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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 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 paper, we investigate a class of the linear evolution process with memory in Banach space by a different approach. Suppose that the linear evolution process is well posed, we introduce a family pair of bounded linear operators, {(G(t),F(t)),t≥0}$\lbrace (G(t), F(t)),t\ge 0\rbrace$, that is, called the resolvent family for the linear evolution process with memory, the F(t)$F(t)$ is called the memory effect family. In this paper, we prove that the families G(t)$G(t)$ and F(t)$F(t)$ are exponentially bounded, and the family (G(t),F(t))$(G(t),F(t))$ associate with an operator pair (A,L)$(A, L)$ that is called generator of the resolvent family. Using (A,L)$(A,L)$, we derive associated differential equation with memory and representation of F(t)$F(t)$ via L. These results give necessary conditions of the well‐posed linear evolution process with memory. To apply the resolvent family to differential equation with memory, we present a generation theorem of the resolvent family under some restrictions on (A,L)$(A,L)$. The obtained results can be directly applied to linear delay differential equation, integro‐differential equation and functional differential equations. [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]

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]

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]

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]

When a real‐life problem is mathematically modeled by differential equations or another type of equation, there are always intrinsic phenomena that are not taken into account and can affect the behavior of such a model. For example, external forces can abruptly change the model; impulses and delay can cause a breakdown of it. Considering these intrinsic phenomena in the mathematical model makes the difference between a simple differential equation and a differential equation with impulses, delay, and nonlocal conditions. So, in this work, we consider a semilinear nonautonomous neutral differential equation under the influence of impulses, delay, and nonlocal conditions. In this paper we study the controllability of these semilinear neutral differential equations with some of these intrinsic phenomena taking into consideration. Our aim is to prove that the controllability of the associated ordinary linear differential equation is preserved under certain conditions imposed on these new disturbances. In order to achieve our objective, we apply Rothe's fixed point Theorem to prove the exact controllability of the system. Finally, our method can be extended to the evolution equation in Hilbert spaces with applications to control systems governed by PDE's equations. [ABSTRACT FROM AUTHOR]

In this study, some new results are obtained on asymptotic behavior and stability analysis of the mixed type differential equation x′t=∫−10xt−r1θdvθ+∫−10xt+r2θdηθ,where xt∈ℝ, r1θ and r2θ are real nonnegative continuous functions on −1,0, and vθ and ηθ are real valued functions of bounded variation on −1,0. These results were obtained by using an appropriate real root of the characteristic equation. Moreover, by the use of two appropriate distinct real roots of the corresponding characteristic equation, a new result on the behavior of solutions is established. Five examples are also given to illustrate our results. We also presented the application of the obtained results in the special case of constant coefficients and have given three different cases in one example. The results obtained in this article show that real roots play an important role. [ABSTRACT FROM AUTHOR]

This study is about getting some conditions that guarantee the existence and uniqueness of the weighted pseudo almost periodic (WPAP) solutions of Mackey–Glass with time‐varying delays. Some adequate conditions have been obtained for the existence and uniqueness of the WPAP solutions of the Mackey–Glass, which we dealt with using some differential equation inequalities, the WPAP theory, and the Banach fixed‐point theorem. Besides, an example is given to demonstrate the accuracy of the conditions of our main results. [ABSTRACT FROM AUTHOR]

An adaptive grid method based on the backward Euler formula for a system of semilinear singularly perturbed initial value problems is studied. Based on the a priori error analysis and mesh equidistribution principle, we prove that the convergence rate of our semidiscrete adaptive grid method is first order, which is robust with respect to the perturbation parameters. Then, in order to construct a fully discrete adaptive grid method, a standard residual‐type a posterior error estimation is constructed by using the linear polynomial interpolation technique. A partly heuristic argument based on this a posteriori error estimator leads to an optimal monitor function, which is used to design an adaptive grid algorithm. Furthermore, we also extend our presented adaptive grid method to a nonlinear system of singularly perturbed problem arising in the modeling of enzyme kinetics and a system of singularly perturbed delay differential equations, respectively. Finally, numerical results are provided to illustrate the effectiveness of our presented adaptive grid method. [ABSTRACT FROM AUTHOR]

We study the global existence and uniqueness of solutions and wellposedness of a general class of abstract second order differential equations with state dependent delay. Some examples are presented. [ABSTRACT FROM AUTHOR]

In this paper, an approximate method for solving delayed differential equations involving pantograph‐type delay is proposed. The method is based upon hybrid function approximations. The properties of hybrid functions of block‐pulse functions and the well‐known Legendre polynomials are provided. The operational matrix of pantograph‐type delay corresponding to the method is presented. The associated operational matrices of delay, derivative, and product are utilized to reduce the original problem into a system of algebraic equations. Illustrative examples are included to demonstrate the applicability and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is devoted to obtaining a new oscillation criterion for a class of neutral delay differential equations of the type u(x)ym−1(x)κ′+q(x)zκγ(x)=0, where m ≥ 4 is an even integer and y(x):=z(x)+g(x)zξ(x). It should be noted here that the results obtained improve and complement those given by Xing et al. (2016). An example illustrating the applicability of the new criterion is presented. [ABSTRACT FROM AUTHOR]

The aims of this manuscript is to establish conditions for obtaining mild solutions to a coupled systems of multipoint boundary value problems (BVPs) of fractional order hybrid differential equations (FHDEs) with nonlinear perturbations of second type. In the concerned problem, we consider a proportional type delay that represent a famous class of differential equation called pantograph equations. Hence, we establish sufficient conditions for at least one mild solution to the coupled system of fractional hybrid pantograph differential equations (FHPDEs) by using Burton and couple‐type fixed‐point theorems. We also provide examples to show the applicability of our results. [ABSTRACT FROM AUTHOR]

