With the rapid popularization of the Internet, computers can enter or leave the Internet increasingly frequently. In fact, no antivirus software can detect and remove all sorts of computer viruses. This implies that viruses would persist on the Internet. To better understand the spread of computer viruses in these situations, a new propagation model is established and analyzed. The unique equilibrium of the model is globally asymptotically stable, in accordance with the reality. A parameter analysis of the equilibrium is also conducted.

We study the following third-order $p$-Laplacian functional dynamic equation on time scales: ${\left[{\mathrm{\Phi }}_p(u^{\Delta \nabla }(t))\right]}^{\nabla }+a(t)f(u(t),u(\mu (t)))=0$, $t\in {\left(0,T\right)}_{\mathbf{T}}$, $u(t)=\phi (t)$, $t\in {\left[-r,0\right]}_{\mathbf{T}}$, $u^{\Delta }(0)=u^{\Delta \nabla }(T)=0$, and $u(T)+B_0(u^{\Delta }(\eta ))=0$. By applying the Five-Functional Fixed Point Theorem, the existence criteria of three positive solutions are established.

In succession to our earlier work, we further provide some new generalized Gronwall inequalities and apply these inequalities to the study of qualitative estimations of solutions to certain fractional differential equations.

We consider the time-oscillating Hartree-type Schrödinger equation ${iu}_t+\Delta u+\theta \left(\omega t\right)\left({\left|x\right|}^{-\gamma }*{\left|u\right|}^2\right)u=0$, where $\theta$ is a periodic function. For the mean value $I(\theta )$ of $\theta$, we show that the solution $u_{\omega }$ converges to the solution of ${iU}_t+\Delta U+I\left(\theta \right)\left({\left|x\right|}^{-\gamma }*{\left|U\right|}^2\right)U=0$ for their local well-posedness and global well-posedness.

This work focuses on the antiperiodic problem of nonautonomous semilinear parabolic evolution equation in the form $u\mathrm{\prime }(t)=A(t)u(t)+f(t,u(t))$, $t\in \mathbb{R}$, $u(t+T)=-u(t)$, $t\in \mathbb{R}$, where ${(A\left(t\right))}_{t\in \mathbb{R}}$ (possibly unbounded), depending on time, is a family of closed and densely defined linear operators on a Banach space $X$. Upon making some suitable assumptions such as the Acquistapace and Terreni conditions and exponential dichotomy on ${(A\left(t\right))}_{t\in \mathbb{R}}$, we obtain the existence results of antiperiodic mild solutions to such problem. The antiperiodic problem of nonautonomous semilinear parabolic evolution equation of neutral type is also considered. As sample of application, these results are applied to, at the end of the paper, an antiperiodic problem for partial differential equation, whose operators in the linear part generate an evolution family of exponential stability.

We propose an alternative framework for total variation based image denoising models. The model is based on the minimization of the total variation with a functional coefficient, where, in this case, the functional coefficient is a function of the magnitude of image gradient. We determine the considerations to bear on the choice of the functional coefficient. With the use of an example functional, we demonstrate the effectiveness of a model chosen based on the proposed consideration. In addition, for the illustrative model, we prove the existence and uniqueness of the minimizer of the variational problem. The existence and uniqueness of the solution associated evolution equation are also established. Experimental results are included to demonstrate the effectiveness of the selected model in image restoration over the traditional methods of Perona-Malik (PM), total variation (TV), and the D-α-PM method.