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We study one-term operator L acting in the space $H_{1}=L_{2}([0,\infty );H)$ generated by the operator-differential expression $\mathcal{L}=-\frac{d}{dx} ( P(x)\frac{d}{dx})$ and the boundary condition $y(0)=0$ . We evaluate the asymptotic number of eigenvalues of the operator L under certain conditions.

A diffusive model of pioneer and climax species interaction is considered. We perform a detailed Hopf bifurcation analysis to the model, and derive conditions for determining the bifurcation direction and the stability of the bifurcating periodic solutions.

In this paper we propose a stable finite difference method to solve the fractional reaction–diffusion systems in a two-dimensional domain. The space discretization is implemented by the weighted shifted Grünwald difference (WSGD) which results in a stiff system of nonlinear ordinary differential equations (ODEs). This system of ordinary differential equations is solved by an efficient compact implicit integration factor (cIIF) method. The stability of the second order cIIF scheme is proved in the discrete $L^{2}$ -norm. We also prove the second-order convergence of the proposed scheme. Numerical examples are given to demonstrate the accuracy, efficiency, and robustness of the method.

The paper is devoted to the study of the measure-driven differential inclusions , for arbitrary finite Borel measure μ. This type of results allows one to treat in a similar manner differential and difference inclusions, as well as impulsive problems and therefore to study the evolution of hybrid systems with very complex (including Zeno) behavior. Our method is based on viewing the Borel measures as Lebesgue-Stieltjes measures. We thus obtain, under very general assumptions, the existence of regulated or bounded variation solutions of the considered problem and we indicate some advantages of our approach. MSC:49N25, 34A60, 93C30, 49J53, 37N35, 34A37.

In recent years, chronic diseases, such as high blood pressure, diabetes, heart attack, and cancer, have increased around the world. Obesity is a common factor that makes individuals susceptible to these diseases. One reason for excessive weight gain is the frequent consumption of fast food. This study examined the impact that fast food has on obesity by analyzing the influence of peer pressure on fast food consumption and the role that exercise plays in weight gain. Two mathematical models were proposed to study each factor separately. The models demonstrated the dynamics of individuals transitioning from one weight class to another according to their body mass index (BMI). The models were governed by systems of nonlinear ordinary differential equations. A qualitative approach for analyzing the systems was utilized, and the numerical simulations supported the qualitative results. The results of this study may contribute to the development of strategies for controlling obesity. In particular, the models showed the importance of resisting peer pressure that drives individuals to eat fast food meals, as well as maximizing the role that quitters can play in convincing obese individuals to stop consuming fast food meals. In addition to the two previous strategies, the research revealed that physical activity has a significant role in reducing weight.

In this paper, stabilization is studied for a two-dimensional delta operator system with time-varying delays and actuator saturation. Both lower and upper bounds of the time-varying delays are considered. An estimate of the domain of attraction for the two-dimensional delta operator system is introduced to analyze stability of the closed-loop system. A state feedback controller is designed via a Lyapunov–Krasovskii functional approach for the two-dimensional delta operator system with time-varying delays and actuator saturation. Two numerical examples are given to illustrate the effectiveness and advantages of the developed techniques.

In this paper, some improved delay-dependent exponential stability criteria of the neutral delayed differential equation are expressed in terms of linear matrix inequalities (LMIs). Compared with some existing ones, our results are derived without the use of the mode transformation method and the bounding technique. Three numerical examples are provided to show the effectiveness of the proposed method and less conservatism of the obtained results.

We use the derivative sampling theorem (Hermite interpolations) to compute eigenvalues of a discontinuous regular Dirac systems with transmission conditions at the point of discontinuity numerically. We closely follow the analysis derived by Levitan and Sargsjan (1975) to establish the needed relations. We use recently derived estimates for the truncation and amplitude errors to compute error bounds. Numerical examples, illustrations and comparisons with the sinc methods are exhibited. Mathematical Subject Classification 2010: 34L16; 94A20; 65L15.

We investigate the uniqueness of solutions for fully anti-periodic fractional boundary value problems of order with nonlinearity depending on lower-order fractional derivatives. Our results are based on some standard fixed point theorems. The paper concludes with illustrative examples. MSC:34A12, 34A40.