A delayed neural network model of two neurons with inertial coupling is dealt with in this paper. The stability is investigated and Hopf bifurcation is demonstrated. Applying the normal form theory and the center manifold argument, we derive the explicit formulas for determining the properties of the bifurcating periodic solutions. An illustrative example is given to demonstrate the effectiveness of the obtained results.

This paper studies the nonlinear two-dimensional difference system with multiple delays $\mathrm{\Delta }(x_n+p_{1n}{x}_{n-{\tau }_1})$ $+$ $f_1(n,x_{a_{1n}},\dots ,x_{a_{hn}},y_{b_{1n}},\dots ,y_{b_{kn}})=q_{1n},\mathrm{\Delta }(y_n+p_{2n}{y}_{n-{\tau }_2})+f_2(n,x_{c_{1n}},\dots ,x_{c_{hn}},y_{d_{1n}},\dots ,y_{d_{kn}})=q_{2n},n\ge n_0$. Using the Banach fixed point theorem and a few new analysis techniques, we show the existence of uncountably many bounded positive solutions for the system, suggest Mann iterative algorithms with errors, and discussthe error estimates between the positive solutions and iterative sequences generated by the Mann iterativealgorithms. Examples to illustrates the results are included.

The asymptotic behaviour of a real two-dimensional differential system $x\prime (t)=\mathsf{A}(t)x(t)+{\sum }_{k=1}^m{\mathsf{B}}_k(t)x({\theta }_k(t))+h(t,x(t),x({\theta }_1(t)),\dots ,x({\theta }_m(t)))$ with unbounded nonconstant delays $t-{\theta }_k(t)\ge 0$ satisfying ${\lim \hspace{0.17em}}_{t\to \infty }{\theta }_k(t)=\infty$ is studied under the assumption of instability. Here, $\mathsf{A}$, ${\mathsf{B}}_{\mathrm{k,}}$ and $h$ are supposed to be matrix functions and a vector function. The conditions for the instable properties of solutions and the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of a Lyapunov-Krasovskii functional and the suitable Ważewski topological principle. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one constant or nonconstant delay were studied.

Let $𝕋\subset ℝ$ be a periodic time scale in shifts ${\delta }_{\pm }$. We use a fixed point theorem due to Krasnosel'skiĭ to show that nonlinear delay in dynamic equations of the form $x^{\mathrm{\Delta }}(t)=-a(t)x^{\sigma }(t)+b(t)x^{\mathrm{\Delta }}({\delta }_{-}(k,t)){\delta }_{-}^{\mathrm{\Delta }}(k,t)+q(t,x(t),x({\delta }_{-}(k,t))),t\in 𝕋$, has a periodic solution in shifts ${\delta }_{\pm }$. We extend and unify periodic differential, difference, $h$-difference, and $q$-difference equations and more by a new periodicity concept on time scales.

The first boundary-value problem for an autonomous second-order system of linear partial differential equations of parabolic type with a single delay is considered. Assuming that a decomposition of the given system into a system of independent scalar second-order linear partial differential equations of parabolic type with a single delay is possible, an analytical solution to the problem is given in the form of formal series and the character of their convergence is discussed. A delayed exponential function is used in order to analytically solve auxiliary initial problems (arising when Fourier method is applied) for ordinary linear differential equations of the first order with a single delay.

The paper deals with nonlinear differential systems with random parameters in a general form. A new method for construction of the Lyapunov functions is proposed and is used to obtain sufficient conditions for $L_2$-stability of the trivial solution of the considered systems.

New oscillation criteria are established for the second-order nonlinear neutralfunctional differential equations of the form ${(r(t){|z^{\prime }(t)|}^{\alpha -1}{z}^{\prime }(t))}^{’}+f(t,x[\sigma (t)])=0$, $t\ge t_0$, where $z(t)=x(t)+p(t)x(\tau (t))$, $p\in C^1([t_0,\infty ),[0,\mathrm{\infty }))$, and $\alpha \ge 1$. Our results improve andextend some known results in the literature. Some examples are also provided to show theimportance of these results.

