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In this paper, we apply Lin's method to study the existence of heteroclinic orbits near the degenerate heteroclinic chain under m$$ m $$‐dimensional periodic perturbations. The heteroclinic chain consists of two degenerate heteroclinic orbits γ1$$ {\gamma}_1 $$ and γ2$$ {\gamma}_2 $$ connected by three hyperbolic saddle points q1,q2,q3$$ {q}_1,{q}_2,{q}_3 $$. Assume that the degeneracy of the unperturbed heteroclinic orbit γi$$ {\gamma}_i $$ is ni$$ {n}_i $$, the splitting index is δi$$ {\delta}_i $$. By applying Lin's method, we construct heteroclinic orbits connected q1$$ {q}_1 $$ and q3$$ {q}_3 $$ near the unperturbed heteroclinic chain. The existence of these orbits is equivalent to finding zeros of the corresponding bifurcation function. The lower order terms of the bifurcation function is the map from ℝn1+n2+m$$ {\mathrm{\mathbb{R}}}^{n_1+{n}_2+m} $$ to ℝn1+n2+δ1+δ2$$ {\mathrm{\mathbb{R}}}^{n_1+{n}_2+{\delta}_1+{\delta}_2} $$. Using the contraction mapping principle, we provide a detailed analysis on how zeros can exist based on different cases of splitting indices δ1$$ {\delta}_1 $$, δ2$$ {\delta}_2 $$ and then obtain the existence of the heteroclinic orbits which backward asymptotic to q1$$ {q}_1 $$ and forward asymptotic to q3$$ {q}_3 $$. [ABSTRACT FROM AUTHOR]

This paper studies two nonlocal diffusion problems with a free boundary and a fixed boundary in heterogeneous environment. The main goal is to understand how the evolution of the two species is affected by the heterogeneous environment. We first prove the existence and uniqueness of a global solution for such systems. Then, for models with Lotka–Volterra type competition or predator–prey growth terms, we establish the spreading‐vanishing dichotomy. Sharp criteria of spreading and vanishing are also obtained. Furthermore, we show that accelerated spreading occurs if and only if the kernel function violates the threshold condition. [ABSTRACT FROM AUTHOR]

In the present paper, we first introduce the concept of double measure r$$ r $$‐mean piecewise pseudo almost periodic for stochastic processes for r≥2$$ r\ge 2 $$. Next, we make extensive use of the exponential dichotomy techniques and a fixed point strategy with stochastic analysis theory to obtain the existence of doubly measure r$$ r $$‐mean piecewise pseudo almost periodic mild solutions for a class of impulsive non‐autonomous partial stochastic evolution equations in Hilbert spaces. In addition, we study the exponential stability of r$$ r $$‐mean piecewise pseudo almost periodic mild solutions. Finally, we give an example to confirm the reliability and feasibility of our findings. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish some sufficient conditions ensuring the existence and uniqueness of positive (Bohr) almost periodic solutions to a class of semilinear evolution equations of the form: u′(t)=A(t)u(t)+f(t,u(t)),t∈ℝ$$ {u}^{\prime }(t)=A(t)u(t)+f\left(t,u(t)\right),t\in \mathbb{R} $$. We assume that the family of closed linear operators (A(t))t∈ℝ$$ {\left(A(t)\right)}_{t\in \mathbb{R}} $$ on a Banach lattice X$$ X $$ satisfies the "Acquistapace–Terreni" conditions, so that the associated evolution family is positive and has an exponential dichotomy on ℝ$$ \mathbb{R} $$. The nonlinear term f$$ f $$, acting on certain real interpolation spaces, is assumed to be almost periodic only in a weaker sense (i.e., in Stepanov's sense) with respect to t$$ t $$, and Lipschitzian in bounded sets with respect to the second variable. Moreover, we prove a new composition result for Stepanov almost periodic functions by assuming only continuity of f$$ f $$ with respect to the second variable (see the condition Lemma 1‐(ii)). Finally, we provide an application to a system of Lotka–Volterra predator–prey type model with diffusion and time–dependent parameters in a generalized almost periodic environment. [ABSTRACT FROM AUTHOR]

In this paper, we study the partial differential equation with piecewise constant argument of the form: xt(t,s)=A(t)x(t,s)+B(t,s)x([t],s)+C(t,s)x(t,[s])+D(t,s)x([t],[s])+f(x(t,[s])),t,s∈IR+=(0,∞),$$ {\displaystyle \begin{array}{cc}\hfill {x}_t\left(t,s\right)=& \kern0.2em A(t)x\left(t,s\right)+B\left(t,s\right)x\left(\left[t\right],s\right)+C\left(t,s\right)x\left(t,\left[s\right]\right)\hfill \\ {}\hfill & +D\left(t,s\right)x\left(\left[t\right],\left[s\right]\right)+f\left(x\left(t,\left[s\right]\right)\right),t,s\in I\kern-0.2em {R}^{+}=\left(0,\infty \right),\hfill \end{array}} $$where A(t)$$ A(t) $$ is a k×k$$ k\times k $$ invertible and continuous matrix function on IR+$$ I\kern-0.2em {R}^{+} $$; B(t,s)$$ B\left(t,s\right) $$, C(t,s)$$ C\left(t,s\right) $$, and D(t,s)$$ D\left(t,s\right) $$ are k×k$$ k\times k $$ continuous and bounded matrix functions on IR+×IR+$$ I\kern-0.2em {R}^{+}\times I\kern-0.2em {R}^{+} $$; [t]$$ \left[t\right] $$ and [s]$$ \left[s\right] $$ are the integral parts of t$$ t $$ and s$$ s $$, respectively; and f:IRk→IRk$$ f:I\kern-0.2em {R}^k\to I\kern-0.2em {R}^k $$ is a continuous function. More precisely, under some conditions on the matrices A(t)$$ A(t) $$, B(t,s)$$ B\left(t,s\right) $$, C(t,s)$$ C\left(t,s\right) $$, and D(t,s)$$ D\left(t,s\right) $$ and the function f$$ f $$, we investigate the asymptotic behavior of the solutions of the above equation. [ABSTRACT FROM AUTHOR]

