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

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 consi...

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Bibliographic Details
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
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Summary: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.
ISSN:2075-1680
2075-1680
DOI:10.3390/axioms13090640