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The Volterra integral equation associated with autonomous Caputo fractional differential equation (FDE) of order α ∈ (0 , 1) in R d was shown by the authors [4] to generate a semi-group on the space C of continuous functions f : R + → R d with the topology uniform convergence on compact subsets. It serves as a semi-dynamical system for the Caputo FDE when restricted to initial functions f(t) ≡ i d x 0 for x 0 ∈ R d . Here it is shown that this semi-dynamical system has a global Caputo attractor in C , which is closed, bounded, invariant and attracts constant initial functions, when the vector field function in the Caputo FDE satisfies a dissipativity condition as well as a local Lipschitz condition. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of differential equations describing a mathematical model of a phase-locked loop with a delay. We introduce the concept of the curvature of a cycle, which is applied to analyzing the proximity of cycles of nonphase and phase systems. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of mild solution for impulsive fractional differential inclusions in Banach spaces involving the Hilfer derivative. Our study is based on the nonlinear alternative of Leray–Schauder type for multivalued maps due to Martelli. An example will be added to illustrate the main result. [ABSTRACT FROM AUTHOR]

This paper is concerned with a new class of coupled implicit systems involving ϱ -fractional derivatives of different orders and anti periodic boundary conditions. We first convert the given implicit problem into a fixed point problem and then apply the fixed point theorems due to Krasnoselskii and Banach to establish the existence and uniqueness of its solutions. Examples are given for the illustration of the main results. [ABSTRACT FROM AUTHOR]

This work investigates the existence and uniqueness of mild solutions to a class of stochastic integral differential equations with various time delay driven by the Rosenblatt process. We can obtain alternative conditions that guarantee mild solutions by using the resolvent operator in the Grimmer sense, stochastic analysis, fixed-point methods, and noncompact measures. We give an example to illustrate the theory. [ABSTRACT FROM AUTHOR]

An elementary inequality is discussed for autonomous Caputo fractional differential equation (FDE) of order α ∈ (0 , 1) in R d for which the vector field satisfies a dissipativity condition. This inequality is fundamental for investigating qualitative and dynamical properties of such equations. Here its use and effectiveness are illustrated to show the global existence and uniqueness of solutions of such equations when the vector field is only locally Lipschitz. In addition, the existence of an absorbing set, which is positively invariant, is established. Finally, it is used to show that an equilibrium solution of a nonlinear Caputo FDE is locally asymptotically stable when the matrix of its linear part is negative definite. [ABSTRACT FROM AUTHOR]

We focus upon the existence of periodic traveling waves in a nonlocal reaction–diffusion model with chemotaxis and nonlocal delay effect. Based on the perturbation method, the existence of periodic traveling waves with large wave speed are obtained by the Lyapunov–Schmidt reduction and the implicit function theorem. The trick of the proof is to treat the wave profile as a solution of an operator in a suitable Banach space. [ABSTRACT FROM AUTHOR]

In this study, an initial-value problem for a nonlinear Volterra functional integro-differential equation on a finite interval was considered. The nonlinear term in the equation contains multiple time delays. In addition to giving some new theorems on the existence and uniqueness of solutions to the equation, the authors also prove the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the equation. The proofs use several different tools including Banach’s fixed point theorem, the construction of a Picard operator, and an application of Pachpatte’s inequality. An example is provided to illustrate the existence, uniqueness, and stability properties of solutions.

In this paper, we propose a methodology for computing the analytic solutions of linear retarded delay-differential equations and neutral delay-differential equations that include Dirac delta function inputs. In numerous applications, the delta function serves as a convenient and effective surrogate for modeling high voltages, sudden shocks, large forces, impulse vaccinations, etc., applied over a short period of time. The solutions are obtained using the Laplace transform method, in conjunction with the Cauchy residue theorem. The accuracy of these solutions are assessed by comparing them with the ones provided by the method of steps. Numerical examples illustrating the methodology are presented and discussed. These examples show that the Laplace transform solution is very reliable for linear retarded delay-differential equations, because the analytic solution, for a single delta function input, is continuous. However, for linear neutral delay-differential equations with a delta function input the analytic solution is discontinuous. Consequently, the well-known Gibbs phenomenon is observed in the vicinity of the discontinuities. However, for neutral delay differential equations, we show that in some cases, the magnitude of the jumps at the discontinuities decrease, as time increases. Therefore, the Gibbs phenomenon of the Laplace solution dissipates. [ABSTRACT FROM AUTHOR]

