Monotone Iterative Technique for a Class of Four Point BVPs with Reversed Ordered Upper and Lower Solutions

2019 ◽  
Vol 17 (09) ◽  
pp. 1950066
Author(s):  
Amit K. Verma ◽  
Nazia Urus ◽  
Mandeed Singh
Keyword(s):  
Existence Of Solutions ◽  
Iterative Scheme ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Monotone Iterative ◽  
Uniformly Convergent

Consider the class of four point nonlinear BVPs [Formula: see text] [Formula: see text] where [Formula: see text] is continuous, [Formula: see text], [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we demonstrate an iterative technique. The iterative scheme is deduced by using quasilinearization. Then we consider upper-lower solutions in well ordered and reverse ordered cases and prove existence of solutions under some sufficient conditions. We show that under certain conditions, generated sequences are monotone, uniformly convergent and converges to the solution of the above problem. We also provide examples which validate that all the conditions derived in this paper, are realistic and can be satisfied. We have also plotted upper and lower solutions for the test examples and have shown that under the conditions, the derived upper and lower solutions are monotonic in nature.

scholarly journals Picard Type Iterative Scheme with Initial Iterates in Reverse Order for a Class of Nonlinear Three Point BVPs

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Mandeep Singh ◽  
Amit K. Verma
Keyword(s):  
Source Term ◽  
Iterative Scheme ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Existence Results ◽  
Monotone Iterative Technique ◽  
Reverse Order ◽  
Monotone Iterative Method ◽  
Iterative Technique ◽  
Monotone Iterative

We consider the following class of three point boundary value problemy′′(t)+f(t,y)=0,0<t<1,y′(0)=0,y(1)=δy(η), whereδ>0,0<η<1, the source termf(t,y)is Lipschitz and continuous. We use monotone iterative technique in the presence of upper and lower solutions for both well-order and reverse order cases. Under some sufficient conditions, we prove some new existence results. We use examples and figures to demonstrate that monotone iterative method can efficiently be used for computation of solutions of nonlinear BVPs.

scholarly journals Existence for Certain Systems of Nonlinear Fractional Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Zhaowen Zheng ◽  
Xiujuan Zhang ◽  
Jing Shao
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence Of Solutions ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Comparison Result ◽  
Nonlinear Fractional Differential Equations ◽  
Monotone Iterative ◽  
Fractional Differential

By establishing a comparison result and using the monotone iterative technique, combining with the method of upper and lower solutions, the existence of solutions for systems of nonlinear fractional differential equations is considered. An example is given to demonstrate the applicability of our results.

scholarly journals Monotone Iterative Technique for Conformable Fractional Differential Equations with Deviating Arguments

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Huiling Chen ◽  
Shuman Meng ◽  
Yujun Cui
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Monotone Iterative Technique ◽  
Extremal Solutions ◽  
Iterative Technique ◽  
Deviating Arguments ◽  
Comparison Principles ◽  
Monotone Iterative ◽  
Fractional Differential

This paper is concerned with the existence of extremal solutions for periodic boundary value problems for conformable fractional differential equations with deviating arguments. We first build two comparison principles for the corresponding linear equation with deviating arguments. With the help of new comparison principles, some sufficient conditions for the existence of extremal solutions are established by combining the method of lower and upper solutions and the monotone iterative technique. As an application, an example is presented to enrich the main results of this article.

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

2015 ◽  
Vol 20 (4) ◽  
pp. 457-470 ◽  
Author(s):  
Amit K. Verma ◽  
Mandeep Singh
Keyword(s):  
Boundary Value Problem ◽  
Present Method ◽  
Iterative Scheme ◽  
Existence Results ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Point Boundary ◽  
Monotone Iterative ◽  
Close Form ◽  
User Friendly

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 &lt; t &lt; 1, y(0) = 0, y(1) = δy(η), where f(I × R, R), I = [0, 1], 0 &lt; η &lt; 1, δ &gt; 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.

