Monotone Iterative Method Research Articles

Extremal system of solutions for a coupled system of nonlinear fractional differential equations by monotone iterative method

Extremal system of solutions for a coupled system of nonlinear fractional differential equations by monotone iterative method

Monotone iterative method for boundary value problems of fuzzy differential equations1

This paper provides some existence results about first-order fuzzy differential equation with two-point boundary value condition. Firstly, we study a class of linear fuzzy differential equation and some necessary conditions for the existence of solutions are provided. Based on these conditions and the upper and lower solutions method, we obtain some existence results about first-order nonlinear differential equations with the same boundary value condition.

The Existence of extremal solutions to nonlinear fractional integro-differential equations with advanced arguments

This paper deals with the existence of extremal solutions for nonlinear fractional integro-differential equations with advanced arguments. Our analysis rely on monotone iterative method based on upper and lower solutions. Also, we give an illustrative example in order to indicate the validity of our assumptions.

Numerical solving nonlinear integro-parabolic equations by the monotone weighted average method

The paper deals with numerical solving nonlinear integro-parabolic problems of Volterra type based on the weighted average method. A monotone iterative method is presented. Construction of initial upper and lower solutions is given. Existence and uniqueness of a solution to the nonlinear difference scheme are established. An analysis of convergence rates of the monotone iterative method is given. Numerical experiments are presented.

Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis

We study an atherosclerosis model described by a reaction-diffusion system of three equations, in one dimension, with homogeneous Neumann boundary conditions. The method of upper and lower solutions and its associated monotone iteration (the monotone iterative method) are used to establish existence, uniqueness and boundedness of global solutions for the problem. Upper and lower solutions are derived for the corresponding steady-state problem. Moreover, solutions of Cauchy problems defined for time-dependent system are presented as alternatives upper and lower solutions. The stability of constant steady-state solutions and the asymptotic behavior of the time-dependent solutions are studied.

Monotone iterative method for nonlinear fractional q-difference equations with integral boundary conditions

This paper investigates the existence of positive solutions for a class of nonlinear fractional q-difference equations with integral boundary conditions. By applying monotone iterative method and some inequalities associated with the Green’s function, the existence results of positive solutions and two iterative schemes approximating the solutions are established. An explicit example is given to illustrate the main result.

Generalized monotone iterative method for integral boundary value problems with causal operators

Generalized monotone iterative method for integral boundary value problems with causal operators

Successive iteration and positive extremal solutions for nonlinear impulsive q k -difference equations

This paper investigates the existence of positive extremal solutions for nonlinear impulsive \(q_{k}\)-difference equations via a monotone iterative method. The main result is well illustrated with the aid of an example.

Monotone iterates for solving nonlinear integro-parabolic equations of Volterra type

The paper deals with numerical solving of nonlinear integro-parabolic problems based on the method of upper and lower solutions. A monotone iterative method is constructed. Existence and uniqueness of a solution to the nonlinear difference scheme are established. An analysis of convergence rates of the monotone iterative method is given. Construction of initial upper and lower solutions is discussed. Numerical experiments are presented.

Existence and uniqueness of positive mild solutions for nonlocal evolution equations

This paper deals with the existence and uniqueness of positive mild solutions for a class of semilinear evolution equations with nonlocal initial conditions on infinite interval. The existence and uniqueness of mild solution for the associated linear evolution equation nonlocal problem is established, and the spectral radius of resolvent operator is accurately estimated. With the aid of the estimation, the existence and uniqueness of positive mild solutions for nonlinear evolution equation nonlocal problem are obtained by using the monotone iterative method without the assumption of lower and upper solutions. The theorems proved in this paper improve and extend some related results in ordinary differential equations and partial differential equations. An example is also given to illustrate that our results are valuable.

Stability for Monostable Wave Fronts of Delayed Lattice Differential Equations

This paper is concerned with the stability of traveling wave fronts for delayed monostable lattice differential equations. We first investigate the existence non-existence and uniqueness of traveling wave fronts by using the technique of monotone iteration method and Ikehara theorem. Then we apply the contraction principle to obtain the existence, uniqueness, and positivity of solutions for the Cauchy problem. Next, we study the stability of a traveling wave front by using comparison theorems for the Cauchy problem and initial-boundary value problem of the lattice differential equations, respectively. We show that any solution of the Cauchy problem converges exponentially to a traveling wave front provided that the initial function is a perturbation of the traveling wave front, whose asymptotic behaviour at \(-\infty \) satisfying some restrictions. Our results can apply to many lattice differential equations, for examples, the delayed cellular neural networks model and discrete diffusive Nicholson’s blowflies equation.