In this paper, we establish sufficient criteria for ensuring the existence of solutions and uniqueness for a class of nonlinear neutral Caputo fractional differential equations supplemented with infinite delay and nonlocal boundary conditions involving fractional derivatives. The theory of infinite delay and standard fixed point theorems are employed to obtain the existence results for the given problem. Examples will be constructed to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

We describe the behaviour of solutions of a scalar Delay Differential Equation (DDE) with delay that periodically switches between two constant values. Such an equation arises naturally from structured vector populations involved in a range of vector-borne diseases spreading in a periodically varying environment. We examine if and how the two different time lags and the switching time influence the existence and patterns of periodic solutions. We pay particular attention to the patterns involving multi-cycles within the prime period of the periodic solutions. [ABSTRACT FROM AUTHOR]

We first consider the finite time stability of second order linear differential systems with pure delay via giving a number of properties of delayed matrix functions. We secondly give sufficient and necessary conditions to examine that a linear delay system is relatively controllable. Further, we apply the fixed-point theorem to derive a relatively controllable result for a semilinear system. Finally, some examples are presented to illustrate the validity of the main theorems. [ABSTRACT FROM AUTHOR]

In this paper we provide sufficient conditions for the existence as well as the uniqueness of solution of general nonlinear functional differential equations, Our results will be obtained using suitable fixed point theorems, We give also some remarks about uniqueness theorem due to Dhage. [ABSTRACT FROM AUTHOR]

In this paper, we present an epidemiological model to study the dynamics of toxoplasmosis in cat and mouse populations under a continuous cat vaccination program. We construct a mathematical model at the cat and mouse populations level that includes the effect of oocysts of the parasite T. gondii which causes the toxoplasmosis infection. We include vertical transmission in both populations. We prove that the basic reproduction number R 0 is a threshold parameter that determines the global dynamics and the outcome of the toxoplasmosis disease in the cat and population. Numerical simulations are presented to support the theoretical results and to show the impact of a vaccination program for cats. In addition, the simulations give insight on the effect of a public health program related to removing the oocysts from the environment. These simulations show the effectiveness of a constant vaccination intervention and a oocysts clearance program. [ABSTRACT FROM AUTHOR]

The first part of the paper is devoted to studying the continuous dependence of the solutions of Carathéodory constant delay differential equations where the vector fields satisfy classical cooperative conditions. As a consequence, when the set of considered vector fields is invariant with respect to the time-translation map, the continuity of the respective induced skew-product semiflows is obtained. These results are important for the study of the long-term behavior of the trajectories. In particular, the construction of semicontinuous semiequilibria and equilibria is extended to the context of ordinary and delay Carathéodory differential equations. Under appropriate assumptions of sublinearity, the existence of a unique continuous equilibrium, whose graph coincides with the pullback attractor for the evolution processes, is shown. The conditions under which such a solution is the forward attractor of the considered problem are outlined. Two examples of application of the developed tools are also provided. [ABSTRACT FROM AUTHOR]

We present a circle criterion which is necessary and sufficient for absolute stability with respect to a natural class of sector-bounded nonlinear causal operators. This generalized circle criterion contains the classical result as a special case. Furthermore, we develop a version of the generalized criterion which guarantees input-to-state stability. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of mild solutions for a nonlinear neutral fractional differential equation with two orders of Caputo’s fractional derivative using the Krasnoselskii and Banach fixed point theorems. We establish four types of Ulam stability: Ulam–Hyers, Ulam–Hyers–Rassias, generalized Ulam–Hyers and generalized Ulam–Hyers–Rassias. Two examples are given to substantiate the usefulness of the obtained results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of frequency-phase locked loop systems (FPLL). The mathematical model of such systems is a system of differential equations with a cylindrical phase space. For FPLL systems, we obtain conditions of latent synchronization. [ABSTRACT FROM AUTHOR]

We develop a Conley index theory for retarded functional differential equations $\dot x=f(x_{t})$ with values in a differentiable manifold and (merely) continuous nonlinearities f. We use this index to establish an existence result for nonconstant full solutions of such equations. [ABSTRACT FROM AUTHOR]

The objective is to study the existence and uniqueness questions for a nonlinear Volterra-Fredholm integro-differential equation, then, develop a numerical approach using the Nyström method coupled with successive approximations algorithm. A numerical example is presented to examine the efficiency of the proposed method according to the existence and uniqueness conditions. [ABSTRACT FROM AUTHOR]

