Vol 29, No 148 (2024)

The paper proposes a mathematical model based on a continuous dynamic system that formalizes the interaction of populations of a greenhouse agricultural insect pest and an entomophage acting as an agent of environmental control of the pest population in the framework of the concept of integrated pest management paradigm. The model parameters are identified based on experimental data obtained in laboratory conditions for the greenhouse system “cucumber — spider mite — acariphage mite”. For the proposed system, an impulse control problem that corresponds to the implementation of the pest population control using an entomophage is formulated, and the issues of the existence of solutions and their continuous dependence on control are investigated. Based on the studied control system, the issues of constructing cost-effective control strategies that allow implementing the mathematical software for automated pest control systems in greenhouses are discussed.

We consider the simplest extrapolation procedure, specifically doubling the step, intended for acceleration of convergence of Newton-type methods to singular solutions of smooth nonlinear equations. We demonstrate that the acceleration effect of this procedure can be different for different Newton-type methods. For linear-quadratic equations we provide theoretical results yielding quantitative estimates of the potential effect of extrapolation for the Newton method, for the Levenberg–Marquardt method, and for the recently proposed LP-Newton method, in some sense explaining the observed difference. Theoretical analysis relies on interpretation of these methods as a perturbed Newton method with the appropriate estimates of perturbations, as well as on sharp results yielding a quantitative characterization of a step of such perturbed method, and its local convergence at a linear rate to singular solutions satisfying the 2-regularity condition in a direction from the null space of the first derivative. Furthermore, we perform numerical experiments with globalized versions of the algorithms in question, equipped with choosing the stepsize parameter, on two sets of test problems. Experimental observations confirm the theoretical results, and also demonstrate that in cases when the equation contains nonlinear and nonquadratic terms, the effect of extrapolation is evened out.

A superdiffusion equation with Riesz fractional derivatives with respect to space with several delay variables is considered. The problem is discretized. For this purpose, an analog of the Crank–Nicolson difference method with piecewise linear interpolation to account for the effect of variable delay and with extrapolation by continuation is constructed in time so that the implicitness of the method becomes finite-dimensional. An analog of a compact scheme with a special replacement of Riesz fractional derivatives by fractional central differences is constructed in space. As a result, the method is reduced to solving a system of linear algebraic equations with symmetric and positive-definite main matrix at each time step. The order of smallness with respect to the discretization time-steps $\Delta$ and space-steps h of the residual of the method without interpolation and with interpolation is studied; it is equal to $O({\Delta }^2+h^4)$. The main result consists in proving that the method converges with the order $O{\Delta }^2+h^4$ in the energy and compact norm of the layered error vector. The results of test examples for superdiffusion equations with constant and variable delay are presented. The computable orders of convergence for each discretization step in the examples turned out to be close to the theoretically obtained orders of convergence for the corresponding discretization steps.

In this paper, we study the question of conditions for the existence and uniqueness of a fixed point of a mapping over a complete metric space. We first discuss the concepts of F-contraction and $F^{*}$-contraction in fixed point theory. These concepts, developed respectively by Wardowski and Piri with Kumam, have catalyzed significant research in various metric spaces. We then propose a generalization of these concepts, $\rho -F$-contraction and $\rho -F^{*}$-contraction, and demonstrate its effectiveness in ensuring the existence and uniqueness of fixed points. This new approach provides greater flexibility by including a function $\rho$ that modulates the contraction, extending the applicability of F- and $F^{*}$-contractions. We conclude the paper with an example of a mapping that is a $\rho -F$-contraction and a $\rho -F^{*}$-contraction, respectively, and has a unique fixed point. However, this mapping does not satisfy the conditions of Wardowski and the conditions of Piri and Kumam.