Global Dynamics of Sixth-Order Fuzzy Difference Equation

Across many fields, such as engineering, ecology, and social science, fuzzy differences are becoming more widely used; there is a wide variety of applications for difference equations in real-life problems. Our study shows that the fuzzy difference equation of sixth order has a nonnegative solution, an equilibrium point and asymptotic behavior. y i + 1 = D y i − 1 y i − 2 / E + F y i − 3 + G y i − 4 + H y i − 5 , i = 0,1,2 , … , where y i is the sequence of fuzzy numbers and the parameter D , E , F , G , H and the initial condition y − 5 , y − 4 , y − 3 , y − 2 , y − 1 , y 0 are nonnegative fuzzy number. The characterization theorem is used to convert each single fuzzy difference equation into a set of two crisp difference equations in a fuzzy environment. So, a pair of crisp difference equations is formed by converting the difference equation. The stability of the equilibrium point of a fuzzy system has been evaluated. By using variational iteration techniques and inequality skills as well as a theory of comparison for fuzzy difference equations, we investigated the governing equation dynamics, such as its boundedness, existence, and local and global stability analysis. In addition, we provide some numerical solutions for the equation describing the system to verify our results.

Existence results involving fractional Liouville derivative

In this paper we investigate the question of existence of nonnegative solution to some fractional liouville equation. Our main tools based on the well known Krasnoselskiis xed point theorem.

Classification of nonnegative solutions to fractional Schrödinger-Hatree-Maxwell type system

<abstract><p>In this paper, we are concerned with the fractional Schrödinger-Hatree-Maxwell type system. We derive the forms of the nonnegative solution and classify nonlinearities by appling a variant (for nonlocal nonlinearity) of the direct moving spheres method for fractional Laplacians. The main ingredients are the variants (for nonlocal nonlinearity) of the maximum principles, i.e., <italic>narrow region principle</italic> (Theorem 2.3).</p></abstract>

A log–exp elliptic equation in the plane

<p style='text-indent:20px;'>In this paper we show the existence of a nonnegative solution for a singular problem with logarithmic and exponential nonlinearity, namely <inline-formula><tex-math id="M1">\begin{document}$ -\Delta u = \log(u)\chi_{\{u&gt;0\}} + \lambda f(u) $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ u = 0 $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M4">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain in <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{R}^{2} $\end{document}</tex-math></inline-formula>. We replace the singular function <inline-formula><tex-math id="M7">\begin{document}$ \log(u) $\end{document}</tex-math></inline-formula> by a function <inline-formula><tex-math id="M8">\begin{document}$ g_\epsilon(u) $\end{document}</tex-math></inline-formula> which pointwisely converges to -<inline-formula><tex-math id="M9">\begin{document}$ \log(u) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M10">\begin{document}$ \epsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>. When the parameter <inline-formula><tex-math id="M11">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> is small enough, the corresponding energy functional to the perturbed equation <inline-formula><tex-math id="M12">\begin{document}$ -\Delta u + g_\epsilon(u) = \lambda f(u) $\end{document}</tex-math></inline-formula> has a critical point <inline-formula><tex-math id="M13">\begin{document}$ u_\epsilon $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M14">\begin{document}$ H_0^1(\Omega) $\end{document}</tex-math></inline-formula>, which converges to a nontrivial nonnegative solution of the original problem as <inline-formula><tex-math id="M15">\begin{document}$ \epsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>.</p>

Radial symmetry of nonnegative solutions for nonlinear integral systems

<p style='text-indent:20px;'>In this paper, we investigate the nonnegative solutions of the nonlinear singular integral system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \left\{ \begin{array}{lll} u_i(x) = \int_{\mathbb{R}^n}\frac{1}{|x-y|^{n-\alpha}|y|^{a_i}}f_i(u(y))dy,\quad x\in\mathbb{R}^n,\quad i = 1,2\cdots,m,\\ 0&lt;\alpha&lt;n,\quad u(x) = (u_1(x),\cdots,u_m(x)),\nonumber \end{array}\right. \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;a_i/2&lt;\alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ f_i(u) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 1\leq i\leq m $\end{document}</tex-math></inline-formula>, are real-valued functions, nonnegative and monotone nondecreasing with respect to the independent variables <inline-formula><tex-math id="M4">\begin{document}$ u_1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ u_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \cdots $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ u_m $\end{document}</tex-math></inline-formula>. By the method of moving planes in integral forms, we show that the nonnegative solution <inline-formula><tex-math id="M8">\begin{document}$ u = (u_1,u_2,\cdots,u_m) $\end{document}</tex-math></inline-formula> is radially symmetric when <inline-formula><tex-math id="M9">\begin{document}$ f_i $\end{document}</tex-math></inline-formula> satisfies some monotonicity condition.</p>

Dynamic of a stochastic delayed one-predator two-prey model with Lévy jumps in polluted environments

This paper is concerned with a stochastic delayed one-predator two-prey model with Lévy jumps in polluted environments. First, under some simple assumptions, we prove that there exists a unique global nonnegative solution which is permanent in time average. Moreover, sufficient criteria for the extinction of each species are obtained. Finally, we carry out some numerical simulations to verify the theoretical results.

Mathematical analysis of information propagation model in complex networks

In virtue of identifying the influence of nodes, the spatial distance of rumor propagation is defined with the partition and clustering in the network. Considering the temporal and spatial propagation characteristics of rumors in online social networks, we establish a delayed rumor propagation model based on the graph theory and partial functional differential equations. Firstly, the unique existence and uniform boundedness of the nonnegative solution are explored. Secondly, we discuss the existence of positive equilibrium points sufficiently. Thirdly, stabilities of the rumor-free and rumor-spreading equilibrium points are investigated according to the linearization approach and Lyapunov function. Finally, we perform several numerical simulations to validate theoretical results and show the influence of time delay on rumor propagation. Experimental results further illustrate that taking forceful actions such as increasing the time delay in the rumor-spreading process can control rumor propagation due to the timely effectiveness of the information.

Estimates of the Solution of the Balance Model Considering the Environmental Factor and Investment

The article explains the nonlinear balance model that considers disposal and recycling of wastes and investments. The suggested model is the equilibrium prices model in which the costs of harmful wastage disposal and recycling are considered. Besides, there are nonlinear interrelations between the branches of production, which allows us to predict the release of useful products, which is necessary for the economistsanalysts who are engaged in forecasting the manufactured products. For the model which is described by a system of differential equations, the conditions are created when the system of differential equations has only one solution. The paper defines the conditions under which this model is solvable and has a nonnegative solution, if at the same time the given values can be negative. For the model the methods of creating bilateral estimated solutions are adapted; the method of improving bilateral estimation is offered. Unlike the methods of searching the precise solution, the application of the method of bilateral estimation facilitates successful solution of tasks with big dimension of the processed models, without resorting to direct integration. The results of this article can be used in the solution of specific tasks of mathematics, economics, biology and other tasks with nonlinear interrelations.

Global Stability for a Discrete Space-Time Lotka–Volterra System with Feedback Control

In this paper, a discrete space-time Lotka–Volterra model with the periodic boundary conditions and feedback control is proposed. By means of a discrete version of comparison theorem, the boundedness of the nonnegative solution of the system is proved. By the combination of the Volterra-type and quadratic Lyapunov functions, the global asymptomatic stability of the unique positive equilibrium is investigated. Finally, numerical simulations are presented to verify the effectiveness of the main results.