACT FROM AUTHOR]

The problem of stabilizing an unstable highly nonlinear hybrid neutral stochastic functional differential equation with infinite delay (HNSFDEwID) is addressed in this article. The objective of this study is to propose a discrete‐time feedback control (DTFC) approach for stabilizing the original system. Sufficient conditions are derived for the existence and uniqueness of the system's solution, as well as its boundedness, prerequisites for studying the system's dynamic behavior. Four distinct criteria for stabilizing the controlled system are provided, including H∞$$ {H}_{\infty } $$ stabilization, asymptotic stabilization, exponential stabilization, and almost surely exponentially stabilization. Additionally, the upper limit of the time interval for discrete observations is calculated based on these criteria. Finally, the proposed theory is validated through an illustrative example. [ABSTRACT FROM AUTHOR]

In this study, the stochastic distributed‐order time‐fractional version of the fourth‐order delay sub‐diffusion equation is defined by employing the Caputo fractional derivative. The orthonormal discrete Legendre polynomials, as a well‐known family of discrete polynomials basis functions, are used to develop a numerical method to solve this equation. To employ these polynomials in constructing the expressed approach, the operational matrices of the classical integration, differentiation (ordinary, fractional and distributed‐order fractional), and stochastic integration of these polynomials are extracted. The established method turns solving the introduced stochastic‐fractional equation into solving a more simple linear algebraic system of equations. In fact, by representing the unknown solution in terms of the introduced polynomials and employing the extracted matrices, this system is obtained. The accuracy of the developed algorithm is numerically checked by solving two examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a numerical method for solving singularly perturbed delay differential equations using Liouville–Green transformation. As an initial step, to approximate the term with negative shift, we use Taylor series, which in turn changes the equation into a singular perturbation problem with the same asymptotic behavior. Then we utilize the Liouville–Green transformation technique to derive a recurrence relation comprising three terms, which can be solved by using the Thomas algorithm. A look into the method's stability and convergence has been done. The MATLAB R2022a mathematical software has been used to obtain the computational results and plots. Almost second‐order accuracy is achieved with the scheme derived. The numerical findings have been compiled into tables and graphs have been provided to visualize the impact of delay on the solution boundary layer and oscillatory behavior. The convergence rate of the present method has been determined theoretically and numerically, and both are shown to agree. In brief, the present method enhances the results of certain previously published numerical approaches. [ABSTRACT FROM AUTHOR]

This paper explores general coupled dynamic equations on time scales with multiple delays and investigates the existence of periodic solutions using the coincidence degree theory approach. The model studied includes the mathematical models of Nicholson, Mackey–Glass, and Lasota–Wazewska as special cases. Furthermore, we demonstrate that the solutions are not only asymptotically stable but also exponentially synchronized. The presented results significantly extend and complement existing findings in the field. The paper concludes with illustrative examples that highlight the practical implications of our analytical discoveries. [ABSTRACT FROM AUTHOR]

In this article, we focussed on a nonlinear viscoelastic Kirchhoff‐type equation with distributed delay and variable‐exponents. The blow‐up solutions of the problem are proved under suitable hypothesis, and by using an integral inequality due to Komornik, the general decay outcome is obtained in the case b=0$$ b=0 $$. [ABSTRACT FROM AUTHOR]

In the realm of modeling biological systems with memory, particularly those involving intricate interactions like tumor-immune responses, the utilization of multiple time delays and Caputo-type fractional-order derivatives represents a cutting-edge approach. In this research paper, we introduce a novel fractional-order model to investigate the dynamic interplay between tumors and macrophages, a key component of the immune system, while incorporating multiple time delays into our framework. Our proposed model comprises a system of three Caputo-type fractional-order differential equations, each representing distinct cell populations: tumor cells, anti-tumor cells (specifically M1-type macrophages with pro-inflammatory properties), and pro-tumor cells (M2-type macrophages with immune-suppressive characteristics). The stability of equilibria is discussed by analyzing the characteristic equations for each case, and the existence conditions for the Hopf bifurcation are obtained according to the critical values of delay parameters. Furthermore, numerical simulations are presented in order to verify the analytical results obtained for stability and Hopf-bifurcation with respect to the two-time delay parameters τ1 and τ2. The analysis shows the rich dynamics of the model according to the fractional-order parameter and the time delay parameters. [ABSTRACT FROM AUTHOR]

