In this paper, a delayed nonresident computer virus model with graded infection rate is considered in which the following assumption is imposed: latent computers have lower infection ability than infectious computers. With the aid of the bifurcation theory, sufficient conditions for stability of the infected equilibrium of the model and existence of the Hopf bifurcation are established. In particular, explicit formulae which determine direction and stability of the Hopf bifurcation are derived by means of the normal form theory and the center manifold reduction for functional differential equations. Finally, a numerical example is given in order to show the feasibility of the obtained theoretical findings.

$[b,T]$ denotes the commutator of generalized Calderón-Zygmund operators $T$ with Lipschitz function $b$, where $b ∈ \rm{Lip}_β(R^n)$, $(0<β≤1)$ and $T$ is a $θ(t)$−type Calderón-Zygmund operator. The commutator $[b,T]$ generated by $b$ and $T$ is defined by 

$$[b,T] f(x)=b(x)Tf(x)−T(bf)(x)=∫k(x,y)(b(x)−b(y))f(y)dy.$$

In this paper, the authors discuss the boundedness of the commutator $[b,T]$ on weighted Hardy spaces and weighted Herz type Hardy spaces and prove that $[b,T]$ is bounded from $H^p(ω^p)$ to $L^q(ω^q)$, and from $H\dot{K}^{α,p}_{q_1}(ω_1,ω^{q_1}_2)$ to $\dot{K}^{α,p}_{q_2}(ω_1,ω^{q_2}_2)$. The results extend and generalize the well-known ones in [7].

We define Bloch-type functions of $\mathcal{C}^1(\mathbb{D})$ on the unit disk of complex plane $\mathbb{C}$ and characterize them in terms of weighted Lipschitz functions. We also discuss the boundedness of a composition operator $C_{\phi}$ acting between two Bloch-type spaces. These obtained results generalize the corresponding known ones to the setting of $\mathcal{C}^1(\mathbb{D})$.

An elementary, but very useful tool for proving inequalities for polynomials with restricted zeros is the Bernstein or Lorentz representation of polynomials. In the present paper, we give two classes of Lorentz polynomials, for which the Erdös-type inequality holds.

In this study, as the domain of four dimensional Euler mean $E(r,s)$ of orders $r$, $s$ in the space $\mathcal{L}_p$ for $0<p<1$, we examine the double sequence space $\varepsilon^{r,s}_p$ and some properties of four dimensional Euler mean. We determine the $α$- and $β(bp)$-duals of the space $\varepsilon^{r,s}_p$, and characterize the classes $(\varepsilon^{r,s}_p:\mathcal{M}_u)$, $(\varepsilon^{r,s}_p:\mathcal{C}_{bp})$ and $(\varepsilon^{r,s}_p:\mathcal{L}_q)$ of four dimensional matrix transformations, where $1≤q<∞$. Finally, we shortly emphasize on the Euler spaces of single and double sequences, and note some further suggestions.

In this work, we study existence and uniqueness of solutions for multi-point boundary value problem of nonlinear fractional differential equations with two fractional derivatives. By using the variety of fixed point theorems, such as Banach’s fixed point theorem, Leray-Schauder’s nonlinear alternative and Leray-Schauder’s degree theory, the existence of solutions is obtained. At the end, some illustrative examples are discussed.

This paper is sequel to the authors paper [18]. By using the generalized Burchnall-Chaundy operator method, the authors are aiming at deriving certain decomposition formulas for some interesting special cases of Kampé de Fériets series of double hypergeometric series $F^{p:q;k}_{l:m;n}$.