Generalized Common Fixed-Point Theorems in Bipolar & Controlled Metric Spaces
DOI:
https://doi.org/10.29070/q324m341Keywords:
Fixed Point Theory, Bipolar Metric Spaces, Controlled Metric Spaces, Common Fixed Points, Weak Compatibility, E.A. Property, Generalized ContractionsAbstract
Fixed point theory has found a wide range of applications in differential equations, optimization, mathematical modeling, & dynamical systems, & is a vital tool in nonlinear analysis. During the past decade, the concept of a generalized metric structure has been developed & has facilitated the generalization of classical fixed point theory to more general & realistic mathematical environments. This paper explores some generalized bipolar metric spaces & controlled metric spaces.
In the bipolar metric setting, new common fixed point results are obtained for covariant & contravariant mappings with the existence of the approximate fixed point (E.A.) property in the weak compatibility setting. Using generalized contractive conditions based on both distance functions, forward & backward, sufficient conditions for the existence & uniqueness of common fixed points are derived. The proposed results considerably challenge the classical common belief of continuity & strong compatibility, & generalize the several Banach-type & Kannan-type fixed point theorems to asymmetric distance functions.
The study also extends the results of fixed point theory for controlled metric spaces, where the classical triangle inequality is replaced by a control function. Two auxiliary lemmas regarding convergence, completeness & Cauchy sequences are obtained, giving extensions of fixed point type theorems of Mlaiki. Hybrid Banach–Kannan contractive conditions are given & existence & uniqueness theorems of fixed points & common fixed points are obtained. The resulting theorems make some of the well-known results in the literature special cases.
The theory is applied to nonlinear integral equations, optimization problems, equilibrium problems & dynamical systems to show the usefulness of the results. The results show that the bipolar & controlled metric spaces offer strong & flexible structures to investigate fixed point problems under weaker structural conditions. As such, the proposed work is a significant step forward in the continued development of generalized fixed point theory & new avenues for future research in nonlinear analysis & applied mathematics.
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