Unique positive solution for a fractional boundary value problem

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Keyu Zhang ◽  
Jiafa Xu
Keyword(s):  
Boundary Value Problem ◽  
Positive Solution ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Fractional Boundary Value Problem ◽  
Unique Positive Solution ◽  
Monotone Iterative ◽  
Liouville Fractional Derivative

AbstractIn this work we consider the unique positive solution for the following fractional boundary value problem $\left\{ \begin{gathered} D_{0 + }^\alpha u(t) = - f(t,u(t)),t \in [0,1], \hfill \\ u(0) = u'(0) = u'(1) = 0. \hfill \\ \end{gathered} \right. $ Here α ∈ (2, 3] is a real number, D 0+α is the standard Riemann-Liouville fractional derivative of order α. By using the method of upper and lower solutions and monotone iterative technique, we also obtain that there exists a sequence of iterations uniformly converges to the unique solution.

scholarly journals Analysis and Computation of Solutions for a Class of Nonlinear SBVPs Arising in Epitaxial Growth

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 774
Author(s):  
Amit K Verma ◽  
Biswajit Pandit ◽  
Ravi P. Agarwal
Keyword(s):  
Differential Equation ◽  
Molecular Beam ◽  
Upper And Lower Solutions ◽  
Elliptic Partial Differential Equation ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Radial Solutions ◽  
Qualitative Properties ◽  
Monotone Iterative ◽  
Singular Ordinary Differential Equation

In this work, the existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. Since we are interested in radial solutions, we focus on the fourth-order singular ordinary differential equation. It is non-self adjoint, it does not have exact solutions, and it admits multiple solutions. Here, λ∈R measures the intensity of the flux and G is stationary flux. The solution depends on the size of the parameter λ. We use a monotone iterative technique and integral equations along with upper and lower solutions to prove that solutions exist. We establish the qualitative properties of the solutions and provide bounds for the values of the parameter λ, which help us to separate existence from nonexistence. These results complement some existing results in the literature. To verify the analytical results, we also propose a new computational iterative technique and use it to verify the bounds on λ and the dependence of solutions for these computed bounds on λ.

scholarly journals Monotone iterations for differential equations with a parameter

1997 ◽  
Vol 10 (3) ◽  
pp. 273-278 ◽  
Author(s):  
Tadeusz Jankowski ◽  
V. Lakshmikantham
Keyword(s):  
Differential Equations ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Extremal Solutions ◽  
Iterative Technique ◽  
Monotone Iterative ◽  
Monotone Iterations

Consider the problem {y′(t)=f(t,y(t),λ),t∈J=[0,b],y(0)=k0,G(y,λ)=0.. Employing the method of upper and lower solutions and the monotone iterative technique, existence of extremal solutions for the above equation are proved.

A class of fractional impulsive functional differential equations with nonlocal conditions

2016 ◽  
Vol 66 (4) ◽  
Author(s):  
Yiliang Liu ◽  
Jiangfeng Han
Keyword(s):  
Differential Equations ◽  
Comparison Theorem ◽  
Existence Of Solutions ◽  
Functional Differential Equations ◽  
Nonlocal Conditions ◽  
Monotone Iterative Technique ◽  
Extremal Solutions ◽  
Iterative Technique ◽  
Monotone Iterative ◽  
Functional Differential

AbstractIn this paper, we deal with the existence of solutions for the fractional impulsive functional differential equations with nonlocal conditions. Then we build a new comparison theorem and obtain the existence of extremal solutions and quasi-solutions by use of the monotone iterative technique and the method of lower and upper solutions.

scholarly journals Monotone Iterative Technique for Fractional Evolution Equations in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Jia Mu
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Evolution Equations ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Initial Value ◽  
Fractional Evolution Equations ◽  
Monotone Iterative ◽  
Noncompactness Measure

We investigate the initial value problem for a class of fractional evolution equations in a Banach space. Under some monotone conditions and noncompactness measure conditions of the nonlinearity, the well-known monotone iterative technique is then extended for fractional evolution equations which provides computable monotone sequences that converge to the extremal solutions in a sector generated by upper and lower solutions. An example to illustrate the applications of the main results is given.

Sign in / Sign up

Export Citation Format

Share Document