Monotone Iterative Technique Research Articles

Monotone iterative technique for impulsive fractional evolution equations with noncompact semigroup

This paper deals with the existence of mild solutions for a class of Caputo fractional impulsive evolution equation with nonlocal condition and noncompact semigroup. By using a monotone iterative technique in the presence of coupled lower and upper L-quasi-solutions and using Sadovskii’s fixed point theorem, some existence theorems are obtained. The discussion is based on operator semigroup theory.

Existence of periodic solutions for impulsive evolution equations in ordered Banach spaces

In this paper, we use the perturbation method and the mixed monotone iterative technique to discuss the existence of periodic solutions for impulsive evolution equations in ordered Banach spaces. Under impulsive functions satisfying broader monotone conditions and without assumption that the lower and upper solutions exist, we obtain the existence results of ω-periodic mild solutions. Moreover, an application is given to illustrate our theoretical results.

Periodic boundary value problem for the graph differential equation and the matrix differential equation

A network can be represented by graph which is isomorphic to its adjacency matrix. Thus the analysis of networks involving rate of change with respect to time reduces to the study of graph differential equations and its associated matrix differential equations. In this paper we develop monotone iterative technique for graph differential equations and its associated matrix differential equations using Periodic boundary value problem.

Monotone iterative technique for second order delayed periodic problem in Banach spaces

In this paper, we deal with the existence of ω-periodic solutions for second-order functional differential equation with delay in E−u′′(t)=f(t,u(t),u(t−τ)),t∈R,where E is an ordered Banach space, f:R×E×E→E is a continuous function which is ω-periodic in t and τ ≥ 0 is a constant. We first build a new maximum principle for the ω-periodic solutions of the corresponding linear equation with delay. With the aid of this maximum principle, under the assumption that the nonlinear function is quasi-monotonicity, we study the existence of the minimal and maximal periodic solutions for abstract delayed equation by combining perturbation method and monotone iterative technique of the lower and upper solutions.

Positive solutions for nonlinear double impulsive differential equations with p-Laplacian on infinite intervals

In this paper, we investigate nonlinear second-order double impulsive differential equations integral boundary value problem with p-Laplacian on an infinite interval with the infinite number of impulsive times. Based on the cone theory and monotone iterative technique, we establish the existence of minimal nonnegative solution and iteration of positive solutions for such a boundary value problem. The main results are new and extend the existing results. At last, some examples are worked out to demonstrate the use of the main results.

Successive iteration and positive solutions for a third-order boundary value problem involving integral conditions

This paper investigates the existence of concave positive solutions and establishes corresponding iterative schemes for 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.

Maximum Principles for Second-Order Impulsive Integro-Differential Equations with Integral Jump Conditions

We are concerned with two new maximum principles for second-order impulsive integro-differential equations with integral jump conditions. The impulse effects or jump conditions of this paper are involved in terms of integral of past states. As an application, we introduce a new definition of lower and upper solutions which leads to the development of the monotone iterative technique for a periodic boundary value problem related to a nonlinear second-order impulsive functional integro-differential equation with integral jump conditions.

A NOTE ON EXISTENCE RESULTS FOR A CLASS OF THREE-POINT NONLINEAR BVPS

This article deals with a computational iterative technique for the following second order three point boundary value problem y''(t) + f(t, y, y' ) = 0, 0 < t < 1, y(0) = 0, y(1) = δy(η), where f(I × R, R), I = [0, 1], 0 < η < 1, δ > 0. We consider simple iterative scheme and develop a monotone iterative technique. Some examples are constructed to show the accuracy of the present method. We show that our technique is quite powerful and some user friendly packages can be developed by using this technique to compute the solutions of the nonlinear three point BVPs whose close form solutions are not known.

Nonlinear fractional differential equations with nonlocal integral boundary conditions

This paper concerns the boundary value problem of a class of fractional differential equations involving the Riemann-Liouville fractional derivative with nonlocal integral boundary conditions. By using 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, especially numerically simulates the conclusion by an example.

Existence and iteration of positive solutions for third-order Sturm–Liouville boundary value problems with p-Laplacian

In this paper, we investigate the existence and iteration of positive solutions for the following third-order Sturm–Liouville boundary value problem with p-Laplacian{(ϕp(u′′(t)))′+q(t)f(t,u(t),u′′(t))=0,t∈(0,1),αu(0)−βu′(0)=0,γu(1)+δu′(1)=0,u′′(0)=0,where ϕp(s)=|s|p−2s,p > 1. By applying a monotone iterative technique, we not only obtain the existence of positive solutions for the above boundary value problem, but also establish iterative schemes for approximating the solutions.

