On ?-asymmetric Geraghty Type Contractive Mappings in Metric Spaces
A New Class of Asymmetric Quasi-Geraghty-Type Contraction Mappings with Applications tomNon-Linear...
DOI:
https://doi.org/10.63746/njtd.v23i2.4125Keywords:
Asymmetric contraction, Convergence, Differential and Integral equations, Fixed point, Geraghty-type mapping, StabilityAbstract
: Fixed point theory has become an essential tool in the study of nonlinear problems arising in mathematics, science and engineering. However, many of the existing contractive conditions are restrictive or insufficient for establishing the existence and uniqueness of solutions for certain classes of nonlinear problems. The paper aims to introduce a new class of asymmetric Geraghty-type contractive mappings in metric spaces and establish its theoretical properties together with selected practical applications. In order to achieve this aim, a novel asymmetric Geraghty-type contractive condition is formulated for self-mappings on complete metric spaces. The newly proposed contractive condition is then employed to derive sufficient conditions for the existence and uniqueness of fixed points by constructing an appropriate iterative sequence and analysing its convergence behaviour. The stability of the iterative sequence associated with the proposed contraction is further established under suitable assumptions. The applicability of the developed fixed-point theorem is demonstrated by proving the existence and uniqueness of solutions for a first-order nonlinear differential equation arising from a population growth model and the analysis of a mass-spring-damper system governed with second order- nonlinear differential equations, which has important applications in vehicle suspension systems, structural vibration control and machine design. The proposed asymmetric Geraghty-type contractive mapping guaranteeing unique solutions and stable iterative approximations for nonlinear problems and extends the applicability of fixed-point theory for analyzing practical problems encountered in science and engineering.
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