Vol. 7, No. 3 (2018): Regular Issue

We consider a general first-order scalar difference equation with and without Allee effect. The model without Allee effect represents asexual reproduction of a species while the model including Allee effect represents sexual reproduction. We analyze global stabilities of both models analytically and compare the results obtained. Numerical simulations are included to support the analytical results. We conclude that Allee effect has a destabilizing effect on the global stability of the model. This result is different from the local stability behaviour of the same fixed point of the model.

We prove global existence and uniqueness of strong solutions for the Schrödinger-Poisson system in the repulsive Coulomb case with relativistic, magnetic kinetic energy.

In this work, Lyapunov functions combining Matrix inequalities or Matrix equations are developed to analyze the asymptotic stability of Caputo fractional singular systems with multiple time-varying delay. Integer-order derivatives of the Lyapunov functions are used to derive asymptotic stability criteria. The main advantage of applying stability criteria is that no need of solving roots of transcendental equations. Some examples are provided to explain the effectiveness of the proposed criteria.

In this paper, we establish four types of Ulam stability, namely Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias and generalized Ulam-Hyers-Rassias stability for differential equation of fractional order.

In this paper, we take into account two coupled Van der Pol equations with impacts. The main novelty is that the degenerated system for the model admits an oscillation with zero impact velocity. To prove presence of oscillations, beside the perturbation method, the newly developed linearization for dynamics with grazing has been applied. As different from the theoretical results such as Nordmark mapping and Zero time discontinuity mapping, the grazing is examined through another method of discontinuous dynamics, which diminishes the role of mappings in the analysis. The rich diversity of changes in the dynamics is observed under regular perturbations, since of the grazing discontinuity. By means of the simulation results, the analytical studies are visualized.

In this paper, we investigate the controllability of impulsive fractional linear time invariant systems with state delay and distributed delays in control. By using the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function , a new set of ufficient conditions are obtained for the considered system to be controllable. Finally, two numerical examples are given to demonstrate the validity of the obtained theoretical results.

The paper considers impulsive systems with singularities. The main novelty of the present research is that impulses (impulsive functions) are singular. This is beside singularity of differential equations. The most general Tikhonov theorem for the impulsive systems s obtained. Illustrative examples with simulations are given to support the theoretical results.

This paper investigates the existence and approximate controllability results for a class of impulsive functional integro-differential evolution systems with nonlocal conditions via resolvent operators in Banach spaces. By making utilization of Banach contraction principle and Schaefer’s fixed point theorem along with resolvent operators and semigroup theory, we build up the desired results. As an application, we also consider an impulsive partial functional integro-differential equations.

The article reviews some nonlinear generalizations of the perturbation theory performed by various authors (including some co-authors of this article) in various years, and can be useful in analyzing complex systems and inverse problems. There is considered the roblem of widening the range of perturbation theory applicability with simultaneous reducing the complexity of computational algorithm. The consideration is based on the duality principle, including adjoint (in the Lagrange sense) operators and importance unctions. The article contains an exposition of perturbation theory of higher orders, variational methods of higher orders, symmetry transformation and some other question. Special attention is paid to such a promising approach as multi-reference methods. It is shown that this approach extends the range of applicability of perturbation theory and saves calculations time. To demonstrate details of the method, some examples of transport problems from reactor engineering and astrophysics are considered.