We present a direct proof of asymptotic consensus in the non‐linear Hegselmann–Krause model with transmission‐type delay, where the communication weights depend on the particle distance in phase space. Our approach is based on an explicit estimate of the shrinkage of the group diameter on finite time intervals and avoids the usage of Lyapunov‐type functionals or results from non‐negative matrix theory. It works for both the original formulation of the model with communication weights scaled by the number of agents, and the modification with weights normalized a'la Motsch–Tadmor. We pose only minimal assumptions on the model parameters. In particular, we only assume global positivity of the influence function, without imposing any conditions on its decay rate or monotonicity. Moreover, our result holds for any length of the delay. [ABSTRACT FROM AUTHOR]

The oscillation of a first order differential equation with several non-monotone delays is studied. A standard technique is improved and used to prove some new oscillation criteria of lim sup type. The applicability and strength of the obtained results are assured using an analytic approach as well as a practical example. [ABSTRACT FROM AUTHOR]

A class of the boundary value problem is investigated in this research work to prove the existence of solutions for the neutral fractional differential inclusions of Katugampola fractional derivative which involves retarded and advanced arguments. New results are obtained in this paper based on the Kuratowski measure of noncompactness for the suggested inclusion neutral system for the first time. On the one hand, this research concerns the set-valued analogue of Mönch fixed point theorem combined with the measure of noncompactness technique in which the right-hand side is convex valued. On the other hand, the nonconvex case is discussed via Covitz and Nadler fixed point theorem. An illustrative example is provided to apply and validate our obtained results. [ABSTRACT FROM AUTHOR]

We present a novel mathematical model of heterogeneous cell proliferation where the total population consists of a subpopulation of slow-proliferating cells and a subpopulation of fast-proliferating cells. The model incorporates two cellular processes, asymmetric cell division and induced switching between proliferative states, which are important determinants for the heterogeneity of a cell population. As motivation for our model we provide experimental data that illustrate the induced-switching process. Our model consists of a system of two coupled delay differential equations with distributed time delays and the cell densities as functions of time. The distributed delays are bounded and allow for the choice of delay kernel. We analyse the model and prove the nonnegativity and boundedness of solutions, the existence and uniqueness of solutions, and the local stability characteristics of the equilibrium points. We find that the parameters for induced switching are bifurcation parameters and therefore determine the long-term behaviour of the model. Numerical simulations illustrate and support the theoretical findings, and demonstrate the primary importance of transient dynamics for understanding the evolution of many experimental cell populations. [ABSTRACT FROM AUTHOR]

The novelty of the present article is to introduce the study of hybrid dynamical system of order ξ ∈ (δ - 1 , δ ] with finite time delay. These hybrid systems are more suitable to deal several dynamical process as particular cases. The importance of this manuscript is to discuss the concept of stability results including Ulam-Hyers stability (UHS), generalized Ulam-Hyers stability (GUHS), Ulam-Hyers Rassias stability (UHRS) and generalized Ulam-Hyers Rassias stability (GUHRS). Meanwhile, we investigate some sufficient conditions for existence result of the solution for proposed work by adopting the application of fixed point theorem of Banach algebra due to Dhage under mixed Lipschitz and Carathéodory conditions. Finally the paper is enriched by two interesting applications to demonstrate our results. [ABSTRACT FROM AUTHOR]

This paper studies the linear fractional-order delay differential equation * D − α C x (t) − p x (t − τ) = 0 , where 0 < α = odd integer odd integer < 1 , p , τ > 0 , D − α C x (t) = − Γ − 1 (1 − α) ∫ t ∞ (s − t) − α x ′ (s) d s . We obtain the conclusion that p 1 / α τ > α / e is a sufficient and necessary condition of the oscillations for all solutions of Eq. (*). At the same time, some sufficient conditions are obtained for the oscillations of multiple delays linear fractional differential equation. Several examples are given to illustrate our theorems. [ABSTRACT FROM AUTHOR]

In this paper, we study controllability results for non-densely defined impulsive fractional differential equation by applying the concepts of semigroup theory, fractional calculus, and Banach Fixed Point Theorem. An example is also discussed to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

