Fixed Point Theorems in Super Metric Spaces with Applications to Integral Equations, Stability Analysis and Boundary Value Problems
DOI:
https://doi.org/10.29070/c4mfjt46Keywords:
Super metric space, Fixed point theorem, Asymptotic regularity, Banach-type contraction, Max-type contraction, Rational contraction, Common fixed point, Ulam–Hyers stability, Volterra integral equation, Boundary value problemAbstract
The present paper is dedicated to the study of fixed point theory in the context of super metric spaces, a relatively new concept which is a generalization of classical metric space, where the usual triangle condition is replaced by a sequence condition with the control of limsup. The study presents some new fixed point theorems in complete super metric spaces under Banach-type, max-type and rational contractive conditions. A new concept of asymptotic regularity of type (AR) is defined and explored which helps in convergence analysis. The presented idea is appropriate for relating an iterative process to the structure of super-metrics that is sequence dependent and also to a valuable instrument for iterative convergence proof of Picard iterates.
The main theoretical results are studied and applied to investigate fundamental topological properties of super metric spaces such as convergence, completeness, uniqueness of limits and Cauchy sequence behavior. A Banach-type fixed point theorem is derived as an extension of the classical contraction principle, then a more general max-type contraction theorem and finally a rational contraction result. Moreover, a common fixed point theorem of weakly compatible mappings with a common contractive condition involving a combined condition is derived.
There are several major applications of the theory developed to show its applicability. Existence and uniqueness of solutions to nonlinear Volterra integral equations are established as well as computable estimates of the errors, using the fixed point results. The framework is also used to examine the Ulam–Hyers stability of functional equations with explicit bounds on the stability. Theory is also applied to nonlinear boundary value problems by using the Green's function techniques, and existence, uniqueness, convergence analysis and numerical error estimates are derived. The outcomes can be used to show that super metric spaces are well-suitable and applicable study environment to develop the classical fixed point theory and still remain applicable to nonlinear mathematical models used in various applied sciences and engineering problems.
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