The interval exponential state estimation and robust exponential stability for the switched interval neural networks with discrete and distributed time delays are considered. Firstly, by combining the theories of the switched systems and the interval neural networks, the mathematical model of the switched interval neural networks with discrete and distributed time delays and the interval estimation error system are established. Secondly, by applying the augmented Lyapunov-Krasovskii functional approach and available output measurements, the dynamics of estimation error system is proved to be globally exponentially stable for all admissible time delays. Both the existence conditions and the explicit characterization of desired estimator are derived in terms of linear matrix inequalities (LMIs). Moreover, a delay-dependent criterion is also developed, which guarantees the robust exponential stability of the switched interval neural networks with discrete and distributed time delays. Finally, two numerical examples are provided to illustrate the validity of the theoretical results.

By applying the least action principle and minimax methods in critical point theory, we prove the existence of periodic solutions for a class of difference systems with p-Laplacian and obtain some existence theorems.

This paper is concerned with a delay logistical model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall's inequality, and Young's inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Also, the relationships between the stochastic permanence and extinction as well as asymptotic estimations of solutions are investigated by virtue of $V$-function technique, $M$-matrix method, and Chebyshev's inequality. Finally, an example is given to illustrate the main results.

This paper is concerned with mean square exponential stability of switched stochastic system with interval time-varying delays. The time delay is any continuous function belonging to a given interval, but not necessary to be differentiable. By constructing a suitable augmented Lyapunov-Krasovskii functional combined with Leibniz-Newton’s formula, a switching rule for the mean square exponential stability of switched stochastic system with interval time-varying delays and new delay-dependent sufficient conditions for the mean square exponential stability of the switched stochastic system are first established in terms of LMIs. Numerical example is given to show the effectiveness of the obtained result.

We introduce a notion of attractivity for delay equations which are defined on bounded time intervals. Our main result shows that linear delay equations are finite-time attractive, provided that the delay is only in the coupling terms between different components, and the system is diagonally dominant. We apply this result to a nonlinear Lotka-Volterra system and show that the delay is harmless and does not destroy finite-time attractivity.

This paper studies the following third order neutral delay discrete equation $\mathrm{\Delta }(a_n{\mathrm{\Delta }}^2(x_n+p_n{x}_{n-\tau }))+f(n,x_{n-d_{1n}},\dots ,x_{n-d_{ln}})=g_n,n\ge n_0$, where $\tau ,l\in \mathbb{N}$, $n_0\in \mathbb{N}\cup \{0\}$, $\{a_n{\}}_{n\in {\mathbb{N}}_{n_0}}$, $\{p_n{\}}_{n\in {\mathbb{N}}_{n_0}}$, $\{g_n{\}}_{n\in {\mathbb{N}}_{n_0}}$ are real sequences with $a_n\ne 0$ for $n\ge n_0$, $\{d_{in}{\}}_{n\in {\mathbb{N}}_{n_0}}\subseteq \mathbb{Z}$ with ${\lim \hspace{0.17em}}_{n\to \infty }(n-d_{in})=+\infty$ for $i\in \{\mathrm{1,2},\dots ,l\}$ and $f\in C({\mathbb{N}}_{n_0}\times {ℝ}^l,ℝ)$. By using a nonlinear alternative theorem of Leray-Schauder type, we get sufficient conditions which ensure the existence of bounded positive solutions for the equation. Three examples are given to illustrate the results obtained in this paper.

