Monotone Iterative Scheme Research Articles

Existence and uniqueness for the evolutionary impulse control problem using an asynchronous algorithms

Impulse control problems pose significant challenges in decision making under uncertainty, particularly in scenarios where actions must be taken at irregular intervals or in response to abrupt changes in the environment. Ensuring both the existence and uniqueness of optimal control policies in such settings is paramount for practical implementation and theoretical understanding. In this study, we investigate the existence and uniqueness of solutions for impulse control problems utilizing asynchronous algorithms. This study presents a novel demonstration concerning the uniqueness and existence of Partially Quasi-Variational Inequalities (PQVIs). Our approach unfolds through a four-step methodology, leveraging distinctive features of a discrete iterative technique. This methodology integrates semi-implicit techniques with respect to the variable t and employs spatial approximation via finite element methods. We establish a pivotal connection between a fixed-point mapping and the discrete EQVIs system, utilizing it to formulate a discrete algorithm featuring a semi-implicit time scheme. Furthermore, we introduce a monotone iterative scheme inspired by Bensoussan’s algorithm and rigorously confirm several of its properties through mathematical proofs. This work contributes to advancing the understanding and application of iterative techniques in solving PQVIs, offering insights into their uniqueness and existence.

Stability of steady‐state solutions of a class of Keller–Segel models with mixed boundary conditions

In this paper, we investigate the existence and stability of non‐trivial steady‐state solutions of a class of chemotaxis models with zero‐flux boundary conditions and Dirichlet boundary conditions on a one‐dimensional bounded interval. By using upper–lower solution and the monotone iteration scheme method, we get the existence of the steady‐state solution of the chemotaxis model. Moreover, by adopting the “inverse derivative” technique and the weighted energy method, we prove the stability of the steady‐state solution of this chemotaxis model.

On a coupled system under coupled integral boundary conditions involving non-singular differential operator

<abstract><p>In this work, a coupled system under coupled integral boundary conditions with Caputo-Fabrizio derivative (CFD) is considered. We intend to derive some necessary and sufficient results for the existence of at least one solution. In addition, we extend our analysis further to develop a monotone iterative scheme coupled with the upper and lower solution method to compute extremal solutions. Therefore, in this regard, Perov's fixed point theorem is applied to study the existing criteria for the solution. Also, results related to at least one solution are derived by using Schauder's fixed point theorem. Finally, we use a monotone iterative procedure together with upper and lower solution methods to study extremal solutions. Graphical presentations of upper and lower solutions are provided for some examples to illustrate our results.</p></abstract>

A time-periodic reaction-diffusion population model of a species subjected to harvesting on the boundary of a protected habitat

The objective in this paper is to determine analytically a maximally sustainable pro-rata density-dependent harvest rate of a hypothetical biological species on the spatial boundary of a protected habitat (i.e. no harvesting of the species is allowed in the interior of the spatial domain). This is achieved by analysing an initial-boundary value model involving a reaction-diffusion equation in which the reaction term is a concave function of the population density, depends periodically on time, and varies spatially. The model is equipped with Robin boundary conditions modelling a continuum of potential pro-rata density-dependent harvest rates on the spatial boundary. A set of necessary and sufficient criteria for the existence of a unique positive periodic attractor of solutions to the model is established by employing the theory of comparison principles and monotone iteration schemes. Thereafter, a long-time asymptotic analysis of the population density is undertaken by invoking classical results from the theory of eigenproblems. In this way, the aforementioned necessary and sufficient conditions are rendered more practical in terms of an upper bound on the maximal allowable pro-rata density-dependent harvest rate. Moreover, important properties of the time-periodic attractor of solutions are established to guarantee the existence of a density pro-rata harvest rate which maximises the total harvest per unit time at equilibrium.

Stratified Radiative Transfer in a Fluid and Numerical Applications to Earth Science

New mathematical results are given for the radiative transfer equations alone and coupled with the temperature equation of a fluid: existence, uniqueness, a maximum principle, and a convergent monotone iterative scheme. Numerical tests for Earth's atmosphere and the heating of a pool by the Sun are included.

An accelerated technique for the coupled system of reaction-diffusion-transport equations arising from catalytic converters

A study on accelerated monotone iterative methods for two systems of non-linear partial differential equations arising from catalytic converter models is presented. Two mathematical models are considered in this work. The first mathematical model consists of a semilinear parabolic partial differential equation and two integral equations. In contrast, the second mathematical model consists of a semilinear parabolic partial differential equation and an integral equation. The proposed modified monotone iterative methods converge to the unique solution of these mathematical models faster than the existing monotone iterative schemes available in the literature. Interesting theoretical justification is provided for the accelerated convergence and the monotone behaviour of the proposed iterative methods. Numerical simulation results support the theoretical claims.