Using Krasnoselskii's fixed‐point theorem, we obtain some results for the existence of periodic solutions to the third‐order neutral differential equations with delay. The results obtained essentially improve, include, and complement the results in the literature. [ABSTRACT FROM AUTHOR]

We study scalar advanced and delayed differential equations with piecewise constant generalized arguments, in short DEPCAG of mixed type, that is, the arguments are general step functions. It is shown that the argument deviation generates, under certain conditions, oscillations of the solutions, which is an impossible phenomenon for the corresponding equation without the argument deviations. Criteria for existence of periodic solutions of such equations are discussed. New criteria extend and improve related results reported in the literature. The efficiency of our criteria is illustrated via several numerical examples and simulations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the stability and Hopf bifurcation in a general class of differential equation with nonlocal delayed feedback that models the population dynamics of a two age structured spices. The existence of Hopf bifurcation is firstly established after delicately analyzing the eigenvalue problem of the linearized nonlocal equation. The direction of the Hopf bifurcation and stability of the bifurcated periodic solutions are then investigated by means of center manifold reduction. Subsequently, we apply our main results to explore the spatial‐temporal patterns of the nonlocal Mackey‐Glass equation. We obtain both spatially homogeneous and inhomogeneous periodic solutions and numerically show that the former is stable while the latter is unstable. We also show that the inhomogeneous periodic solutions will eventually tend to homogeneous periodic solutions after transient oscillations and increasing of the immature mobility constant will shorten the transient oscillation time. [ABSTRACT FROM AUTHOR]

Mathematical models of interacting populations are often constructed as systems of differential equations, which describe how populations change with time. Below we study such a model connected to the nonlinear dynamics of a system of populations in presence of time delay. The consequence of the presence of the time delay is that the nonlinear dynamics of the studied system become more rich, eg, new orbits in the phase space of the system arise, which are dependent on the time‐delay parameters. In more detail, we introduce a time delay and generalize the model system of differential equations for the interaction of 3 populations based on generalized Volterra equations in which the growth rates and competition coefficients of populations depend on the number of members of all populations. Then we solve numerically the system with and without time delay. We use a modification of the method of Adams for the numerical solution of the system of model equations with time delay. By appropriate selection of the parameters and initial conditions, we show the impact of the delay time on the dynamics of the studied population system. [ABSTRACT FROM AUTHOR]

This study studies how impulses affect the bounded behaviours of delay system with bounded disturbance. By employing the method of Lyapunov function, several criteria of finite‐gain L∞ stability from disturbance to output are established. It is shown that a linear delay differential system can be finite‐gain L∞ stabilised from disturbance to output using impulsive feedback control even there is an unstable matrix. Moreover, delay differential equations also maybe finite‐gain L∞ stable from disturbance to output under an appropriate sequence of impulses treated as disturbances. Two examples and simulations are also given to illustrate our results. [ABSTRACT FROM AUTHOR]

Networked systems are often subject to environmental uncertainties and communication delays, which make timely and accurate information exchange among neighbours difficult or impossible. This study investigates the distributed consensus problem of dynamical networks of multi‐agents in which each agent can only obtain noisy and delayed measurements of the states of its neighbours. The authors consider consensus protocols that take into account both the noisy measurements and the communication time delays, and introduce the notions of almost sure average‐consensus and p th moment average‐consensus. Using a convergence theorem for continuous‐time semimartingales and moment inequality techniques for stochastic delay differential equations, the authors establish sufficient conditions for both almost sure and moment average‐consensus. These results naturally generalise to networks with arbitrary and Markovian switching topologies. The consensus protocol considered here can be applied to networks with arbitrary bounded communication delays, which appears to the first consensus algorithm that is both average preserving and robust to arbitrarily sized delays. Numerical simulations are also provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider whether or not the feedback control has influence on a nonlinear population equation with several time delays. The general criteria of integrable form on the permanence are established, and we note that the feedback control has no influence to the permanence of system. By constructing suitable Lyapunov functionals, a set of easily verifiable sufficient conditions are derived for global asymptotic stability of any positive solutions to the system. [ABSTRACT FROM AUTHOR]

Abstract: The Wnt/β-catenin signalling pathway is evolutionarily conserved across many species and plays important roles during embryogenesis. The Lee–Heinrich model (to recognize the contributions of R. Heinrich we will refer to the work proposed by Lee et al. [Lee, E., Salic, A., Krüger, R., Heinrich, R., Kirschner, M.W. (2003) The roles of APC and axin derived from experimental and theoretical analysis of the Wnt pathway. PloS Biol. 1, 116–132] as the Lee–Heinrich model) describes this pathway by use of coupled ordinary differential equations. Here, we extend this model by introducing negative feedback loops of the pathway using time-delay differential equations. Single- and multiple-parameter perturbations suggest a very robust behaviour of this pathway that can also demonstrate oscillatory behaviour. These findings are of biological significance as Wnt pathway components show oscillations during vertebrate somitogenesis. [Copyright &y& Elsevier]

In this paper, we give a necessary and sufficient condition for the oscillation of all solutions of a class of superlinear second order forced neutral differential equations in the form $${d^2 \over dt^2} (x(t)-px(t-\tau))+Q(t) G(x(t-\sigma))=f(t),$$ where t ∈ R+ = [0, ∞), 0 ≤ p p < 1, τ, σ ∈ (0, ∞), Q(t), f(t) are real valued continuous functions on R+ and G : R → R such that xG(x) > 0 for x ≠ 0, G is nondecreasing, differentiable, Lipschitzian, and satisfies the superlinear conditions $$-infty for every positive real K. [ABSTRACT FROM AUTHOR]