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.

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 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.

Ordinary Differential Equations with Nonlinear Boundary Conditions

2002 ◽  
Vol 9 (2) ◽  
pp. 287-294
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Existence Results ◽  
Monotone Iterative Technique ◽  
Lower And Upper Solutions ◽  
Iterative Technique ◽  
Monotone Iterative

Abstract The method of lower and upper solutions combined with the monotone iterative technique is used for ordinary differential equations with nonlinear boundary conditions. Some existence results are formulated for such problems.

scholarly journals Region of existence of multiple solutions for a class of Robin type four-point BVPs

2021 ◽  
Vol 41 (4) ◽  
pp. 571-600
Author(s):  
Amit K. Verma ◽  
Nazia Urus ◽  
Ravi P. Agarwal
Keyword(s):  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Source Term ◽  
Lipschitz Constant ◽  
Sufficient Conditions ◽  
Existence Of Solution ◽  
Monotone Iterative Technique ◽  
Nonlinear Boundary Value Problems ◽  
Existence Of A Solution ◽  
Monotone Iterative

This article aims to prove the existence of a solution and compute the region of existence for a class of four-point nonlinear boundary value problems (NLBVPs) defined as \[\begin{gathered} -u''(x)=\psi(x,u,u'), \quad x\in (0,1),\\ u'(0)=\lambda_{1}u(\xi), \quad u'(1)=\lambda_{2} u(\eta),\end{gathered}\] where \(I=[0,1]\), \(0\lt\xi\leq\eta\lt 1\) and \(\lambda_1,\lambda_2\gt 0\). The nonlinear source term \(\psi\in C(I\times\mathbb{R}^2,\mathbb{R})\) is one sided Lipschitz in \(u\) with Lipschitz constant \(L_1\) and Lipschitz in \(u'\) such that \(|\psi(x,u,u')-\psi(x,u,v')|\leq L_2(x)|u'-v'|\). We develop monotone iterative technique (MI-technique) in both well ordered and reverse ordered cases. We prove maximum, anti-maximum principle under certain assumptions and use it to show the monotonic behaviour of the sequences of upper-lower solutions. The sufficient conditions are derived for the existence of solution and verified for two examples. The above NLBVPs is linearised using Newton's quasilinearization method which involves a parameter \(k\) equivalent to \(\max_u\frac{\partial \psi}{\partial u}\). We compute the range of \(k\) for which iterative sequences are convergent.

scholarly journals MultiPoint BVPs for Second-Order Functional Differential Equations with Impulses

2009 ◽  
Vol 2009 ◽  
pp. 1-16
Author(s):  
Xuxin Yang ◽  
Zhimin He ◽  
Jianhua Shen
Keyword(s):  
Differential Equations ◽  
Upper And Lower Solutions ◽  
Functional Differential Equations ◽  
Existence Results ◽  
Monotone Iterative Technique ◽  
Second Order ◽  
Monotone Iterative ◽  
Functional Differential ◽  
Extreme Solutions ◽  
Multipoint Boundary Value Problem

This paper is concerned about the existence of extreme solutions of multipoint boundary value problem for a class of second-order impulsive functional differential equations. We introduce a new concept of lower and upper solutions. Then, by using the method of upper and lower solutions introduced and monotone iterative technique, we obtain the existence results of extreme solutions.

Monotone iterative method for a class of nonlinear fractional differential equations

2012 ◽  
Vol 15 (2) ◽  
Author(s):  
Guotao Wang ◽  
Dumitru Baleanu ◽  
Lihong Zhang
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Fractional Differential Equations ◽  
Monotone Iterative Technique ◽  
Monotone Iterative Method ◽  
Iterative Technique ◽  
Nonlinear Fractional Differential Equations ◽  
Monotone Iterative ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

AbstractBy applying the monotone iterative technique and the method of lower and upper solutions, this paper investigates the existence of extremal solutions for a class of nonlinear fractional differential equations, which involve the Riemann-Liouville fractional derivative D q x(t). A new comparison theorem is also build. At last, an example is given to illustrate our main 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.

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