Monotone Iterative Technique Research Articles

On stable iterative solutions for a class of boundary value problem of nonlinear fractional order differential equations

In this article, sufficient conditions for the existence of extremal solutions to nonlinear boundary value problem (BVP) of fractional order differential equations (FDEs) are provided. By using the method of monotone iterative technique together with upper and lower solutions, conditions for the existence and approximation of minimal and maximal solutions to the BVP under consideration are constructed. Some adequate results for different kinds of Ulam stability are investigated. Maximum error estimates for the corresponding solutions are given as well. Two examples are provided to illustrate the results.

Existence criteria of solutions for a fractional nonlocal boundary value problem and degeneration to corresponding integer-order case

In this paper, we mainly discuss the existence and uniqueness results of solutions to fractional differential equations with multi-strip boundary conditions. When the fractional order α becomes integer, the existence theorem of positive solutions can be established by a monotone iterative technique. Also, some examples are presented to illustrate the main results.

Rumor spreading of a SEIR model in complex social networks with hesitating mechanism

In the process of rumor spreading, controlling and killing rumor problem is of great importance on social networks. In this paper, we present a new SEIR (susceptible-exposed-infected-removed) rumor spreading model with hesitating mechanism. By using mean-field theory, the equilibrium of the model and the basic reproduction number R_{0} are obtained. The global stability of the rumor-free equilibrium and the permanence are proved in detail, and the global attractivity of the rumor-prevailing equilibrium is proved by using a monotone iterative technique. Furthermore, the modified model with feedback mechanism on social networks is introduced. The feedback mechanism cannot change the basic reproductive number but it can reduce the continuous level and the spread of rumor. Numerical simulations confirm the analytical results.

CONVERGENCE ANALYSIS OF ITERATIVE SCHEME AND ERROR ESTIMATION OF POSITIVE SOLUTION FOR A FRACTIONAL DIFFERENTIAL EQUATION

In this paper, we focus on the iterative scheme and error estimation of positive solutions for a class of p-Laplacian fractional order differential equation subject to Riemann-Stieltjes integral boundary condition. Under a weaker growth condition of nonlinearity, by using a monotone iterative technique, we first establish a new result on the sufficient condition for the existence of a unique positive solution to the above problem, then construct an iterative scheme which converges to the unique positive solution, and then present an error estimation and the exact convergence rate of the approximate solution.

A Class of Coupled Causal Differential Equations

In this thesis, we consider the existence of extreme solutions to a class of coupled causal differential equations. By utilizing two comparison theorems and a monotone iterative technique, we have obtained sufficient conditions under which the equations have extreme solutions. One practical example is presented.

Positive solutions for periodic boundary value problem of fractional differential equation in Banach spaces

This paper discusses the existence and uniqueness of positive solutions for a periodic boundary value problem of a fractional differential equation in an ordered Banach space E. The existence and uniqueness results of solutions for the associated linear periodic boundary value problem of the fractional differential equation are established, and the norm estimation of resolvent operator is accurately obtained. With the aid of this estimation, the existence and uniqueness results of positive solutions are obtained by using a monotone iterative technique.

Existence of solutions for several higher-order Hadamard-type fractional differential equations with integral boundary conditions on infinite interval

In this paper, we investigate the existence of solutions for several higher-order integral boundary value problems of Hadamard-type fractional differential equations on an infinite interval by using the monotone iterative technique and Mawhin’s continuation theorem. The results enrich and extend some known conclusions of Hadamard-type fractional boundary value problems. Moreover, we give two concrete examples to illustrate the theoretical results.

Global stability of endemic equilibrium for a SIQRS epidemic model on complex networks

We study the global stability of endemic equilibrium of a SIQRS epidemic model on complex networks. This model was proposed by Li et al. (2014). The local asymptotic stability of disease-free equilibrium and the permanence of the disease were proved by them. By use of a combination of Lyapunov function method and a monotone iterative technique, we prove that the endemic equilibrium is globally asymptotically stable. Numerical simulations indicate that quarantine strategy is an effective measure in preventing and controlling epidemic spreading.

Fixed-Point Theorems for Systems of Operator Equations and Their Applications to the Fractional Differential Equations

We study the existence and uniqueness of positive solution for a class of nonlinear binary operator equations systems by means of the cone theory and monotone iterative technique, under more general conditions. Also, we give the iterative sequence of the solution and the error estimation of the system. Moreover, we use this new result to study the existence and uniqueness of the solutions for fractional differential equations systems involving integral boundary value conditions in ordered Banach spaces as an application. The results obtained in this paper are more general than many previous results and complement them.