Monotone iterative ADI method for semilinear parabolic problems

The paper deals with numerical solving semilinear parabolic problems based on a nonlinear alternating direction implicit (ADI) scheme. The convergence of the nonlinear ADI scheme to the continuous solution is proved. The existence and uniqueness of a solution of the nonlinear ADI scheme are established. A monotone iterative ADI method is constructed. An analysis of convergence of the monotone iterative ADI method to the solution of the nonlinear ADI scheme on the whole time interval is given. Numerical experiments are presented.

Existence of Extremal Solutions for a Nonlinear Fractional q-Difference System

In this paper, we study the boundary value problem of a fractional q-difference system with nonlocal integral boundary conditions involving the fractional q-derivatives of the Riemann–Liouville type. Using the properties of the Green function, and monotone iterative method, the extremal solutions were obtained. Finally, an example is presented to illustrate our main results.

A reaction–diffusion model of the Darien Gap Sterile Insect Release Method

The Sterile Insect Release Method (SIRM) is used as a biological control for invasive insect species. SIRM involves introducing large quantities of sterilized male insects into a wild population of invading insects. A fertile/sterile mating produces offspring that are not viable and the wild insect population will eventually be eradicated. A U.S. government program maintains a permanent sterile fly barrier zone in the Darien Gap between Panama and Columbia to control the screwworm fly (Cochliomyia Hominivorax), an insect that feeds off of living tissue in mammals and has devastating effects on livestock. This barrier zone is maintained by regular releases of massive quantities of sterilized male screwworm flies from aircraft. We analyze a reaction–diffusion model of the Darien Gap barrier zone. Simulations of the model equations yield two types of spatially inhomogeneous steady-state solutions representing a sterile fly barrier that does not prevent invasion and a barrier that does prevent invasion. We investigate steady-state solutions using both phase plane methods and monotone iteration methods and describe how barrier width and the sterile fly release rate affects steady-state behavior.

Generalized monotone iterative method for nonlinear boundary value problems with causal operators

This paper discusses nonlinear boundary value problems for causal differential equations where the right-hand side is the sum of two monotone functions. We develop the monotone iterative technique and establish the existence results of the extremal solutions. The results obtained include several special cases and extend previous results; two examples satisfying the assumptions are also presented. MSC: 34B15; 39B12

Positive solutions of a Lotka–Volterra competition model with cross-diffusion

This paper concerns a Lotka–Volterra competition reaction–diffusion system with nonlinear diffusion effects. We first briefly discuss the stability of trivial solutions by spectrum analysis. Based on the boundedness of the solutions, the existence of the positive steady state solution is also investigated by the monotone iteration method.

SUCCESSIVE ITERATIONS FOR POSITIVE EXTREMAL SOLUTIONS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS ON A HALF-LINE

AbstractIn this paper, positive solutions of fractional differential equations with nonlinear terms depending on lower-order derivatives on a half-line are investigated. The positive extremal solutions and iterative schemes for approximating them are obtained by applying a monotone iterative method. An example is presented to illustrate the main results.

Монотонний ітераційний метод для розв’язування задач комбінаторної оптимізації ігрового типу на переставленнях

Монотонний ітераційний метод для розв’язування задач комбінаторної оптимізації ігрового типу на переставленнях

Time periodic solutions of the diffusive Nicholson blowflies equation with delay

This paper is concerned with the Nicholson blowflies equation with nonlinear diffusion and time delay subject to the homogeneous Dirichlet boundary condition in a bounded domain. We establish the existence of nontrivial periodic solutions of the time-periodic problem under general conditions by constructing a coupled upper–lower solution pair and by applying the Schauder fixed point theorem. The attractivity of the periodic solutions is also discussed by using the monotone iteration method.

Inexact block monotone methods for solving nonlinear elliptic problems

The paper deals with numerical solving of semilinear elliptic problems based on the method of upper and lower solutions. Inexact block monotone iterative methods are constructed, where monotone linear systems are solved by the block Jacobi or block Gauss–Seidel methods only approximately. The inexact block monotone methods combine the quadratic monotone iterative method at outer iterations and the block iterative methods at inner iterations, and possess global monotone convergence. Results of numerical experiments, implemented in the framework of an inexact Newton method, are presented, where iteration counts and CPU times of the inexact block monotone methods are compared with the block monotone iterative methods, whose convergence rate is linear.