In this work, we extend previous research on exponential integrators for stiff semilinear delay differential equations to the nonlinear case. In addition to the stiffness, there are two new issues that should be handled properly: nonlinear term and delay term. For nonlinear problems, a badly chosen linearization can cause a severe step size restriction. In this work, we linearize the equation along the numerical solution in each step. For the delay term, the interpolation based on the numerical values at the mesh points rather than inner stage values is adopted to significantly reduce the number of stiff order conditions. We focus on the construction and convergence analysis of high‐order exponential Rosenbrock methods for nonlinear stiff delay differential equations. The main result of this paper is that under the framework of strongly continuous semigroup, the explicit exponential Rosenbrock method is proved to be stiffly convergent of order p$$ p $$ even if the order conditions of order p$$ p $$ hold in a weak form. Moreover, by pointing that there does not exist fifth‐order method with less than or equal to four stages, we present the construction of a fifth‐order method with five stages. Finally, numerical tests are carried out to validate the theoretical results and to demonstrate the superiority of high‐order methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonlinear second‐order functional differential equations with piecewise constant arguments with applications to a thermostat that is controlled by the introduction of functional terms in the temperature and the speed of change of the temperature at some fixed instants. We first prove some comparison results for boundary value problems associated to linear delay differential equations that allow to give a priori bounds for the derivative of the solutions, so that we can control not only the values of the solutions but also their rate of change. Then, we develop the method of upper and lower solutions and the monotone iterative technique in order to deduce the existence of solutions in a certain region (and find their approximations) for a class of boundary value problems, which include the periodic case. In the approximation process, since the sequences of the derivatives for the approximate solutions are, in general, not monotonic, we also give some estimates for these derivatives. We complete the paper with some examples and conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we study stability of hybrid stochastic differential equations (SDEs) with multiple delays by intermittent control based on time‐varying delay observations. By constructing an auxiliary non‐delayed hybrid stochastic differential equation and using M$$ M $$‐matrix theory, Lyapunov method, and the comparison principle, sufficient criterions to guarantee the q$$ q $$th moment exponential stability and almost sure exponential stability of hybrid SDEs are derived. Finally, a simulation example is shown to illustrate the validity of theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents the computational analysis of fractional differential equations of variable‐order delay systems. To the proposed problem, the existence of solutions is derived using Arzela‐Ascoli theorem, and the Banach fixed point theorem is used for uniqueness results. To investigate and address the computational solutions, Adams‐Bashforth‐Moulton technique is established. To demonstrate the method's efficiency, computational simulations of chaotic behaviors in several one‐dimensional delayed systems with distinct variable orders are employed. The numerical solution of the proposed problem gives high precision approximations. [ABSTRACT FROM AUTHOR]

In this work, we discuss the existence and approximate controllability results for impulsive fractional differential systems of order 1

In this paper, we establish global fractional Halanay‐type inequalities with distributed delays in differential and integral forms and consequently give a simple sufficient condition for Mittag–Leffler stability of solutions to fractional differential equations with this type of delays. Our essential tool is the submultiplicative property of the Mittag–Leffler functions which imitates the multiplicative property of the exponential functions in the classical theory of differential equations. The estimation with explicit decaying rate and amplitude constant is a faithful extension of the classical Halanay inequality. [ABSTRACT FROM AUTHOR]

In this work, we establish a null controllability property of a dispersive age‐structured population dynamics model with finite‐time delay on the mortality rate term. We assume that the dispersion term includes an interior degeneracy satisfying some suitable assumptions. To achieve our main results, we act according to the classical procedure, which is the Carleman estimates of a suitable adjoint system, and then its associated observability inequality. It is well known that the basis of Carleman's estimates is an adjusted weight function and one of the originalities of our article is the expression of the function as a function of age and time variables. [ABSTRACT FROM AUTHOR]

