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We present two sufficient conditions for the oscillatory behaviour of a class of noncanonical fourth-order dynamic equations on arbitrary time scales. An illustrative example is included to show that new criteria improve related results reported in the literature.

Abstract Background Although studies have shown that sleep quality (duration) is associated with health-related quality of life (HRQoL), most of these studies have been small-sized and targeted at young and middle-aged adults. In addition, few studies have explored the path mechanism of sleep disorders leading to impaired HRQoL. Objectives This study aimed to determine the association between sleep quality and duration and HRQoL among the elderly in the United Kingdom, assess whether depression mediated the association, and explore the role of physical activity (PA) in the path association. Methods Data were extracted from the baseline survey of the UK Biobank, a large prospective cohort study enrolling more than 500,000 participants, of which 52,551 older adults (aged ≥60 years) were included in the study. HRQoL was assessed using the European Quality of Life-5 Dimensions. Tobit and multivariate logistic regression models were used to determine the association between sleep quality and duration and HRQoL. The mediating and moderated mediation models were estimated using the PROCESS macro and MEDCURVE macro. Results The Tobit model showed that the elderly with short or long sleep duration (β = − 0.062, 95% confidence interval [CI] = − 0.071 to − 0.053; β = − 0.072, 95% CI = − 0.086 to − 0.058) had worse HRQoL after adjusting potential covariates. In the logistic regression models, we found an inverted U-shaped association between sleep duration and HRQoL. Moreover, a significant positive association was observed between sleep quality and HRQoL (all P

ObjectivesTo investigate the relationship between multimorbidity and health-related quality of life (HRQoL), and explore the effects of functional status and cognitive function on Chinses elderly behind this relationship.MethodsThe Multivariate logistic regression and Tobit regression models were used to determine the influence of multimorbidity on HRQoL. Bootstrap analysis was used to probe the mediating effects of functional status and the moderating role of cognition on multimorbidity and HRQoL.ResultsResults of the 2,887 participants age ≥ 60 years included in the analysis, 51.69% had chronic diseases. Stroke (β = −0.190; 95% confidence interval [CI], −0.232, −0.149; p

Purpose The Liyang cohort study on chronic diseases and risk factors monitoring in China (Liyang Study) is a prospective population-based study which aims to investigate and identify the determinants of the most prevalent chronic non-communicable diseases (NCDs) and to evaluate the impact of demographic characteristics, lifestyle, dietary habits, cognition, disability and NCDs on the health-related quality of life.Participants Between March 2019 and June 2020, 10 056 individuals aged ≥18 years were administered a baseline survey through a multistage cluster random sampling in Liyang City, southern Jiangsu Province, China.Findings to date The Liyang Study included detailed sociodemographic, anthropometric and health-related behaviour, common NCDs and blood sample information. Moreover, the study gathered a series of data on specific scales including the activities of daily living, instrumental activities of daily living, abbreviated mental test, Food Frequency Questionnaire and EuroQol 5-Dimensions 5-Levels Scale. Of the 10 056 participants, 52.92% (n=5322) were female and 92.26% (n=9278) came from rural areas. The mean age was 49.9±16.2 years. Men were more likely to have a higher level of education, annual income and a paid job than women (p

Abstract We investigate the existence of different types of nonoscillatory solutions to a class of higher-order nonlinear neutral dynamic equations on a time scale. Two examples are provided to show the significance of the conclusions.

Abstract We investigate non-Gaussian noise second-order filtering and fixed-order smoothing problems for non-Gaussian stochastic discrete systems with packet dropouts. We present a novel Kalman-like nonlinear non-Gaussian noise estimation approach based on the packet dropout probability distribution and polynomial filtering technique. By means of properties of Kronecker product we first introduce a second-order polynomial extended system and then analyze the means and variances of the Kronecker powers of the extended system noises. To generate noise estimators in forms of filtering and smoothing, we use the innovation approach. We give an example to illustrate that the presented algorithm has better robustness against packet dropouts than conventional linear minimum variance estimation.

