Dynamics of an Impulsive Stochastic Predator–Prey System with the Beddington–DeAngelis Functional Response

Taking impulsive effects into account, an impulsive stochastic predator–prey system with the Beddington–DeAngelis functional response is proposed in this paper. First, the impulsive system is transformed into an equivalent system without pulses. Then, by constructing suitable functionals and applying the extreme-value theory of quadratic functions, sufficient conditions on the existence of periodic Markovian processes are provided. The uniform continuity and global attractivity of solutions are also investigated. Additionally, we investigate the extinction and permanence in the mean of all species with the help of comparison methods and inequality techniques. Sufficient conditions on the existence and ergodicity of the stationary distribution of solutions for the autonomous and non-impulsive case are given. Finally, numerical simulations are performed to illustrate the main results.

An explicit extragradient algorithm for solving variational inequality problem with application

In this paper, we introduce a new explicit extragradient algorithm for solving Variational Inequality Problem (VIP) in Banach spaces. The proposed algorithm uses a linesearch method whose inner iterations are independent of any projection onto feasible sets. Under standard and mild assumption of pseudomonotonicity and uniform continuity of the VIP associated operator, we establish the strong convergence of the scheme. Further, we apply our algorithm to find an equilibrium point with minimal environmental cost for a model in electricity production. Finally, a numerical result is presented to illustrate the given model. Our result extends, improves and unifies other related results in the literature.

METRIC SPACE AND ITS APPLICATIONS

This paper gives a short review of some basics of metric spaces, the concept of continuity of these metric spaces, with a comparison of it with uniform continuity and convergence. In addition to this, it includes a discussion of some past work of researchers on metric space, its generalization, and useful results made by them with applications using these spaces.

Preliminaries

In addition to giving a very brief reminder of set notation and basic set operations, this chapter provides a brief refresher on basic mathematical concepts. The natural, rational and real number systems are taken for granted. However, it does develop at length the Cauchy criterion and its equivalence to the completeness of the real line, and the Bolzano-Weierstrass theorem, as well as the complex number field, including its completeness. Embryonic manifestations of completeness and compactness can be seen in this chapter. Examples include the nested interval theorem and the uniform continuity of continuous functions on compact intervals, and the proof of the Heine-Borel theorem in chapter 4 is squarely based on the Bolzano-Weierstrass property of bounded sets.