In this paper, a viscoelastic equation with nonlinear distributed delay and nonlinear damping effects is considered. Under very general conditions that do not involve differential inequalities for the kernel function, a general decay criterion is established through the perturbed energy method and properties of convex functions. We extend the stability result of wave equation with linear delay and linear damping to the nonlinear case. [ABSTRACT FROM AUTHOR]

In this work, we obtain an expression, given in terms of some special functions, for the exact numerical solution of the second order delay differential equation x′′(t)=ax(t)+bx(t−τ),$$ {x}^{\prime \prime }(t)= ax(t)+ bx\left(t-\tau \right), $$ with general initial condition x(t)=f(t)$$ x(t)=f(t) $$ for −τ≤t≤0$$ -\tau \le t\le 0 $$. Based on this exact solution, we propose a family of nonstandard finite difference methods that combine high order accuracy and efficient computational properties. We also show that the numerical solutions obtained with the new schemes are consistent with asymptotic stability properties of the exact solutions. [ABSTRACT FROM AUTHOR]

This manuscript aims to establish a new set of sufficient conditions for the existence and approximate controllability of a neutral‐type fractional integro‐differential system employed with noninstantaneous impulse and finite delay in Hilbert space. In order to bring off the main results, the author used fractional calculus, semigroup theory, and Schauder's fixed‐point theorem. At the end, we provide an example to illustrate the main results. [ABSTRACT FROM AUTHOR]

We establish the existence and uniqueness of mild solutions of a first‐order neutral differential equation with state‐dependent delay and almost sectorial operator by using the theory of almost sectorial operators and fixed point theorems and show that under certain conditions mild solution becomes the classical solution. Later, we also discuss the local well‐posedness of the associated Cauchy problem. [ABSTRACT FROM AUTHOR]

The asymptotic stability and long‐time decay rates of solutions to linear Caputo time‐fractional ordinary differential equations (FODEs) with order both 0<α<1$$ 0&lt;\alpha &lt;1 $$ and 1<α<2$$ 1&lt;\alpha &lt;2 $$ are known to be completely determined by the eigenvalues of the coefficient matrices. Very different from the exponential decay of solutions to classical ordinary differential equations, solutions of FODEs decay only polynomially, leading to the so‐called Mittag–Leffler stability, which was already extended to linear fractional delay differential equations (FDDEs) with order 0<α<1$$ 0&lt;\alpha &lt;1 $$. However, the study on the long‐time decay rates of the solutions for FDDEs with order 1<α<2$$ 1&lt;\alpha &lt;2 $$ is scarce. Hence, this paper is devoted to the asymptotical stability and long‐time decay rates for a class of FDDEs with 1<α<2$$ 1&lt;\alpha &lt;2 $$, that is, fractional delay diffusion‐wave equations. We reformulate the fractional delay diffusion‐wave equations to infinite‐dimensional FDDEs by the eigenvalue decomposition method. Through a detailed analysis for the characteristic equations, a novel necessary and sufficient stability condition is established in a coefficient criterion. Moreover, we are able to deal with the awful singularities caused by delay and fractional exponent by introducing a novel integral path, and hence to give an accurate estimation for the fractional resolvent operators. We show that the long‐time decay rates of the solutions for both linear FDDEs and fractional delay diffusion‐wave equations are O(t1−α)$$ \mathcal{O}\left({t}&amp;#x0005E;{1-\alpha}\right) $$. Finally, an example of an application model with feedback control is provided to show the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study a system of multiple positive solutions and stability results for nonlinear fractional delay differential equations involving p$$ p $$‐Laplacian operator. We derived adequate conditions to ensure that at least three nonnegative solutions exist by applying the conditions of the Leggett–Williams fixed‐point theory and some Green function properties. Due to a small change in time‐delay, we analyzed the Hyers–Ulam stability‐type of the equation. We used Riemann–Liouville fractional differential definition, and we assumed that nonzero delay ϑ>0$$ \vartheta >0 $$. In addition, for application purpose, comprehensive examples are given to ensure the effectiveness and feasibility of the results in this paper. Our proposed equation generalize some literature in the system. [ABSTRACT FROM AUTHOR]