Abstract We consider the corrosion detection problem in terms of the Laplace equation and study a simply connected bounded domain with Wentzell-type GIBC boundary condition. We derive the systems of integral equations and establish the equivalence to the inverse shape problem in a Sobolev space setting. For the direct problem, we use potential theory to simulate the Neumann data from Dirichlet data on Dirichlet boundary. Then we propose a Newton iterative approach based on the boundary integral equations derived from Green’s representation theorem. After describing the linearization and the iteration scheme for the inverse shape, we compute the Fréchet derivatives with respect to the unknowns. We conclude by presenting several numerical examples for shape reconstructions to show the validity of the proposed method.

Abstract We study the existence of nonoscillatory solutions tending to zero of a class of third-order nonlinear neutral dynamic equations on time scales by employing Krasnoselskii’s fixed point theorem. Two examples are given to illustrate the significance of the conclusions.

Abstract This paper is concerned with the convergence of solutions for a class of functional integro-differential equations with nonlinear boundary conditions. New comparison principles are obtained. By using the comparison principles and quasilinearization method, we present two monotone iterative sequences uniformly and monotonically converging to the unique solution with rate of order 2. Meanwhile, an example is given to demonstrate applications of the result reported.

Abstract This paper discusses the synchronization problem of N-coupled fractional-order chaotic systems with ring connection via bidirectional coupling. On the basis of the direct design method, we design the appropriate controllers to transform the fractional-order error dynamical system into a nonlinear system with antisymmetric structure. By choosing appropriate fractional-order Lyapunov functions and employing the fractional-order Lyapunov-based stability theory, several sufficient conditions are obtained to ensure the asymptotical stabilization of the fractional-order error system at the origin. The proposed method is universal, simple, and theoretically rigorous. Finally, some numerical examples are presented to illustrate the validity of theoretical results.

Abstract The present paper focuses on the oscillation of the third-order nonlinear neutral differential equations with damping and distributed delay. The oscillation of the third-order damped equations is often discussed by reducing the equations to the second-order ones. However, by applying the Riccati transformation and the integral averaging technique, we give an analytical method for the estimation of Riccati dynamic inequality to establish several oscillation criteria for the discussed equation, which show that any solution either oscillates or converges to zero. The results make significant improvement and extend the earlier works such as (Zhang et al. in Appl. Math. Lett. 25:1514–1519 2012). Finally, some examples are given to demonstrate the effectiveness of the obtained oscillation results.

Abstract The oscillatory behavior of the solutions to a differential equation with several non-monotone arguments and nonnegative coefficients is studied, and some new oscillation criteria are given. More precisely, sufficient conditions in terms of limsup and liminf are established, which essentially improve several known criteria existing in the literature. The results are illustrated by examples numerically solved in MATLAB.

In this article we study first-order impulsive integro-differential equations with integral boundary conditions, employing the method of quasilinearization with reversed ordering upper and lower solutions. We obtain two monotone sequences of iterates converging uniformly and quadratically to the unique solution of the problem. Two examples are given to illustrate the applications of the established results.

Abstract In the paper, fourth-order delay differential equations of the form (r3(r2(r1y′)′)′)′(t)+q(t)y(τ(t))=0 $$ \bigl(r_{3} \bigl(r_{2} \bigl(r_{1}y' \bigr)' \bigr)' \bigr)'(t) + q(t) y \bigl( \tau (t) \bigr) = 0 $$ under the assumption ∫t0∞dtri(t)

Abstract We study almost periodic solutions for a class of nonlinear second-order differential equations involving reflection of the argument. We establish existence results of almost periodic solutions as critical points by a variational approach. We also prove structure results on the set of strong almost periodic solutions, existence results of weak almost periodic solutions, and a density result on the almost periodic forcing term for which the equation possesses usual almost periodic solutions.

In this paper we study existence and uniqueness of solutions for a coupled system consisting of fractional differential equations of Caputo type, subject to Riemann–Liouville fractional integral boundary conditions. The uniqueness of solutions is established by Banach contraction principle, while the existence of solutions is derived by Leray–Schauder’s alternative. We also study the Hyers–Ulam stability of mentioned system. At the end, examples are also presented which illustrate our results.