Monotone iterative solutions for nonlinear fractional differential systems with deviating arguments

By means of the monotone iterative technique, we consider the nonlinear fractional differential systems with deviating arguments, introduce two well-defined monotone sequences of solutions which converge uniformly to the solutions of the systems, and then the existence result of solution for the systems is established. A numerical iterative scheme is introduced to obtain an accurate approximate solution for the systems. As an application, an example is presented to demonstrate the accuracy and efficiency of the new approach.

Anti-periodic boundary value problem for first order impulsive delay difference equations

In this paper, we investigate the anti-periodic boundary value problem for first order impulsive delay difference equations. To begin with, we establish two comparison theorems. Then, by using these theorems, we prove the existence and uniqueness of solutions for the linear problem. Finally, by using the method of upper and lower solutions coupled with the monotone iterative technique, we obtain the new existence results of extremal solutions. Meanwhile, an example is given to illustrate the results obtained.

Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval

By employing the monotone iterative technique, we not only establish the existence of the unique solution for a fractional integral boundary value problem on semi-infinite intervals, but also develop an explicit iterative sequence for approximating the solution and give an error estimate for the approximation, which is an important improvement of existing results.

Extremal solutions for p-Laplacian fractional integro-differential equation with integral conditions on infinite intervals via iterative computation

We study the extremal solutions of a class of fractional integro-differential equation with integral conditions on infinite intervals involving the p-Laplacian operator. By means of the monotone iterative technique and combining with suitable conditions, the existence of the maximal and minimal solutions to the fractional differential equation is obtained. In addition, we establish iterative schemes for approximating the solutions, which start from the known simple linear functions. Finally, an example is given to confirm our main results.

Monotone iterative technique by upper and lower solutions with initial time difference

Monotone iterative technique by upper and lower solutions with initial time difference

Existence and Iteration of Positive Solutions to Third-Order BVP for a Class ofp-Laplacian Dynamic Equations on Time Scales

We investigate the existence and iteration of positive solutions for the following third-orderp-Laplacian dynamic equations on time scales:(ϕp(uΔΔ(t)))∇+q(t)f(t,u(t),uΔΔ(t))=0, t∈[a,b],αu(ρ(a))-βuΔ(ρ(a))=0, γu(b)+δuΔ(b)=0, uΔΔ(ρ(a))=0,whereϕp(s)isp-Laplacian operator; that is,ϕp(s)=sp-2s, p>1, ϕp-1=ϕq, and1/p+1/q=1.By applying the monotone iterative technique and without the assumption of the existence of lower and upper solutions, we not only obtain the existence of positive solutions for the problem, but also establish iterative schemes for approximating the solutions.

Monotone iterative technique for nonlocal fractional differential equations with finite delay in Banach space

Monotone iterative technique for nonlocal fractional differential equations with finite delay in Banach space

Singular nonlinear three point BVPs arising in thermal explosion in a cylindrical reactor

In this work we deal with a nonlinear three point singular boundary value problems (SBVPs), when the nonlinearity depends upon derivative. We establish the maximum principles for linear model. Prove some new inequalities based on Bessel and modified Bessel functions. Finally by using the Monotone Iterative Technique, we obtain some new existence results with well order and reverse order upper and lower solutions. The method developed in this paper can be used in computer algebra to compute solutions of real life problems.

On global stability of an HIV pathogenesis model with cure rate

In this paper, applying both Lyapunov function techniques and monotone iterative techniques, we establish new sufficient conditions under which the infected equilibrium of an HIV pathogenesis model with cure rate is globally asymptotically stable. By giving an explicit expression for eventual lower bound of the concentration of susceptible CD4+ T cells, we establish an affirmative partial answer to the numerical simulations investigated in the recent paper [Liu, Wang, Hu and Ma, Global stability of an HIV pathogenesis model with cure rate, Nonlinear Analysis RWA (2011) 12: 2947–2961]. Our monotone iterative techniques are applicable for the small and large growth rate in logistic functions for the proliferation rate of healthy and infected CD4+ T cells. Copyright © 2014 John Wiley & Sons, Ltd.

Existence and monotone iteration of symmetric positive solutions for integral boundary-value problems with ϕ-Laplacian operator

The purpose of this paper is to investigate the existence and iteration of symmetric positive solutions for integral boundary-value problems. An existence result of positive, concave and symmetric solutions and its monotone iterative scheme are established by using the monotone iterative technique. An example is worked out to demonstrate the main result.MSC:34B10, 34B18, 39A10.