scholarly journals Periodic solutions of quasi-differential equations

1996 ◽  
Vol 9 (1) ◽  
pp. 11-20
Author(s):  
Abdelkader Boucherif ◽  
Eduardo García-Río ◽  
Juan J. Nieto
Keyword(s):  
Maximum Principle ◽  
Differential Operator ◽  
Periodic Solutions ◽  
Solution Method ◽  
Lower Solution ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Monotone Iterative ◽  
Carathéodory Function ◽  
Lower Solution Method

Existence principles and theorems are established for the nonlinear problem Lu=f(t,u) where Lu=−(pu′)′+hu is a quasi-differential operator and f is a Carathéodory function. We prove a maximum principle for the operator L and then we show the validity of the upper and lower solution method as well as the monotone iterative technique.

scholarly journals Periodic problem for non-instantaneous impulsive partial differential equations

2021 ◽  
Vol 7 (3) ◽  
pp. 3345-3359
Author(s):  
Huanhuan Zhang ◽  
◽  
Jia Mu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Mild Solutions ◽  
Periodic Problem ◽  
Lower Solution ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Partial Differential ◽  
Monotone Iterative ◽  
Impulsive Equation

<abstract><p>We obtain a new maximum principle of the periodic solutions when the corresponding impulsive equation is linear. If the nonlinear is quasi-monotonicity, we study the existence of the minimal and maximal periodic mild solutions for impulsive partial differential equations by using the perturbation method, the monotone iterative technique and the method of upper and lower solution. We give an example in last part to illustrate the main theorem.</p></abstract>

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 of Solutions for a Class of Nonlinear Neumann BVPs in the Presence of Upper and Lower Solutions

2021 ◽  
Author(s):  
Nazia Urus ◽  
Amit Verma
Keyword(s):  
Maximum Principle ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Upper And Lower Solutions ◽  
Lower Solution ◽  
Specific Region ◽  
Monotone Iterative Technique ◽  
Monotone Iterative ◽  
Strict Condition ◽  
Sign Restriction

In this article, we explore the monotone iterative technique (MI-technique) to study the existence of solutions for a class of nonlinear Neumann 4-point, boundary value problems (BVPs) defined as, \begin{eqnarray*} \begin{split} -\z^{(2)}(\y)=\x(\y,\z,\z^{(1)}),\quad 0<\y<1,\\ \z^{(1)}(0)=\lambda \z^{(1)}(\beta_1 ),\quad \z^{(1)}(1)=\delta \z^{(1)}(\beta_2), \end{split} \end{eqnarray*} where $ 0<\beta_1 \leq \beta_2 <1$ and $\lambda$, $\delta\in (0,1)$. The nonlinear term $ \x(\y,\z,\z^{(1)}): \Omega\rightarrow \mathbb{R} $, where $\Omega =[0,1]\times \mathbb{R}^2 $, is Lipschitz in $ \z^{(1)}(\y)$ and one sided Lipschitz in $ \z(\y)$. Using lower solution $l(\y)$ and upper solutions $u(\y)$, we develop MI-technique, which is based on quasilinearization. To construct the sequences of upper and lower solutions which are monotone, we prove maximum principle as well as anti maximum principle. Then under certain assumptions, we prove that these sequence converges uniformly to the solution $ \z(\y)$ in the specific region, where $ \frac{\partial\x}{\partial\z}<0 $ or $ \frac{\partial\x}{\partial\z}>0 $. To demonstrate that the proposed technique is effective, we compute the solution of the nonlinear multi-point BVPs. We don’t require sign restriction which is very common and very strict condition.

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.

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.

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