We study the following second order mixed nonlinear impulsive differential equations with delay ${\left(r\left(t\right){\Phi }_{\alpha }\left(x^{\prime }\left(t\right)\right)\right)}^{\prime }+p_0\left(t\right){\Phi }_{\alpha }\left(x\left(t\right)\right)+{\sum }_{i=1}^n{p}_i\left(t\right){\Phi }_{{\beta }_i}\left(x\left(t-\sigma \right)\right)=e\left(t\right),t\ge t_0,t\ne {\tau }_k,x\left({\tau }_k^{+}\right)=a_{k}x\left({\tau }_k\right),x\text{'}\left({\tau }_k^{+}\right)=b_{k}x\text{'}\left({\tau }_k\right),k=\mathrm{1,2},\dots$, where ${\Phi }_{*}\left(u\right)=|u{|}^{*-1}u$, $\sigma$ is a nonnegative constant, $\left\{{\tau }_k\right\}$ denotes the impulsive moments sequence, and ${\tau }_{k+1}-{\tau }_k>\sigma$. Some sufficient conditions for the interval oscillation criteria of the equations are obtained. The results obtained generalize and improve earlier ones. Two examples are considered to illustrate the main results.

This paper studies the adaptive fault estimation problems for stochastic Markovian jump systems (MJSs) with time delays. With the aid of the selected Lyapunov-Krasovskii functional, the adaptive fault estimation algorithm based on adaptive observer is proposed to enhance the rapidity and accuracy performance of fault estimation. A sufficient condition on the existence of adaptive observer is presented and proved by means of linear matrix inequalities techniques. The presented results are extended to multiple time-delayed MJSs. Simulation results illustrate that the validity of the proposed adaptive faults estimation algorithms.

Population control has become a major problem in many wildlife species. Sterility control through contraception has been proposed as a method for reducing population size. In this paper, the single species with sterility control and feedback controls is considered. Sufficient conditions are obtained for the permanence and extinction of the system. The results show that the feedback controls do not influence the permanence of the species.

We consider a parametric semilinear Dirichlet problem with an unbounded and indefinite potential. In the reaction we have the competing effects of a sublinear (concave) term and of a superlinear (convex) term. Using variational methods coupled with suitable truncation techniques, we prove two multiplicity theorems for small values of the parameter. Both theorems produce five nontrivial smooth solutions, and in the second theorem we provide precise sign information for all the solutions.

This paper deals with solvability of the third-order nonlinear partial difference equation with delays ${\Delta }_n\left(a_{m,n}{\Delta }_m^2\right(x_{m,n}+b_{m,n}{x}_{m-{\tau }_0,n-{\sigma }_0}\left)\right)+f(m,n,x_{m-{\tau }_{1,m},n-{\sigma }_{1,n}},\dots ,x_{m-{\tau }_{k,m},n-{\sigma }_{k,n}})=c_{m,n},m\ge m_0,n\ge n_0$. With the help of the Banach fixed-point theorem, the existence results of uncountably many bounded positive solutions for the partial difference equation are given; some Mann iterative schemes with errors are suggested, and the error estimates between the iterative schemes and the bounded positive solutions are discussed. Three nontrivial examples illustrating the results presented in this paper are also provided.

We study the uniqueness problems on entire functions and their difference operators or shifts. Our main result is a difference analogue of a result of Jank-Mues-Volkmann, which is concerned with the uniqueness of the entire function sharing one finite value with its derivatives. Two relative results are proved, and examples are provided for our results.

We first show the existence and uniqueness of (pseudo) almost periodic solutions of some types of parabolic equations. Then, we apply the results to a type of Cauchy parabolic inverse problems and show the existence, uniqueness, and stability.

This paper is concerned with the $n$th-order forced nonlinear neutral differential equation $\left[x\right(t)-p(t\left)x\right(\tau \left(t\right)){]}^{\left(n\right)}+{\sum }_{i=1}^m{q}_i(t\left)f_i\right(x\left({\sigma }_{i1}\right(t\left)\right),x\left({\sigma }_{i2}\right(t\left)\right),\dots ,x\left({\sigma }_{i{k}_i}\right(t\left)\right))=g(t),t\ge t_0$. Some necessary and sufficient conditions for the oscillation of bounded solutions and several sufficient conditions for the existence of uncountably many bounded positive and negative solutions of the above equation are established. The results obtained in this paper improve and extend essentially some known results in the literature. Five interesting examples that point out the importance of our results are also included.

