AN ALGORITHMIC APPROACH TO LOCAL SOLUTION OF THE NONLINEAR HIGHER ORDER ORDINARY HYBRID DIFFERENTIAL EQUATIONS 


By

Janhavi B. Dhage, Shyam B. Dhage and Bapurao C. Dhage 

Kasubai, Gurukul Colony, Thodga Road, Ahmedpur, Distr. Latur, Maharashtra, India-413515 

Email: jbdhage@gmail.com, sbdhage4791@gmail.com, bcdhage@gmail.com 

(Received: October 08, 2023; In format: October 25, 2023; Revised: February 15, 2024; Accepted: February 28, 2024)


DOI: https://doi.org/10.58250/jnanabha.2024.54104 


 

Abstract

In this paper, we introduce a new notion of local or neighborhood solutions and establish a couple of approximation results for local existence and uniqueness of the solution of an IVP of nonlinear higher order ordinary hybrid differential equations by using the Dhage monotone iteration method based on the recent hybrid fixed point theorems of Dhage (2023). An approximation result for Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established. Our main abstract results of this paper are also illustrated with a couple of numerical examples. Finally, we compare our existence and uniqueness results with those existing in the literature via other operator theoretic methods from nonlinear functional analysis. We claim that the method and the results of this paper are new to the literature. 


2020 Mathematical Sciences Classification: 34A12, 34A34, 34A45, 47H10 

Keywords and Phrases: Ordinary differential equation; Dhage iteration method; Approximation theorems; Local existence and uniqueness; Ulam-Hyers stability. 


[Download PDF File]