This article investigates the exact and approximate controllability, finite time stability, and B$$ \mathcal{B} $$–Hyers–Ulam–Rassias stability of a neutral impulsive system of differential equations with two delays. Fixed point theory is associated in the analysis of controllability of the system. The stability analysis is carried out with the help of Grönwall type inequality. [ABSTRACT FROM AUTHOR]

In this work, we investigate complete stability behaviors of the Rosenzweig–MacArthur predator–prey model with delayed harvesting of the prey. The unharvested system exhibits either a steady‐state or an oscillatory dynamics for the coexistence of the species. We explore how the delayed harvesting affects the dynamics of these two modes by analyzing the system stability in effort‐delay bi‐parameter plane. Some novel dynamical scenarios and intricate dynamics are obtained. Analytical conditions for different stability scenarios are derived by examining the associated quasi‐polynomial eigenvalue equation. For invariant harvesting effort, the time delay induces four stability scenarios: stability invariance, instability invariance, stability change, and stability switching. On the other hand, the effort instigates five stability scenarios: stability invariance, instability invariance, stability change, instability change, and instability switching, when the delay strength is fixed. Majority of literatures on harvesting reported that harvesting stabilizes predator–prey interactions. However, we will show that the delayed harvesting can destabilize the system. One of the novelties of the study is to unveil the occurrence of effort‐induced chaos via period‐doubling mechanism. Interestingly, the effort‐induced switching phenomena and chaos do not occur for non‐delayed harvesting. [ABSTRACT FROM AUTHOR]

The focal point of this paper is to further enhance the existing stochastic epidemic models by incorporating several new disease characteristics, such as the validation time of the vaccination procedure, the stages of vaccine required to gain a long‐period immunity together with the time separating each stage, the deaths linked to the vaccine, and finally, the sudden environmental noise which is exhibited by sociocultural changes, such as antivaccination movements. To incorporate all the aforementioned characteristics, we extend the standard Susceptible‐Vaccinated‐Infected‐Recovered (SVIR) epidemic model to a new mathematical model, which is governed by a system of coupled stochastic delay differential equations, in which the disease transmission rates are driven by Gaussian noise and Lévy‐type jump stochastic process. First, under suitable conditions on the jump intensities, we address the mathematical well‐posedness and biological feasibility of the model, by virtue of the Lyapunov method and the stopping‐time technique. Then, by choosing an adequate positively invariant set for the considered model, we establish sufficient conditions guaranteeing the disease extinction and persistence. Lastly, to support the theoretical results, we provide the outcome of several numerical simulations which, together with our conducted analysis, indicate that the spread of the disease can be majorly altered by all the new considered characteristics. [ABSTRACT FROM AUTHOR]

We study a mathematical model of gene transcription and protein synthesis with negative feedback. We consider a system of equations taking into account the formation of dimers (i.e., complex formed by two protein monomers), the way in which dimers bind to DNA and time delay in translation process. For the model consisting of three ordinary differential equations with time delay, we derive conditions for stability of the positive steady state and for the existence of the Hopf bifurcation. We check the influence of delay in transcription process on the model dynamics, and we formulate conditions determining the type of the Hopf bifurcation and stability of arising limit cycle. [ABSTRACT FROM AUTHOR]

This paper addresses the existence of stability results for Ulam–Hyers (UHS) and Ulam–Hyers–Rassias (UHRS) in the setting of Caputo–Hadamard fractional functional stochastic differential equations with delay (FFSDEwD). We first prove existence and uniqueness using Banach fixed point theorem coupled with standard stochastic analysis techniques. Then, we deal with UHS and UHRS results of Caputo–Hadamard FFSDEwD through an application of Gronwall inequality. Our theoretical findings are corroborated with two numerical examples. [ABSTRACT FROM AUTHOR]