The present paper is concerned with the asymptotic behavior of solutions to a class of noncanonical third-order Emden–Fowler delay differential equations with a superlinear neutral term. Using a Riccati-type transformation as well as integral criteria, we establish some new sufficient conditions guaranteeing that every solution of the equation considered either oscillates or converges to zero asymptotically. The results are illustrated with two examples.

Abstract In this paper, we study the asymptotic behavior of the solutions of a new class of difference equations xn+1=axn−l+bxn−k+f(xn−l,xn−k), $$x_{n+1}=ax_{n-l}+bx_{n-k}+f ( x_{n-l},x_{n-k} ), $$ where l and k are nonnegative integers, a and b are nonnegative real numbers, the initial values x−s,x−s+1,…,x0 $x_{-s}, x_{-s+1},\ldots, x_{0}$ are positive real numbers, s=max{l,k} $s=\max\{l,k\}$, and f(u,v):(0,∞)2→(0,∞) $f (u,v ): ( 0,\infty ) ^{2}\rightarrow ( 0,\infty ) $ is a continuous and homogeneous real function of degree zero. We consider the stability, boundedness, and periodicity of the solutions of this equation which is the most general form of linear difference equations. Thus, the results in this paper apply to several other equations that are special cases of the studied equation. Moreover, we present a new method to study periodic solutions of period two.

Abstract The paper is devoted to the study of oscillation of solutions to a class of second-order half-linear neutral differential equations with delayed arguments. New oscillation criteria are established, and they essentially improve the well-known results reported in the literature, including those for non-neutral differential equations. The adopted approach refines the classical Riccati transformation technique by taking into account such part of the overall impact of the delay that has been neglected in the earlier results. The effectiveness of the obtained criteria is illustrated via examples.

Abstract In this paper, we review indispensable properties and characterizations of almost periodic functions and asymptotically almost periodic functions in Banach spaces. Special accent is put on the Stepanov generalizations of almost periodic functions and asymptotically almost periodic functions. We also recollect some basic results regarding equi-Weyl-almost periodic functions and Weyl-almost periodic functions. The class of asymptotically Weyl-almost periodic functions, introduced in this work, seems to be not considered elsewhere even in the scalar-valued case. We actually introduce eight new classes of asymptotically almost periodic functions and analyze relations between them. In order to make a picture as complete and clear as possible, several illustrating examples and counter-examples are given. It is worth noting that the topics dealt with in this paper seem to be of an intrinsic connection with the problem of existence and uniqueness of solutions of differential and difference equations, in both determinist and stochastic cases.

Abstract We study a two-dimensional boundary value problem described by a tensorial equation in a bounded domain. Once its more general definition is given, we conclude that its analysis is linked to the resolution of an overdetermined hyperbolic problem and hence present some discussions and considerations. Secondly, for a simplified version of the original formulation, which leads to a degenerate problem on a rectangle, we prove the existence and uniqueness of a solution under proper assumptions on the data.

Abstract We investigate oscillatory behavior of solutions to a class of second-order nonlinear neutral delay dynamic equations with nonpositive neutral coefficients. In particular, we study the corresponding noncanonical neutral differential equations. New oscillation criteria are established that complement and improve related contributions to the subject. An example is given to illustrate the main results.

Abstract Sufficient conditions, involving limsup and liminf, for the oscillation of all solutions of differential equations with several not necessarily monotone deviating arguments and nonnegative coefficients are established. Corresponding differential equations of both delayed and advanced type are studied. We illustrate the results and the improvement over other known oscillation criteria by examples, numerically solved in MATLAB.