We study a homogeneous partial differential equation and get its entire solutions represented in convergent series of Laguerre polynomials. Moreover, the formulae of the order and type of the solutions are established.

In this paper, two classes of first-order neutral functional differential equations with periodic delays are considered. Some results on the existence of positive periodic solutions for the equations are obtained by using the Krasnoselskii fixed point theorem. Four examples are included to illustrate our results.

The object of investigation of the paper is a special type of functional differential equations containing the maximum value of the unknown function over a past time interval. An improved algorithm of the monotone-iterative technique is suggested to nonlinear differential equations with “maxima.” The case when upper and lower solutions of the given problem are known at different initial time is studied. Additionally, all initial value problems for successive approximations have both initial time and initial functions different. It allows us to construct sequences of successive approximations as well as sequences of initial functions, which are convergent to the solution and to the initial function of the given initial value problem, respectively. The suggested algorithm is realized as a computer program, and it is applied to several examples, illustrating the advantages of the suggested scheme.

This paper is concerned with stabilization of impulsive stochastic delay differential systems. Based on the Razumikhin techniques and Lyapunov functions, several criteria on pth moment and almost sure exponential stability are established. Our results show that stochastic functional differential systems may be exponentially stabilized by impulses.

By Faà di Bruno’s formula, using the fixed-point theorems of Schauder and Banach, we study the existence and uniqueness of smooth solutions of an iterative functional differential equation $x^{\prime }\left(t\right)=1/{\mathrm{(c}}_0{x}^{\left[0\right]}\left(t\right)+c_1{x}^{\left[1\right]}\left(t\right)+\cdots +c_m{x}^{\left[m\right]}\left(t\right))$.

This paper is concerned with the existence of solutions for the discrete second-order boundary value problem ${\Delta }^{2}u(t-1)+{\lambda }_{1}u\left(t\right)+g\left(\Delta u\right(t\left)\right)=f\left(t\right)$, $t\in \{\mathrm{1,2},\dots ,T\}$, $u\left(0\right)=u(T+1)=0$, where $T>1$ is an integer, $f:\{1,\dots ,T\}\to \mathbb{R}$, $g:\mathbb{R}\to \mathbb{R}$ is bounded and continuous, and ${\lambda }_1$ is the first eigenvalue of the eigenvalue problem ${\Delta }^{2}u(t-1)+\lambda u\left(t\right)=0$, $t\in \mathbb{T}$, $u\left(0\right)=u(T+1)=0$.

By using Schaeffer's theorem and Lyapunov functional, sufficient conditions of the existence and globally exponential stability of positive periodic solution to an impulsive neural network with time-varying delays are established. Applications, examples, and numerical analysis are given to illustrate the effectiveness of the main results.

We study a frequently investigated class of linear difference equations $\Delta v\left(n\right)=-p\left(n\right)v\left(n-k\right)$ with a positive coefficient $p\left(n\right)$ and a single delay k. Recently, it was proved that if the function $p\left(n\right)$ is bounded above by a certain function, then there exists a positive vanishing solution of the considered equation, and the upper bound was found. Here we improve this result by finding even the lower bound for the positive solution, supposing the function $p\left(n\right)$ is bounded above and below by certain functions.

We investigate first the existence of periodic solution in general Cohen-Grossberg BAM neural networks with multiple time-varying delays by means of using degree theory. Then using the existence result of periodic solution and constructing a Lyapunov functional, we discuss global exponential stability of periodic solution for the above neural networks. Our result on global exponential stability of periodic solution is different from the existing results. In our result, the hypothesis for monotonicity ineqiality conditions in the works of Xia (2010) Chen and Cao (2007) on the behaved functions is removed and the assumption for boundedness in the works of Zhang et al. (2011) and Li et al. (2009) is also removed. We just require that the behaved functions satisfy sign conditions and activation functions are globally Lipschitz continuous.