In this paper, by developing important properties on the composition of functions with reflection, using some exponential dichotomy properties and an application of the fixed‐point theorem, several new sufficient conditions for the existence and the uniqueness of an pseudo almost automorphic solutions with measure for some general‐type reflection integro‐differential equations. We suppose that the nonlinear part is measure pseudo almost automorphic and in which we distinguish the two constant and variable cases for the Lipschitz coefficients of the functions associated with this part. It is assumed that the linear part of the equation considered admits an exponential dichotomy. Finally, an application is given on the very interesting model of Markus and Yamabe. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the existence, uniqueness, and conditional stability of bounded and periodic solutions for a class of noninstantaneous impulsive linear and semilinear equations with evolution family and exponential dichotomy. We utilize the weak * convergence analysis in the conjugate space and the Banach‐Alaoglu theorem to derive the existence result, and then we use the principle of compressed image to prove the uniqueness. In addition, we study the conditional stability of periodic solution with the help of the Grownwall‐Coppel inequality. Finally, we present an example of a noninstantaneous impulsive partial differential equation, which is transferred into an abstract impulsive evolution equation. [ABSTRACT FROM AUTHOR]

In this paper, the stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment is studied. It is shown that there exist 3 equilibria. The sufficient conditions for local asymptotic stability of the infection-free equilibrium and no-immune equilibrium are given. We also discussed the local stability of positive equilibrium and the existence of Hopf bifurcation. Moreover, the direction and stability of Hopf bifurcation is obtained by using standard form theory and the center manifold theorem. Finally, numerical simulations are performed to verify the theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate the functions of ( μ, ν)-pseudo S-asymptotically ω-periodic of class r(class infinity). We systematically explore the properties of these functions in Banach space including composition theorems. As applications, we establish some sufficient criteria for ( μ, ν)-pseudo S-asymptotic ω-periodicity of (nonautonomous) semilinear integro-differential equations with finite or infinite delay. Finally, some interesting examples are presented to illustrate the main findings. [ABSTRACT FROM AUTHOR]

In this article, a new approach for pseudo almost periodic solution under the measure theory, under Acquistpace-Terreni conditions. We make extensive use of interpolation spaces and exponential dichotomy techniques to obtain the existence of μ-pseudo almost periodic solutions to some classes of nonautonomous partial evolution equations. For illustration, we propose some application to a nonautonomous heat equation. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of neutral-type BAM neural networks with distributed leakage delays. By applying the exponential dichotomy of linear differential equations, Lyapunov functional method and contraction mapping principle, we establish some sufficient conditions which ensure the existence and exponential stability of almost periodic solutions for such BAM neural networks. An example is given to illustrate the effectiveness of the theoretical findings. The results obtained in this article are completely new and complement the previously known studies. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we first propose a concept of weighted pseudo-almost periodic functions on time scales and study some basic properties of weighted pseudo-almost periodic functions on time scales. Then, we establish some results about the existence of weighted pseudo-almost periodic solutions to linear dynamic equations on time scales. Finally, as an application of our results, we study the existence and global exponential stability of weighted pseudo-almost periodic solutions for a class of cellular neural networks with discrete delays on time scales. The results of this paper are completely new. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a model of hematopoiesis with an oscillating circulation loss rate is investigated. By applying the exponential dichotomy theory, contraction mapping fixed-point theorem, and differential inequality techniques, a set of sufficient conditions are obtained for the existence and exponential stability of positive pseudo almost periodic solutions of the model. Some numerical simulations are carried out to support the theoretical findings. Our results improve and generalize those of the previous studies. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, it is the first time that some important inequalities are obtained to estimate the exponential type of upper and lower bounds of solutions for the three representative classes of homogeneous impulsive dynamic systems on time scales. Based on these, some new criteria are established for admitting an exponential dichotomy of the impulsive dynamic systems. The obtained results are essentially new, even the time scale T D R or T D Z. In addition, in applications, we apply the obtained results to discuss the almost periodic problems of a class of integro-differential systems, and the numerical simulations are given to illustrate that our timescalemethods are feasible and effective. Finally, we present the conclusion and further discussion related to this topic. [ABSTRACT FROM AUTHOR]