The article considers a nonlinear boundary value problem for a linear delay differential equation. To solve the problem, the idea of parametrization method, namely, the interval at which the problem is being considered, is divided into subintervals whose lengths do not exceed the values of the constant delay; constant parameters are introduced at the left ends of these intervals; a new unknown function is introduced at each subinterval. Thus, the problem under consideration is reduced to an equivalent multipoint boundary value problem for differential equations with delay containing parameters. Auxiliary Cauchy problems without delay with zero initial conditions at the left ends of the subintervals are consistently considered on each of the subintervals. Using an analog of the Cauchy formula to represent the solution of a system of linear differential equations on each of the subintervals and given nonlinear boundary conditions, an algebraic system with respect to unknown parameters is obtained. The article proposes an algorithm for finding a solution to a multipoint boundary value problem for differential equations with a delay containing parameters. At each step of the algorithm, a system of nonlinear algebraic equations is solved to determine the values of the parameters, and an analog of the Cauchy formula is used to obtain solutions to auxiliary Cauchy problems. The results obtained are demonstrated on a test problem. [ABSTRACT FROM AUTHOR]

This paper focuses on exponential stability of numerical solutions of neutral stochastic delay differential equations. A novel approach is introduced to study numerical approximation methods of neutral stochastic differential equations with time‐dependent delay. In contrast to the advances in the literature, this work provides a new method and new criteria for which the Euler‐Maruyama approximation method and the backward Euler‐Maruyama approximation method can reproduce exponential stability in mean square and almost sure exponential stability for sufficiently small step sizes. Two examples are provided to demonstrate the effectiveness of our criteria. [ABSTRACT FROM AUTHOR]

Wood plays a very important role for daily household needs. In this paper, we have taken three parameters, forest biomass density FB, wood‐based industries density WI, and synthetic‐based industries density SI in the proposed mathematical modeling. The forest biomass is assumed to grow logistically, which requires a fixed time to achieve full growth. The excessive cutting of wood for the growth of wood‐based industries not only depletes these forest resources, but there comes a delay in the growth of forest biomass as well. On the contrary, the growth of synthetic‐based industry is independent of forest biomass, yet there is a competition between wood‐based products and synthetic‐based products as per market demands. With the help of differential equation, we studied the stability of the system. Delay plays a significant role in the system to understand the complex behavior of the system about non‐zero equilibrium point. Hopf‐bifurcation observed and directional analysis is studied using normal form theory and manifold reduction. MATLAB software is used to graphical representation of the system. [ABSTRACT FROM AUTHOR]

This paper first finishes off the exact solutions of Riemann‐Liouville fractional differential time‐delay oscillatory system of order ρ∈(1,2)$$ \rho \in \left(1,2\right) $$ by using two newly defined delayed perturbations of Mittag‐Leffler matrix functions and constant variation method. In the light of the exact solutions, we explore the finite time stability of the nonhomogeneous fractional oscillatory differential equations with pure delay. Ultimately, an example is cited to verify the rationality of the results. Through our method, the public problems left by Mahmudov in 2022 were partially solved. [ABSTRACT FROM AUTHOR]

In this research work, we investigate a class of Atangana–Baleanu–Caputo (ABC) differential equations (DEs) having proportional and discrete delay terms accompanied by integral boundary conditions. We study the two important aspects, that is, the existence of the solution and stability analysis for the problem mentioned above. We confirm the existence of at least one solution via Schaefer's fixed‐point theorem and its uniqueness by Banach's fixed‐point theorem. Moreover, for stability analysis, we use the concept of Hyers–Ulam stability. For application purposes, we apply the main results to a numerical problem for specific values of the parameters involved. [ABSTRACT FROM AUTHOR]

In this study, we principally investigate a fractional‐order stage‐structured predator–prey system including distributed time delays and discrete time delays. Taking advantage of transformation of the variable, we obtain an isovalent version of the considered fractional‐order stage‐structured predator–prey system including distributed time delays and discrete time delays. The isovalent version includes fractional‐order and integer‐order equations. Utilizing the stability criterion and bifurcation theory of fractional‐order differential equation, a novel delay‐independent bifurcation condition to ensure the appearance of Hopf bifurcation for the fractional‐order stage‐structured predator–prey system is set up. The impact of time delay on the stability and bifurcation is clearly revealed. Numerical simulation figures are presented to sustain the rationality of the derived key conclusions. [ABSTRACT FROM AUTHOR]

