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Keyword: Monotone Iterative Technique (4)	24 Mar 2025	Run
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articleOpen Access
THE EXISTENCE OF THE EXTREMAL SOLUTION FOR THE BOUNDARY VALUE PROBLEMS OF VARIABLE FRACTIONAL ORDER DIFFERENTIAL EQUATION WITH CAUSAL OPERATOR
- JINGFEI JIANG,
- JUAN L. G. GUIRAO, and
- TAREQ SAEED
Fractals01 Sep 2020
Preview Abstract
In this study, the two-point boundary value problem is considered for the variable fractional order differential equation with causal operator. Under the definition of the Caputo-type variable fractional order operators, the necessary inequality and the existence results of the solution are obtained for the variable order fractional linear differential equations according to Arzela–Ascoli theorem. Then, based on the proposed existence results and the monotone iterative technique, the existence of the extremal solution is studied, and the relative results are obtained based on the lower and upper solution. Finally, an example is provided to illustrate the validity of the theoretical results.
articleNo Access
Monotone Iterative Technique for a Class of Four Point BVPs with Reversed Ordered Upper and Lower Solutions
- Amit K. Verma,
- Nazia Urus, and
- Mandeed Singh
International Journal of Computational Methods17 Sep 2019
Preview Abstract
Consider the class of four point nonlinear BVPs
$- w^{″} (x) = f (x, w, w^{'}), x \in I,$
$w^{'} (0) = 0, w (1) = δ_{1} w (η_{1}) + δ_{2} w (η_{2}),$
where $f \in (I \times ℝ \times ℝ, ℝ)$ is continuous, $I = [0, 1]$ , $η_{1}, η_{2} \in (0, 1)$ such that $η_{1} \leq η_{2}$ and $δ_{1}, δ_{2} \geq 0$ . 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.
articleNo Access
INTEGRO-DIFFERENTIAL EQUATIONS ON UNBOUNDED DOMAINS IN BANACH SPACES
- DAJUN GUO
Chinese Annals of Mathematics01 Oct 1999
Preview Abstract
This paper investigates the maximal and minimal solutions of initial value problem for n-th order nonlinear integro-differential equations of Volterra type on an infinite interval in a Banach space by establishing a comparison result and using the monotone iterative technique.
articleNo Access
Monotone domain decomposition iterative technique for boundary value problems of a nonlinear fractional differential equation
- Myong-Gil Rim and
- Chol-Guk Choe
Asian-European Journal of Mathematics01 Feb 2019
Preview Abstract
In this paper, we present a monotone domain decomposition iterative technique for boundary value problems of a nonlinear fractional differential equation. We construct two monotone domain decomposition sequences of upper and lower solutions which converge uniformly to the actual solution of the problem. The accuracy and efficiency of our new approach are demonstrated through two examples.