This paper deals with stability and Hopf bifurcation analyses of a mathematical model of HIV infection of $\text{CD}{\mathrm{4}}^{+}$ T-cells. The model is based on a system of delay differential equations with logistic growth term and antiretroviral treatment with a discrete time delay, which plays a main role in changing the stability of each steady state. By fixing the time delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. We study the effect of the time delay on the stability of the endemically infected equilibrium. We derive explicit formulae to determine the stability and direction of the limit cycles by using center manifold theory and normal form method. Numerical simulations are presented to illustrate the results.

This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial condition. The class of differential equations is assumed to be approximated by well-posed truncated Taylor series expansions up to a certain order obtained about certain, in general nonperiodic, sampling points $t_i\in [t_0,t_J]$ for $i=\mathrm{0,1},\dots ,J$ of the solution. Two examples are provided.

This paper establishes variation of parameters formula for impulsive differential equations with initial time difference. As an application, one of the results is used to investigate stability properties of solutions.

We obtain two existence results about multiple positive periodic solutions for a class of functional difference system. Two examples are given to illustrate our results.

The paper is devoted mainly to the study of the existence of solutions depending on two variables of a nonlinear integral equation of Volterra-Stieltjes type. The basic tool used in investigations is the technique of measures of noncompactness and Darbo’s fixed point theorem. The results obtained in the paper are applicable, in a particular case, to the nonlinear partial integral equations of fractional orders.

We study the second-order neutral delay differential equation ${[r(t){\Phi }_{\gamma }(z^{\prime }(t))]}^{\prime }+q(t){\Phi }_{\beta }(x(\sigma (t)))=0$, where ${\mathrm{\Phi }}_{\alpha }(t)=|t{|}^{\alpha -\mathrm{1}}t$, $\alpha \ge \mathrm{1}$ and $z(t)=x(t)+p(t)x(\tau (t))$. Based on the conversion into a certain first-order delay differential equation we provide sufficient conditions for nonexistence of eventually positive solutions of two different types. We cover both cases of convergent and divergent integral ${\int \infty r}^{-\mathrm{1}/\gamma }(t)\text{d}t$. A suitable combination of our results yields new oscillation criteria for this equation. Examples are shown to exhibit that our results improve related results published recently by several authors. The results are new even in the linear case.

We consider system of integral equations related to the weighted Hardy-Littlewood-Sobolev (HLS) inequality in a half space. By the Pohozaev type identity in integral form, we present a Liouville type theorem when the system is in both supercritical and subcritical cases under some integrability conditions. Ruling out these nonexistence results, we also discuss the positive solutions of the integral system in critical case. By the method of moving planes, we show that a pair of positive solutions to such system is rotationally symmetric about $x_n$-axis, which is much more general than the main result of Zhuo and Li, 2011.

We consider a modified Leslie-Gower predator-prey model with the Beddington-DeAngelis functional response and feedback controls as follows: $\dot{x}\left(t\right)=x\left(t\right)\left(a_1\left(t\right)-b\left(t\right)x\left(t\right)-c\left(t\right)y\left(t\right)/\left(\alpha \left(t\right)+\beta \left(t\right)x\left(t\right)+\gamma \left(t\right)y\left(t\right)\right)-e_1\left(t\right)u\left(t\right)\right)$, $\dot{u}\left(t\right)=-d_1\left(t\right)u\left(t\right)+p_1\left(t\right)x\left(t-\tau \right)$, $\dot{y}\left(t\right)=y\left(t\right)\left(a_2\left(t\right)-r\left(t\right)y\left(t\right)/\left(x\left(t\right)+k\left(t\right)\right)-e_2\left(t\right)\nu \left(t\right)\right)$, and $\dot{\nu }(t)=-d_2(t)\nu (t)+p_2(t)y(t-\tau )$. Sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained.

We discuss the stability of solutions to a kind of scalar Liénard type equations with multiple variable delays by means of the fixed point technique under an exponentially weighted metric. By this work, we improve some related results from one delay to multiple variable delays.

We establish the existence and uniqueness of a positive solution $u$ for the following fractional boundary value problem${\mathrm{:D}}^{\alpha }u(x)=-a(x)u^{\sigma }(x)$, $x\in (0,1)$ with the conditions $\mathrm{}{\lim }_{x\to 0^{+}}$$x^{2-\alpha }u(x)=0$, $u(1)=0$, where , $\sigma \in (-1,1)$, and $a$ is a nonnegative continuous function on $(0,1)$ that may be singular at $x=0$ or $x=1$. We also give the global behavior of such a solution.

We establish the existence and uniqueness of a positive solution to the following fourth-order value problem: $u^{(4)}(x)=a(x)u^{\sigma }(x)$, $x\in (\mathrm{0,1})$ with the boundary conditions $u(0)=u(1)=u\mathrm{\text{'}}(0)=u\mathrm{\text{'}}(1)=0$, where $\sigma \in (-\mathrm{1,1})$ and $a$ is a nonnegative continuous function on (0, 1) that may be singular at $x=0$ or $x=1$. We also give the global behavior of such a solution.

This paper is concerned with the existence of multiple periodic solutions for discrete Nicholson’s blowflies type system. By using the Leggett-Williams fixed point theorem, we obtain the existence of three nonnegative periodic solutions for discrete Nicholson’s blowflies type system. In order to show that, we first establish the existence of three nonnegative periodic solutions for the $n$-dimensional functional difference system $y\left(k+1\right)=A\left(k\right)y\left(k\right)+f\left(k, y\left(k-\tau \right)\right), k\in \mathbb{Z}$, where $A\left(k\right)$ is not assumed to be diagonal as in some earlier results. In addition, a concrete example is also given to illustrate our results.

We give some sufficient conditions to guarantee convergence of solutions to a nonlinear vector differential equation of third order. We prove a new result on the convergence of solutions. An example is given to illustrate the theoretical analysis made in this paper. Our result improves and generalizes some earlier results in the literature.

The release of transgenic mosquitoes to interact with wild ones is a promising method for controlling malaria. How to effectively release transgenic mosquitoes to prevent malaria is always a concern for researchers. This paper investigates two methods of releasing transgenic mosquitoes and proposes two epidemic models involving malaria patients, anopheles, wild mosquitoes, and transgenic mosquitoes based on system of continuous differential equations. A basic reproduction number ${\mathbf{R}}_0$ is defined for the models and it serves as a threshold parameter that predicts whether malaria will spread. By theoretical analysis of the dynamic behaviors of the models and numerical simulations, it is verified that malaria can be effectively controlled by the opportune release of transgenic mosquitoes; that is, when ${\mathbf{R}}_0\le 1$, malaria will disappear; when ${\mathbf{R}}_0>1$, malaria will become an endemic disease in the target field.