Influenced by Xiao et al. (J Integral Equations Appl 30(1):197–218, 2018), collocation methods are developed to study strong convergence orders of numerical solutions for nonlinear stochastic Volterra integral equations under the Lipschitz condition in this paper. Some properties of exact solutions are discussed. These properties include the mean-square boundedness, the Hölder condition, and conditional expectations. In addition, this paper considers the solvability, the mean-square boundedness, and strong convergence orders of numerical solutions. At last, we validate our conclusions by numerical experiments. [ABSTRACT FROM AUTHOR]

The present work considers the approximation of solutions of a type of fractional-order Volterra–Fredholm integro-differential equations, where the fractional derivative is introduced in Caputo sense. In addition, we also present several applications of the fractional-order differential equations and integral equations. Here, we provide a sufficient condition for existence and uniqueness of the solution and also obtain an a priori bound of the solution of the present problem. Then, we discuss about the higher-order model equation which can be written as a system of equations whose orders are less than or equal to one. Next, we present an approximation of the solution of this problem by means of a perturbation approach based on homotopy analysis. Also, we discuss the convergence analysis of the method. It is observed through different examples that the adopted strategy is a very effective one for good approximation of the solution, even for higher-order problems. It is shown that the approximate solutions converge to the exact solution, even for higher-order fractional differential equations. In addition, we show that the present method is highly effective compared to the existed method and produces less error. [ABSTRACT FROM AUTHOR]

This article is concerned with the study of chaos transition of the duffing oscillator in the presence of fractional-order derivative and third-order polynomial delayed position feedback. The main aim of the work is focused on proving that the existence of time delay parameter has the ability to change the dynamic state of the fractional duffing oscillator from regular to chaotic in the absence of damping and external excitation. The concluded theoretical results are verified numerically using self-implemented MATLAB codes for some special cases of the fractional duffing oscillator with a delayed feedback. [ABSTRACT FROM AUTHOR]

We consider a system of frequency-phase self-tuning whose mathematical model is a system of differential equations. In this paper, existence conditions for asynchronous modes of a phase system are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of differential equations with a cylindrical phase space, which is a mathematical model of a phase-locked loop system. Conditions of the existence of limit cycle of the first kind are obtained. [ABSTRACT FROM AUTHOR]

We discuss the controllability for a damped fractional differential system with impulses and state delay, which involves Caputo fractional derivatives. Deriving the condition based on the Gramian matrix defined by the Mittag-Leffler matrix function and Laplace transformation, we establish necessary and sufficient conditions of controllability criteria. Finally, we construct two numerical examples to support the result. [ABSTRACT FROM AUTHOR]

In this study, we are currently investigating the controllability of nonlinear fractional differential control systems with delays in the state function. The solution representations of fractional delay differential equations have been established by using the delayed Mittag-Leffler function. Firstly we obtain the result of the controllability of a linear fractional control system with delay. Then, for the controllability criteria of nonlinear fractional delay system, we establish the set of sufficient conditions of nonlinear fractional differential systems with delay in their state function by using Schauder's fixed point theorem. In the end, a numerical example is constructed to support the results. [ABSTRACT FROM AUTHOR]

In this paper, we approximate the solution of Fredholm integro-differential equations of the second kind by using exponential spline function. The proposed method reduces to the system of algebraic equations. The convergence analysis of the method has been discussed. The numerical examples are presented to illustrate the applications of the method and to compare the computed results with the method in Chen et al. [A multiscale Galerkin method for second-order boundary value problems of Fredholm integro-differential equation, J. Comput. Appl. Math. 290 (2015), pp. 633–640]. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new scheme based on the exponential spline function for solving linear second-order Fredholm integro-differential equations. Our approach consists of reducing the problem to a set of linear equations. We prove the convergence analysis of the method applied to the solution of integro-differential equations. The method is described and illustrated with numerical examples. The results reveal that the method is accurate and easy to apply. Moreover, results are compared with the method in (J. Comput. Appl. Math. 290:633–640, 2015). [ABSTRACT FROM AUTHOR]

Three control problems for the system of two coupled differential equations governing the dynamics of an energy harvesting model are studied. The system consists of the equation of an Euler–Bernoulli beam model and the equation representing the Kirchhoff's electric circuit law. Both equations contain coupling terms representing the inverse and direct piezoelectric effects. The system is reformulated as a single evolution equation in the state space of 3‐component functions. The control is introduced as a separable forcing term g(x)f(t) on the right‐hand side of the operator equation. The first control problem deals with an explicit construction of f(t) that steers an initial state to zero on a time interval [0, T]. The second control problem deals with the construction of f(t) such that the voltage output is equal to some given function v(t) (with g(x) being given as well). The third control problem deals with an explicit construction of both the force profile, g(x), and the control, f(t), which generate the desired voltage output v(t). Interpolation theory in the Hardy space of analytic functions is used in the solution of the second and third problems. [ABSTRACT FROM AUTHOR]

