DHAGE ITERATION METHOD FOR AN ALGORITHMIC APPROACH TO THE LOCAL SOLUTION OF THE NONLINEAR SECOND ORDER ORDINARY HYBRID DIFFERENTIAL EQUATIONS.

It is known that the Dhage iteration method is very much useful for proving the existence and approximation results for nonlinear hybrid differential and integral equations. In this paper, we introduce a notion of the local solution for the initial value problems of nonlinear second order ordinary differential equations and establish a couple of approximation results for local existence and uniqueness of the solution via Dhage monotone iteration method. Again, various hybrid fixed point theorems are involved in the Dhage iteration method as per the demand of the nonlinear hybrid equations. Here, we base our Dhage monotone iteration method on the recent hybrid fixed point theorems of Dhage (2022) and Dhage et al. (2022). There are different notions of stability of the nonlinear equations. Here we discuss the Ulam-Hyers stability of the local solution of the considered hybrid differential equation is also established by construction of an algorithm. Finally, our main abstract results are also illustrated with a couple of numerical examples. [ABSTRACT FROM AUTHOR]