Abstract We prove that the system θ ˙ ( t ) = Λ ( t ) θ ( t ) $\dot{\theta}(t) =\Lambda(t)\theta(t)$ , t ∈ R + $t\in\mathbb{R}_{+}$ , is Hyers-Ulam stable if and only if it is uniformly exponentially stable under certain conditions; we take the exact solutions of the Cauchy problem ϕ ˙ ( t ) = Λ ( t ) ϕ ( t ) + e i γ t ξ ( t ) $\dot{\phi}(t)=\Lambda(t)\phi(t)+e^{i\gamma t}\xi(t)$ , t ∈ R + $t\in\mathbb{R}_{+}$ , ϕ ( 0 ) = θ 0 $\phi(0)=\theta_{0}$ as the approximate solutions of θ ˙ ( t ) = Λ ( t ) θ ( t ) $\dot{\theta}(t)=\Lambda(t)\theta(t)$ , where γ is any real number, ξ is a 2-periodic, continuous, and bounded vectorial function with ξ ( 0 ) = 0 $\xi(0)=0$ , and Λ ( t ) $\Lambda(t)$ is a 2-periodic square matrix of order l.

Abstract We study the asymptotic properties of the solutions of a class of even-order damped differential equations with p-Laplacian like operators, delayed and advanced arguments. We present new theorems that improve and complement related contributions reported in the literature. Several examples are provided to illustrate the practicability, maneuverability, and efficiency of the results obtained. An open problem is proposed.

New sufficient conditions for the oscillation of all solutions to a class of third-order Emden–Fowler differential equations with unbounded neutral coefficients are established. The criteria obtained essentially improve related results in the literature. In particular, as opposed to known results, new criteria can distinguish solutions of third-order differential equations with different behaviors. Examples are also provided to illustrate the results.

We study the oscillatory behavior of differential equations with nonmonotone deviating arguments and nonnegative coefficients. New oscillation criteria, involving lim sup and lim inf, are obtained based on an iterative method. Examples, numerically solved in MATLAB, are given to illustrate the applicability and strength of the obtained conditions over known ones.

This paper addresses the problem of projective exponential synchronization for a class of complex spatiotemporal networks with multiple time delays satisfying the homogeneous Neumann boundary conditions, where the network is modeled by coupled partial differential-difference equations (PDDEs). A distributed proportional-spatial derivative (P-sD) controller is designed by employing Lyapunov’s direct method and Kronecker product. The controller ensures the projective exponential synchronization of the PDDE network. The main result of this paper is presented in terms of standard linear matrix inequality (LMI). A numerical example is provided to show the effectiveness of the proposed design method.

In this paper the differential transformation method (DTM) is employed to solve a system of linear differential equations derived for energy optimal control theory and nonlinear differential equations and their systems for oscillators near Hopf bifurcation. They are given by differential equations in a complex plane. Different types of forced terms are considered. The approximate solutions are given in the form of series with required accuracy. Some numerical examples are provided.

This paper is concerned with the existence results of nonlocal problems for a class of fractional impulsive integrodifferential equations in Banach spaces. We define a piecewise continuous control function to obtain the results on controllability of the corresponding fractional impulsive integrodifferential control systems. The results are obtained by means of fixed point methods. An example to illustrate the applications of our main results is given.

We establish several oscillation criteria for a class of third-order nonlinear dynamic equations with a damping term and a nonpositive neutral coefficient by using the Riccati transformation. Two illustrative examples are presented to show the significance of the results obtained.

We study the oscillatory behavior of solutions to second-order neutral differential equations. We show that under certain conditions, all solutions are oscillatory.

For a class of nonlinear higher-order neutral dynamic equations on a time scale, we analyze the existence and asymptotic behavior of nonoscillatory solutions on the basis of hypotheses that allow applications to equations with different integral convergence and divergence of the reciprocal of the coefficients. Two examples are presented to demonstrate the efficiency of new results.

This article studies the oscillatory behavior of second-order half-linear differential equations with several neutral terms. Some criteria are presented that include those reported in [22].

This work is concerned with oscillation of a class of fourth-order nonlinear delay dynamic equations on a time scale. A new comparison theorem is established that improves related results reported in the literature.

This paper is concerned with the asymptotic properties of solutions to a third-order nonlinear neutral delay differential equation with distributed deviating arguments. Several new theorems are obtained which ensure that every solution to this equation either is oscillatory or tends to zero. Two illustrative examples are included.

