Qualitative Outcomes on Monotone Iterative Technique and Quasilinearization Method on Time Scale

• Published in: Axioms 2024-09, Vol.13 (9), p.640
• Main Authors: Çetin, Şahap, Yılmaz, Yalçın, Yakar, Coşkun
• Format: Article
• Language: English
• Subjects: Boundary value problems; Differential equations, Nonlinear; extremal solutions; Iterative methods (Mathematics); Nonlinear dynamics; quadratic convergence; quasilinearization; Theorems; Time; time scale; weakly convergence
• ISSN: 2075-1680; 2075-1680
• DOI: 10.3390/axioms13090640

In this paper, a nonlinear dynamic equation with an initial value problem (IVP) on a time scale is considered. First, applying comparison results with a coupled lower solution (LS) and an upper solution (US), we improved the quasilinearization method (QLM) for the IVP. Unlike other studies, we consider the LS and US pair of the seventh type instead of the natural type. It was determined that the solutions of the dynamic equation converge uniformly and monotonically to the unique solution of the IVP, and the convergence is quadratic. Moreover, we will use the delta derivative (Δγ) instead of the classical derivative (dγ) in the proof because it studies a time scale. In the second part of the paper, we applied the monotone iterative technique (MIT) coupled with the LS and US, which is an effective method, proving a clear analytical representation for the solution of the equation when the relevant functions are monotonically non-decreasing and non-increasing. Then an example is given to illustrate the results obtained.