In this article, we study existence, uniqueness, and stability analysis in Hyer–Ulam settings of delay fractional differential equations with nonlinear singular p‐Laplacian operator ϕp. The fractional operators are taken in the Caputo sense. The fractional differential equation is converted into an integral form with the Green's theorem, also a fixed point approach is studied for this article. We include an example with specific parameter and assumptions to show the results of the proposal. Our results generalize some of the works in the literature. [ABSTRACT FROM AUTHOR]

A qualitative analysis for a nonlinear system of fractional pantograph evolution differential equations (FPEDEs) has been established. Conditions about the solution of the underlying equations have been established in the direction of the existence and uniqueness of the solution. Using results of fixed‐point theorems of Krasnoselskii iterations in Banach spaces, we develop the required results. This research work is enriched by establishing the results related to Ulam–Hyers (UH) stability of the problem under consideration via the tools of nonlinear analysis. To validate the reliability of the obtained results, some pertinent examples are given in the manuscript. [ABSTRACT FROM AUTHOR]

Novel coronavirus (COVID‐19), a global threat whose source is not correctly yet known, was firstly recognised in the city of Wuhan, China, in December 2019. Now, this disease has been spread out to many countries in all over the world. In this paper, we solved a time delay fractional COVID‐19 SEIR epidemic model via Caputo fractional derivatives using a predictor–corrector method. We provided numerical simulations to show the nature of the diseases for different classes. We derived existence of unique global solutions to the given time delay fractional differential equations (DFDEs) under a mild Lipschitz condition using properties of a weighted norm, Mittag–Leffler functions and the Banach fixed point theorem. For the graphical simulations, we used real numerical data based on a case study of Wuhan, China, to show the nature of the projected model with respect to time variable. We performed various plots for different values of time delay and fractional order. We observed that the proposed scheme is highly emphatic and easy to implementation for the system of DFDEs. [ABSTRACT FROM AUTHOR]

Using operator‐valued Lp$$ {L}^p $$‐Fourier multiplier theorem, weighted Sobolev spaces, and the Carleman transform, we characterize the well‐posedness of second‐order degenerate differential equations with finite delay (Mu)′′(t)=Au(t)+Fut+Gut′+f(t)$$ {(Mu)}^{\prime \prime }(t)= Au(t)+F{u}_t+G{u}_t^{\prime }+f(t) $$ on Lp(ℝ;X)$$ {L}^p\left(\mathbb{R};X\right) $$, where A:D(A)⊆X→X$$ A:D(A)\subseteq X\to X $$ and M:D(M)⊆X→X$$ M:D(M)\subseteq X\to X $$ are closed linear operators defined on a Banach space X$$ X $$, the operators F$$ F $$ and G$$ G $$ are in B(Lp([−τ,0];X);X)$$ B\left({L}^p\left(\left[-\tau, 0\right];X\right);X\right) $$ for some fixed τ>0$$ \tau >0 $$, and ut(s)=u(t+s),ut′(s)=u′(t+s)$$ {u}_t(s)=u\left(t+s\right),{u}_t^{\prime }(s)={u}^{\prime}\left(t+s\right) $$ when t∈ℝ$$ t\in \mathbb{R} $$ and s∈[−τ,0]$$ s\in \left[-\tau, 0\right] $$. These results are used to study the well‐posedness of the associated second‐order neutral degenerate differential equations. [ABSTRACT FROM AUTHOR]