The existence of positive solutions for Neumann boundary value problem of second-order impulsive differential equations $-u″\left(t\right)+Mu\left(t\right)=f(t,u\left(t\right)$, $t\in J$, $t\ne t_k$, $-\Delta u\text{'}{|}_{t=t_k}=I_k\left(u\right(t_k\left)\right)$, $k=\mathrm{1,2},\dots ,m$, $u\text{'}\left(0\right)=u\text{'}\left(1\right)=\theta$, in an ordered Banach space $E$ was discussed by employing the fixed point index theory of condensing mapping, where $M>0$ is a constant, $J=\left[\mathrm{0,1}\right]$, $f\in C(J\times K,K)$, $I_k\in C(K,K)$, $k=\mathrm{1,2},\dots ,m$, and $K$ is the cone of positive elements in $E$. Moreover, an application is given to illustrate the main result.

we show the existence and multiplicity of positive solutions of the nonlineardiscrete fourth-order boundary value problem ${\Delta }^{4}u\left(t-2\right)=\lambda h\left(t\right)f\left(u\left(t\right)\right)$, $t\in {\mathbb{T}}_2$, $u\left(1\right)=u\left(T+1\right)={\Delta }^{2}u\left(0\right)={\Delta }^{2}u\left(T\right)=0$, where $\lambda >0$, $h:{\mathbb{T}}_2\to (0,\infty )$ is continuous, and $f:\mathbb{R}\to [0,\infty )$ is continuous, $T>4$, ${\mathbb{T}}_2=\left\{\mathrm{2,3},\dots ,T\right\}$. The main tool is the Dancer's global bifurcation theorem.

Using inequality techniques and coincidence degree theory, new results are provided concerning the existence and uniqueness of periodic solutions for the Duffing type $p$-Laplacian equation with multiple constant delays of the form ${\left({\phi }_p\left(x^{\prime }\left(t\right)\right)\right)}^{\prime }+C{x}^{\prime }\left(t\right)+g_0\left(t,x\left(t\right)\right)+{\sum }_{k=1}^n{g}_k\left(t,x\left(t-{\tau }_k\right)\right)=e\left(t\right).$ Moreover, an example is provided to illustrate the effectiveness of the results in this paper.

We provide optimal conditions for the existence and uniqueness of solutions to a nonlocal boundary value problem for a class of linear homogeneous second-order functional differential equations with piecewise constant arguments. The nonlocal boundary conditions include terms of the state function and the derivative of the state function. A similar nonhomogeneous problem is also discussed.

We present some results to the existence and uniqueness of the periodic solutions for the hematopoiesis models which are described by the functional differential equations with multiple delays. Our methods are based on the equivalent norm techniques and a new fixed point theorem in the continuous function space.

The paper contains some suffcient conditions for the existence of positive solutions which are bounded below and above by positive functions for the nonlinear neutral differential equations of higher order. These equations can also support the existence of positive solutions approaching zero at infinity.

We study the existence of at least one monotonic positive solution for the nonlocal boundary value problem of the second-order functional differential equation $x^{\mathrm{\prime \prime }}\left(t\right)=f(t,x(\varphi \left(t\right)\left)\right)$, $t\in \left(\mathrm{0,1}\right)$, with the nonlocal condition ${\sum }_{k=1}^m{a}_{k}x\left({\tau }_k\right)=x_0$, $x^{\prime }\left(0\right)+{\sum }_{j=1}^n{b}_j{x}^{\prime }\left({\eta }_j\right)=x_1$, where ${\tau }_k\in (a,d)\subset \left(\mathrm{0,1}\right)$, ${\eta }_j\in (c,e)\subset \left(\mathrm{0,1}\right)$, and $x_0,x_1>0$. As an application the integral and the nonlocal conditions ${\int }_a^{d}x\left(t\right)dt=x_0$, $x^{\prime }\left(0\right)+x\left(e\right)-x\left(c\right)=x_1$ will be considered.

We prove the existence of one-signed periodic solutions of second-order nonlinear difference equation on a finite discrete segment with periodic boundary conditions by combining some properties of Green's function with the fixed-point theorem in cones.