In this article, we use the Riccati transformation technique and some inequalities, to establish oscillation theorems for all solutions to even-order quasilinear neutral differential equation $$ Big(ig[ig(x(t)+p(t)x(au(t))ig)^{(n-1)}ig]^gammaBig)' +q(t)x^gammaig(sigma(t)ig)=0,quad tgeq t_0. $$ Our main results are illustrated with examples.

In this paper new oscillation criteria for the second order neutral differential equations of the form \begin{equation*} \left(r(t)\left[x(t)+p(t)x(\tau(t))\right]'\right)'+q(t)x(\sigma(t))+v(t)x(\eta(t))=0 \tag{$E$}\end{equation*} are presented. Gained results are based on the new comparison theorems, that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation. Obtained comparison principles essentially simplify the examination of the studied equations. We cover all possible cases when arguments are delayed, advanced or mixed.

In this paper, the authors study the oscillatory properties of third order quasilinear neutral difference equation of the form $$\Delta(a_{n}(\Delta^{2}(x_{n} + p_{n}x_{n-\delta}))^{\alpha}) + q_{n} {x^{\alpha}_{n-\tau}} = 0,\quad n\geq 0, \tag{E}$$ where $\alpha > 0, 0\leq p_{n}\leq p < \infty. $ By using Riccati transformation we estabilish some new sufficient conditions which ensure that every solution of equation (E) is either oscillatory or converges to zero. These results improve some known results in the literature. Examples are provided to illustrate the main results.

In this article, we obtain several comparison theorems for the second-order neutral dynamic equation $$ Big(r(t)ig([x(t)+p(t)x(au(t))]^Deltaig)^gammaBig)^Delta +q_1(t)x^lambda(delta(t))+q_2(t)x^eta(eta(t))=0, $$ where $gamma,lambda, eta$ are ratios of positive odd integers. We compare such equation with the first-order dynamic inequalities in the sense that the absence of the eventually positive solutions of these first-order inequalities implies the oscillation of the studied equation.

In this note, we establish some new comparison theorems and Philos-type criteria for oscillation of solutions to the odd-order nonlinear neutral functional differential equation $$ [x(t)+p(t)x(au(t))]^{(n)}+q(t)x^alpha(sigma(t))=0, $$ where $0leq p(t)leq p_0

Three comparison theorems are established for the oscillation of the second-order neutral differential equations of mixed type $$ ig(r(t)[x(t)+p_1(t)x(t-sigma_1)+p_2(t)x(t+sigma_2)]'ig)' +q_1(t)x(t-sigma_3) +q_2(t)x(t+sigma_4)=0. $$ Our results are new even when $p_2(t)=q_2(t)=0$. An example is provided to illustrate the main results.

In this paper, by defining a class of functions, we establish some oscillation criteria for the second order nonlinear dynamic equations with forced term $$ x^{\Delta\Delta}(t)+a(t)f(x(q(t)))=e(t) $$ on a time scale $\mathbb{T}.$ Our results unify the oscillation of the second order forced differential equation and the second order forced difference equation. An example is considered to illustrate the main results.

We analyze the oscillatory behavior of solutions to a class of second-order nonlinear neutral delay differential equations. Our theorems improve a number of related results reported in the literature.

We study oscillatory behavior of solutions to a class of second-order nonlinear neutral differential equations under the assumptions that allow applications to differential equations with delayed and advanced arguments. New theorems do not need several restrictive assumptions required in related results reported in the literature. Several examples are provided to show that the results obtained are sharp even for second-order ordinary differential equations and improve related contributions to the subject.

We study asymptotic behavior of solutions to a class of higher-order quasilinear neutral differential equations under the assumptions that allow applications to even- and odd-order differential equations with delayed and advanced arguments, as well as to functional differential equations with more complex arguments that may, for instance, alternate indefinitely between delayed and advanced types. New theorems extend a number of results reported in the literature. Illustrative examples are presented.

We prove that the discrete system ζn+1=Anζn is uniformly exponentially stable if and only if the unique solution of the Cauchy problem ζn+1=Anζn+eiθn+1zn+1, n∈Z+, ζ0=0, is bounded for any real number θ and any p-periodic sequence z(n) with z(0)=0. Here, An is a sequence of bounded linear operators on Banach space X.