This paper investigates the fractional optimal control problem with a single delay in the state by using the operator theoretic approach. In this approach, we first reduce the delay fractional dynamical system into an equivalent operator equation, and then, by providing sufficient conditions to the operators, the existence of an optimal pair is proved for the abstract system. The optimality system for the quadratic cost functional is derived by using the Frechet derivative. Then we relate the operator theoretic approach optimality system to a Hamiltonian system of Pontryagin's minimum principle. The primary goal of this article is to demonstrate the existence of an optimal pair for the fractional‐order delay dynamical system by using a functional minimization theorem in functional analysis. Likewise, the optimality systems for the fractional‐order delay dynamical system are derived by using the functional (either Gateaux or Frechet) derivatives. Finally, we provide two numerical examples that support our theoretical findings. [ABSTRACT FROM AUTHOR]

In a former work, we introduced a new class of functions, which we call measure pseudo S$$ \mathcal{S} $$‐asymptotically omega periodic functions and give some properties of those functions as the topological properties of the space of such functions, the invariance by translation of this space, and the convolution product in this space. In this paper, we consider the same class of functions and give contributions as the stability by the integral operator; we present some examples and composition results. Then we apply the obtained results to investigate the existence of measure pseudo S$$ \mathcal{S} $$‐asymptotically omega periodic mild solutions for some nonlinear infinite delay integral equations arising in epidemiology; exactly, we consider the nonlinear infinite delay integral equation: x(t)=α(t)x(t−β)+∫−∞ta(t−s)f(s,x(s))ds+g(t,x(t))fort∈ℝ.$$ x(t)=\alpha (t)x\left(t-\beta \right)+{\int}_{-\infty}^ta\left(t-s\right)f\left(s,x(s)\right) ds+g\left(t,x(t)\right)\kern0.5em \mathrm{for}\kern0.4em t\in \mathbb{R}. $$Then we apply these results to a finite delay integral equation when the delay is time dependent x(t)=α(t)x(t−β)+∫t−γ(t)tf(s,x(s))ds+g(t,x(t))fort∈ℝ.$$ x(t)=\alpha (t)x\left(t-\beta \right)+{\int}_{t-\gamma (t)}^tf\left(s,x(s)\right) ds+g\left(t,x(t)\right)\kern0.5em \mathrm{for}\kern0.4em t\in \mathbb{R}. $$ [ABSTRACT FROM AUTHOR]

In this paper, the asymptotic stability and synchronization of fractional order Hopfield neural networks (FOHNNs) with time‐varying delay are investigated. First, we propose an extended fractional Halanay inequality, which plays an essential role in the proof of the main results. Using the Banach fixed point theorem, the existence and uniqueness of the equilibrium point of the system is proved. Through the extended Halanay inequality and a useful fractional derivative inequality, the sufficient conditions of the asymptotic stability and synchronization are given for FOHNNs. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

We obtain the exact solutions to the higher fractional‐order nonhomogeneous delayed differential equations with Caputo‐type fractional derivative by using a set of newly defined generalized delayed Mittag‐Leffler matrix functions. The Laplace transform and inductive construction are taken up as the major solving approaches. Thereafter, we consider some special cases and prove that the new exact solutions are suitable for the delayed differential equations with arbitrary order α>0$$ \alpha >0 $$. Additionally, we propose some criteria on the finite time stability of the higher fractional‐order delay differential equations. Finally, an illustrative example is presented to test the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

Weyl almost automorphy is a natural generalization of Bochner almost automorphy and Stepanov almost automorphy. However, the space composed of Weyl almost automorphic functions is not a Banach space. Therefore, the results of the existence of Weyl almost automorphic solutions of differential equations are few, and the results of the existence of Weyl almost automorphic solutions of difference equations are rare. Since the study of dynamic equations on time scales can unify the study of differential equations and difference equations. Therefore, in this paper, we first propose a concept of Weyl almost automorphic functions on time scales and then take the Clifford‐valued shunt inhibitory cellular neural networks with time‐varying delays on time scales as an example of dynamic equations on time scales to study the existence and global exponential stability of their Weyl almost automorphic solutions. We also give a numerical example to illustrate the feasibility of our results. [ABSTRACT FROM AUTHOR]