On scale functions for Lévy processes with negative phase-type jumps

We provide a novel expression of the scale function for a Lévy process with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and it is shown that the error decays exponentially fast. Our numerical examples suggest that this algorithm allows us to employ phase-type distributions with hundreds of phases, which is problematic when using the known formula for the scale function in terms of roots. Extensions to other distributions, such as matrix-exponential and infinite-dimensional phase-type, can be anticipated.

An Integral Boundary Value Problem of Fractional Differential Equations with a Sign-Changed Parameter in Banach Spaces

This paper is to investigate the existence and uniqueness of solutions for an integral boundary value problem of new fractional differential equations with a sign-changed parameter in Banach spaces. The main used approach is a recent fixed point theorem of increasing Ψ − h , r -concave operators defined on ordered sets. In addition, we can present a monotone iterative scheme to approximate the unique solution. In the end, two simple examples are given to illustrate our main results.

A Diffusive One-Prey and Two-Cooperative-Predators Model with C–M Functional Response

This paper is dedicated to a study of a diffusive one-prey and two-cooperative-predators model with C–M functional response subject to Dirichlet boundary conditions. We first discuss the existence of positive steady states by the fixed point index theory and the degree theory. In the meantime, we analyze the uniqueness and stability of coexistence states under conditions that one predator’s consumer rate is small and the effect of interference intensity of another predator is large. Then, steady-state bifurcations from two strong semi-trivial steady states (provided that they uniquely exist under some conditions) and from one weak semi-trivial steady state are investigated in detail by the Crandall–Rabinowitz bifurcation theorem, the technique of space decomposition and the implicit function theorem. In addition, we study the asymptotic behaviors including the extinction and permanence of the time-dependent system by the comparison principle, upper-lower solution method and monotone iteration scheme. Finally, numerical simulations are done not only to validate the theoretical conclusions, but also to further clarify the impacts of parameters on the three species.

Optimal Dividend Payment in De Finetti Models: Survey and New Results and Strategies

We consider optimal dividend payment under the constraint that the with-dividend ruin probability does not exceed a given value α. This is done in most simple discrete De Finetti models. We characterize the value function V(s,α) for initial surplus s of this problem, characterize the corresponding optimal dividend strategies, and present an algorithm for its computation. In an earlier solution to this problem, a Hamilton-Jacobi-Bellman equation for V(s,α) can be found which leads to its representation as the limit of a monotone iteration scheme. However, this scheme is too complex for numerical computations. Here, we introduce the class of two-barrier dividend strategies with the following property: when dividends are paid above a barrier B, i.e., a dividend of size 1 is paid when reaching B+1 from B, then we repeat this dividend payment until reaching a limit L for some 0≤L≤B. For these strategies we obtain explicit formulas for ruin probabilities and present values of dividend payments, as well as simplifications of the above iteration scheme. The results of numerical experiments show that the values V(s,α) obtained in earlier work can be improved, they are suboptimal.

Wave propagation for a cooperative model with nonlocal dispersal under worsening habitats

In this paper, we study a Lotka–Volterra cooperative system with nonlocal dispersal under worsening habitats. By constructing appropriate vector super-/subsolutions combined with the monotone iteration scheme, we obtain the existence of bounded and positive forced waves connecting zero equilibrium to the coexistence state of the limiting system corresponding to the most favorable resource with the wave speed at which the habitat is worsening. Compared with the existence result of traveling wave to the homogeneous system, we find that the wave phenomenon is more likely to occur in such shifting environment. Here, we remove the common assumption that the dispersal kernels are compactly supported and symmetric. Further, we investigate the tail behavior of the forced waves, especially for the rate of convergence to the extinction state, and derive the long-time dynamics of two species. Our result shows that weak interspecies cooperation and nonlocal diffusion pattern cannot prevent species from disappearing eventually under such a worsening habitat.

Existence and uniqueness of forced waves in a delayed reaction–diffusion equation in a shifting environment

The existence and uniqueness of forced waves in a general reaction–diffusion equation with time delay under climate change is concerned in this paper. By using upper and lower solutions method, monotone iteration scheme combined with the strong maximum principle, we show that there exists a nondecreasing and unique wave front with the speed consistent with the habitat shifting speed. Our results indicate the propagation of both the leading and trailing edges of the comoving population wavefront lag behind the climate envelope, which drives the species to extinction. Three examples and their corresponding numerical simulations are also given to illustrate the universality of analytical conclusions.