Monotone iterative technique for periodic problem involving Riemann\u2013Liouville fractional derivatives in Banach spaces

In this paper, we use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence and uniqueness of periodic solutions for a class of fractional differential equations in an ordered Banach space E. Under some monotonicity conditions and noncompactness measure conditions of nonlinearity, we obtain the existence of extremal solutions and a unique solution between lower and upper solutions.

Monotone iterative technique and positive solutions to a third-order differential equation with advanced arguments and Stieltjes integral boundary conditions

We treat the existence of monotonic iteration positive solutions to a third-order boundary value problem with advanced arguments and Stieltjes integral boundary conditions. In our work, the main tool is a monotone iterative technique. Meanwhile, at the end of this paper, an example is presented to show that this method can be well used to get the main results.

Existence and uniqueness of solutions for systems of fractional differential equations with Riemann\u2013Stieltjes integral boundary condition

In this article, we first establish an existence and uniqueness result for a class of systems of nonlinear operator equations under more general conditions by means of the cone theory and monotone iterative technique. Furthermore, the iterative sequence of the solution and the error estimation of the system are given. Then we use this new result to study the existence and uniqueness of the solution for boundary value problems of systems of fractional differential equations with a Riemann–Stieltjes integral boundary condition in real Banach spaces. The results obtained in this paper are more general than many previous results and complement them.

Monotone iterative technique for a coupled system of nonlinear Hadamard fractional differential equations

In this paper, we investigate the extremal solutions for a coupled system of nonlinear Hadamard fractional differential equations with Cauchy initial value conditions. By using the comparison principle and the monotone iterative technique combined with the method of upper and lower solutions, we obtain the existence and iterative methods of extremal solution to the system. Finally, an example with numerical simulation is given to show the effectiveness of our main results.

Positive solutions to a third-order boundary value problem with Riemann Stieltjes integral boundary conditions

This paper investigates the existence of concave positive solutions and establishes corresponding iterative schemes to a third-order boundary value problem with Riemann Stieltjes integral boundary conditions. The main tool is a monotone iterative technique. Meanwhile, an example is worked out to demonstrate the main results.

Iterative techniques for the initial value problem for Caputo fractional differential equations with non-instantaneous impulses

Two types of algorithms for constructing monotone successive approximations for solutions to initial value problems for a scalar nonlinear Caputo fractional differential equation with non-instantaneous impulses are given. The impulses start abruptly at some points and their action continue on given finite intervals. Both algorithms are based on the application of lower and upper solutions to the problem. The first one is a generalization of the monotone iterative technique and it requires an application of the Mittag-Leffler function with one and two parameters. The second one is easier from a practical point of view and is applicable when the right hand sides of the equation are monotone. We prove that the functional sequences are convergent and their limits are minimal and maximal solutions of the problem. An example is given to illustrate the results.

Existence of extremal solutions to interval-valued delay fractional differential equations via monotone iterative technique

Existence of extremal solutions to interval-valued delay fractional differential equations via monotone iterative technique

Positive solutions for nonlinear fractional differential equation with nonlocal boundary conditions

In this paper, we study the boundary value problem of a class of fractional differential equations involving the Riemann-Liouville fractional derivative with nonlocal integral boundary conditions. To establish the existence results for the given problems, we use the properties of the Green’s function and the monotone iteration technique, one shows the existence of positive solutions and constructs two successively iterative sequences to approximate the solutions. The results are illustrated with an example.

The Construction of Iterative Solutions to A kind of BVP

We focus on the positive iterative solution to the problem of a kind of BVP with the p-Laplace operator. We will study its positive solutions. The specific methods used in our work is the combination of fixed point theorem and monotone iterative technique. What’s more, at the end of this work, an example is presented to show that this method can be well used to get the main results.

Pseudo Symmetric Solutions to Two Classes of BVPs

We focus on the positive iterative solution to the problem of two kinds of BVPs with p-Laplace operator. We will study their pseudo symmetric positive solutions. The specific methods used in our work is the combination of fixed point theorem and monotone iterative technique. What’s more, at the end of this work, an example is presented to show that this method can be well used to get the main results.

Extremal solutions for p-Laplacian fractional differential systems involving the Riemann-Liouville integral boundary conditions

In this paper, we investigate the existence of extremal solutions for fractional differential systems involving the p-Laplacian operator and Riemann-Liouville integral boundary conditions. We derive our results based on the monotone iterative technique, combined with the method of upper and lower solutions. An example is added to illustrate the main result.