This study purposes to present some new comparison theorems that guarantee the oscillation of all solutions of even order functional differential equations with a neutral term including both advanced and delayed arguments. The obtained results are based on comparisons with associated first‐order delay differential inequalities and first‐order delay differential equations, and they are applicable to the both cases where the neutral coefficients are unbounded and/or bounded. In the last section, several examples showing the applicability of the results are provided. [ABSTRACT FROM AUTHOR]

In this work, we obtain necessary and sufficient conditions for the oscillation of solutions of a second‐order neutral differential equation with impulses. Two examples are provided to show the effectiveness and feasibility of the main results. Our main tool is the Lebesgue's Dominated Convergence theorem. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of the present work is to develop a new wavelet method, named as Krawtchouk wavelets method, for solving both Caputo fractional and Caputo–Hadamard fractional differential equations on a semi‐infinite domain. Design/methodology/approach: We have utilized the discrete Krawtchouk orthogonal polynomial for the construction of Krawtchouk wavelets method. The supporting analysis of the method such as construction of operational matrices, procedure of implementation, and convergence analysis of the method are being provided. We have also proposed a method by combining the Krawtchouk wavelets method with the method of step for the solution of Caputo–Hadamard fractional delay differential equations. Findings: We have provided the orthogonality condition for the Krawtchouk wavelets. We have derived and constructed the Krawtchouk wavelets matrix, Krawtchouk wavelets operational matrix of Riemann–Liouville and Hadamard‐type fractional‐order integration, and Krawtchouk wavelets operational matrix of Riemann–Liouville and Hadamard‐type fractional‐order integration for boundary value problems. These matrices are successfully utilized for the solution of Caputo and Caputo–Hadamard fractional differential equations. Operational matrices contains many zero entries, which makes the present method more efficient. Furthermore, we workout on the procedure of implementation of the method for the Caputo fractional differential equations as well as for the Caputo–Hadamard fractional differential equations. We also derived the convergence analysis of the Krawtchouk wavelets method, which completes the theoretical analysis of the proposed method. We have applied the Krawtchouk wavelets method for the numerical solutions of several Caputo fractional differential equations and Caputo–Hadamard fractional differential equations and compare the obtained results with the analytical solutions. The comparison shows the effectiveness of the present numerical method. In this paper, we have considered initial value problem, boundary value problem, and delay problem. According to the numerical results, the present method is more efficient and accurate. Originality/value: Since fractional differential equation is a latest and emerging field, many engineers and scientists can utilize the present method for solving their fractional models. To the best of the author's knowledge, the present wavelets method has never been introduced and implemented for Caputo fractional and Caputo–Hadamard fractional differential equations. [ABSTRACT FROM AUTHOR]

In this article, we investigate the existence and uniqueness of solution of controlled hybrid neutral stochastic differential equations with infinite delay (HNSFDEswID). It is known that the time lag generated by the controller in each discrete observation must be different. The controlled HNSFDEswID are affected by the variable delay induced by the control function, the infinite time delay, and the highly nonlinear coefficients of the systems itself, which makes our problem more sophisticated. Different from the classical Khasminskii‐type conditions in the literature, we provide some sufficient conditions for the mean square and almost sure exponential stability for controlled HNSFDEswID by using the M$$ M $$‐matrix technique and a suitable Lyapunov functional. In this sense, the main contributions of our results, compared with those in the literature, are the infinite time delay, the neutral term, and the new Lyapunov functions. Finally, we illustrate our results by a numerical example. [ABSTRACT FROM AUTHOR]

This work presents the generalized nonlinear multi‐terms fractional variable‐order delay differential equation with periodic and anti‐periodic conditions. In this paper, a novel shifted Jacobi operational matrix technique is introduced to solve a class of these equations mentioned, so that the main problem becomes a system of algebraic equations that we can solve numerically. The suggested technique is successfully developed for the aforementioned problem. Comprehensive numerical tests are provided to demonstrate the generality, efficiency, accuracy of presented scheme and the flexibility of this technique. The numerical experiments compared it with the true solution, indicating the validity and efficiency of this scheme. Note that the procedure is easy to implement and this technique will be considered as a generalization of many numerical schemes. Furthermore, the error and its bound are estimated. [ABSTRACT FROM AUTHOR]