Existence and uniqueness of periodic solutions for a system of differential equations via operator methods

This article investigates the existence and uniqueness of periodic solutions for a new system of differential equations. By employing fixed point theorems for increasing φ-(h,tau )-concave operators, we establish the existence of unique periodic solution for our differential system and then give a monotone iterative scheme to approximate the unique periodic solution. Some examples are presented in the end to illustrate the validity of our main results.

Monotone iterative schemes for positive solutions of a fractional differential system with integral boundary conditions on an infinite interval

In this paper, using the monotone iterative technique and the Banach contraction mapping principle, we study a class of fractional differential system with integral boundary on an infinite interval. Some explicit monotone iterative schemes for approximating the extreme positive solutions and the unique positive solution are constructed.

Solutions to fractional differential equations involving integral boundary conditions

ABSTRACT In this article, we discuss the existence and uniqueness of solutions to a class of integral boundary value problem of fractional differential equations. Different fractional order derivations are included in the nonlinear term and boundary conditions. The unique solution is derived by a new fixed point theorem of increasing -concave operators defined on ordered sets. Further, we construct a monotone iterative scheme to approximate the unique solution. An example is given to illustrate the validity of our main results.

Monotone Iterative Technique and Ulam-Hyers Stability Analysis for Nonlinear Fractional Order Differential Equations with Integral Boundary Value Conditions

In this manuscript, the monotone iterative scheme has been extended to the nature of solution to boundary value problem of fractional differential equation that consist integral boundary conditions. In this concern, some sufficient conditions are developed in this manuscript. On the base of sufficient conditions, the monotone iterative scheme combined with lower and upper solution method for the existence, uniqueness, error estimates and various view plots of the extremal solutions to boundary value problem of nonlinear fractional differential equations have been studied. The obtain results have clarified the nature of the extremal solutions. Further, the Ulam--Hyers and Ulam--Hyers--Rassias stability have been investigated for the considered problem. Two illustrative examples of the BVP of the nonlinear fractional differential equations have been provided to justify our contribution.

Iterative methods for a fractional-order Volterra population model

We prove an existence and uniqueness theorem for a fractional-order Volterra population model via an efficient monotone iterative scheme. By coupling a spectral method with the proposed iterative scheme, the fractional-order integrodifferential equation is solved numerically. The numerical experiments show that the proposed iterative scheme is more efficient than an existing iterative scheme in the literature, the convergence of which is very sensitive to various parameters, including the fractional order of the derivative. The spectral method based on our proposed iterative scheme shows greater flexibility with respect to various parameters. Sufficient conditions are provided to select the initial guess that ensures the quadratic convergence of the quasilinearization scheme.

Forced waves and their asymptotics in a Lotka–Volterra cooperative model under climate change

The existence and asymptotics of forced traveling wave front in a Lotka–Volterra cooperative system under climate change are concerned in this paper. By constructing appropriate upper and lower solutions combined with the monotone iteration scheme, we show that for any given positive speed of the shifting habitat edge, there exists a nondecreasing wave front with the speed consistent with the habitat shifting speed, which indicates that no matter how moderate of the habitat shifts, two species will still be driven to extinction as the mutualistic effects exist but weak. Numerical simulations are also given to illustrate our results.

Some New Existence Results for a Class of Four Point Nonlinear Boundary Value Problems

In this paper we consider the following class of four point boundary value problems—y"(x) = f (x, y), 0 less than x lessthan 1, y'(0) = 0, y(1) = 1y(1) + 2)7(2)’where 1, 2  0 lesstahn 1, 2 less than 1, and f (x, y), is continuous in one sided Lipschitz in y. We propose a monotone iterative scheme and show that under some sufficient conditions this scheme generates sequences which converges uniformly to solution of the nonlinear multipint boundary value problem.

Asymptotic Periodicity in the Fecally-Orally Epidemic Model in a Heterogeneous Environment

To understand the influence of seasonal periodicity and environmental heterogeneity on the transmission dynamics of an infectious disease, we consider asymptotic periodicity in the fecally-orally epidemic model in a heterogeneous environment. By using the next generation operator and the related eigenvalue problems, the basic reproduction number is introduced and shows that it plays an important role in the existence and non-existence of a positive T-periodic solution. The sufficient conditions for the existence and non-existence of a positive T-periodic solution are provided by applying upper and lower solutions method. Our results showed that the fecally-orally epidemic model in a heterogeneous environment admits at least one positive T-periodic solution if the basic reproduction number is greater than one, while no T-periodic solution exists if the basic reproduction number is less than or equal to one. By means of monotone iterative schemes, we construct the true positive solutions. The asymptotic behavior of periodic solutions is presented. To illustrate our theoretical results, some numerical simulations are given. The paper ends with some conclusions and future considerations.