Stieltjes differential equations, which contain equations with impulses and equations on time scales as particular cases, simply consist in replacing usual derivatives by derivatives with respect to a nondecreasing function. In this paper we prove new results for the existence of extremal solutions for discontinuous Stieltjes differential equations. In particular, we prove that the pointwise infimum of upper solutions of a Stieltjes differential equation is the minimal solution under certain hypotheses. These results can be adapted to the context of both difference equations and impulsive differential equations. [ABSTRACT FROM AUTHOR]

In this research work, we investigate the controllability of linear fractional differential control systems with state and control delay. By using an explicit solution formula, a rank criterion for controllability is established. For the controllability criteria, we establish necessary and sufficient conditions of a fractional differential systems with state and control delay. In the end, a numerical example is constructed to support the results. [ABSTRACT FROM AUTHOR]

In this paper, we give sufficient conditions for the existence and uniqueness of Pseudo S-Asymptotically ω-periodic solutions for a differential equation with piecewise constant argument in a Banach space. This result is obtained using the Pseudo S-asymptotically ω-periodic sequences. [ABSTRACT FROM AUTHOR]

The use of multiple vaccine doses has proven to be essential in providing high levels of protection against a number of vaccine-preventable diseases at the individual level. However, the effectiveness of vaccination at the population level depends on several key factors, including the dose-dependent protection efficacy of vaccine, coverage of primary and booster doses, and in particular, the timing of a booster dose. For vaccines that provide transient protection, the optimal scheduling of a booster dose remains an important component of immunization programs and could significantly affect the long-term disease dynamics. In this study, we developed a vaccination model as a system of delay differential equations to investigate the effect of booster schedule using a control parameter represented by a fixed time-delay. By exploring the stability analysis of the model based on its reproduction number, we show the disease persistence in scenarios where the booster dose is sub-optimally scheduled. The findings indicate that, depending on the protection efficacy of primary vaccine series and the coverage of booster vaccination, the time-delay in a booster schedule can be a determining factor in disease persistence or elimination. We present model results with simulations for a vaccine-preventable bacterial disease, Heamophilus influenzae serotype b, using parameter estimates from the previous literature. Our study highlights the importance of timelines for multiple-dose vaccination in order to enhance the population-wide benefits of herd immunity. [ABSTRACT FROM AUTHOR]

Consider the delay differential equation x ˙ (t) = a ∫ 0 r x (t - s) d η (s) - g (x (t)) and the neutral type differential equation y ˙ (t) = a ∫ 0 r y ˙ (t - s) d μ (s) - g (y (t)) where a > 0 , g : R → R is smooth, u g (u) > 0 for u ≠ 0 , ∫ 0 s g (u) d u → ∞ as | s | → ∞ , r > 0 , η and μ are nonnegative functions of bounded variation on [ 0 , r ] , η (0) = η (r) = 0 , ∫ 0 r η (s) d s = 1 , μ is nondecreasing, μ does not have a singular part, ∫ 0 r d μ = 1 . Both equations can be interpreted as price models. Global asymptotic stability of y = 0 is obtained, in case a ∈ (0 , 1) , for the neutral equation by using a Lyapunov functional. Then this result is applied to get global asymptotic stability of x = 0 for the (non-neutral) delay differential equation provided a ∈ (0 , 1) . As particular cases, two related global stability conjectures are solved, with an affirmative answer. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and controllability for a class of second-order evolution differential inclusions without compactness in Banach spaces. By applying the technique of weak topology and Glicksberg–Ky Fan fixed point theorem, we prove our main results without the hypotheses of compactness on the operator generated by the linear part and any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. Further, we extend our study to existence and controllability of second-order evolution differential inclusions with nonlocal conditions and impulses. Finally, an example is given for the illustration of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we consider a class of neutral shunting inhibitory cellular neural networks with mixed delays. We study the existence, uniqueness, and the exponential stability of the measure pseudo almost periodic (or μ-pseudo almost periodic) solutions from some models for shunting inhibitory cellular neural networks with mixed delays. An example is provided to illustrate the theory developed in this work. [ABSTRACT FROM AUTHOR]