scholarly journals Monotone Iterative Technique for the Periodic Solutions of High-Order Delayed Differential Equations in Abstract Spaces

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 449
Author(s):  
He Yang ◽  
Yongxiang Li
Keyword(s):  
Periodic Solutions ◽  
Existence And Uniqueness ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Order Ordinary Differential Equation ◽  
Monotone Iterative ◽  
Delayed Differential Equations ◽  
Abstract Spaces ◽  
Fixed Delay

This paper deals with the existence of ω-periodic solutions for nth-order ordinary differential equation involving fixed delay in Banach space E. Lnu(t)=f(t,u(t),u(t−τ)),t∈R, where Lnu(t):=u(n)(t)+∑i=0n−1aiu(i)(t), ai∈R, i=0,1,⋯,n−1, are constants, f(t,x,y):R×E×E⟶E is continuous and ω-periodic with respect to t, τ>0. By applying the approach of upper and lower solutions and the monotone iterative technique, some existence and uniqueness theorems are proved under essential conditions.

Monotone iterative technique for impulsive Riemann-Liouville fractional differential equations

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3381-3395 ◽  
Author(s):  
Renu Chaudhary ◽  
Dwijendra Pandey
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Measure Of Noncompactness ◽  
Semigroup Theory ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Monotone Iterative ◽  
Fractional Differential

In this article, Monotone iterative technique coupled with the method of lower and upper solutions is employed to discuss the existence and uniqueness of mild solution to an impulsive Riemann-Liouville fractional differential equation. The results are obtained using the concept of measure of noncompactness, semigroup theory and generalized Gronwall inequality for fractional differential equations. At last, an example is given to illustrate the applications of the main results.

scholarly journals Perturbation Results and Monotone Iterative Technique for Fractional Evolution Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-18
Author(s):  
Jia Mu
Keyword(s):  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Ordered Banach Space ◽  
Fractional Evolution Equations ◽  
Existence And Uniqueness Results ◽  
Monotone Iterative ◽  
Fractional Evolution Equation ◽  
Uniqueness Results

We mainly study the fractional evolution equation in an ordered Banach space , , , . Using the monotone iterative technique based on lower and upper solutions, the existence and uniqueness results are obtained. The necessary perturbation results for accomplishing this approach are also developed.

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.

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.

scholarly journals Existence Results and the Monotone Iterative Technique for Nonlinear Fractional Differential Systems with Coupled Four-Point Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Yujun Cui ◽  
Yumei Zou
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Point Boundary ◽  
Differential Systems ◽  
Monotone Iterative ◽  
Fractional Differential Systems ◽  
Fractional Differential

By establishing a comparison result and using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence of solutions for nonlinear fractional differential systems with coupled four-point boundary value problems.

scholarly journals A generalized upper and lower solutions method for nonlinear second order ordinary differential equations

1992 ◽  
Vol 5 (2) ◽  
pp. 157-165 ◽  
Author(s):  
Juan J. Nieto ◽  
Alberto Cabada
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Nonlinear Boundary ◽  
Upper And Lower Solutions ◽  
Monotone Iterative Technique ◽  
Second Order ◽  
Iterative Technique ◽  
Monotone Iterative ◽  
Carathéodory Function ◽  
Minimal And Maximal Solutions

The purpose of this paper is to study a nonlinear boundary value problem of second order when the nonlinearity is a Carathéodory function. It is shown that a generalized upper and lower solutions method is valid, and the monotone iterative technique for finding the minimal and maximal solutions is developed.

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.

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