Fixed Point Results for Wardowski Type Interpolative Cyclic Contractions

Abstract

This thesis presents a unified framework for fixedpoint theory by introducing Wardowskitype interpolative cyclic contractions in both metric and partial metric spaces. We begin by defining a novel class of selfmaps T on a complete space (X,d), partitioned into cyclic subsets, that satisfy an interpolative condition

where FFF is a Wardowski gaugefunction,  is an interpolation parameter, and  encodes cyclic intersubset distances. Under completeness and mild continuity assumptions, we prove that the Picard iteration ​ generates a Cauchy sequence converging to a unique fixed point. These results extend classical Banach, Kannan, Chatterjea, and Meir Keeler theorems by capturing cyclic and nonlinear contractive behavior in a single inequality. We also extend our main theorems to partial metric spaces where self distance may not disappear thus extending the applicability to computational domains and domain theory. A hybrid of implicit relation proof technique is introduced to simplify the convergence proof by taking advantage of the function F’s monotonicity and divergence properties, and an orbital continuity technique is developed to weaken the full continuity assumptions on T, which, in both cases, are not satisfied by dynamic process, such as a continuously iterated operator.Constant examples are presented to show that the boundary value and integral operators become cyclic interpolative contractions of F, leading to constructive existence and uniqueness results with explicit convergence rates. Sensitivity study indicates that intermediate interpolation weight is the most effective in accelerating iterative convergence, which provides useful implementation hints to algorithm. The contributions of this work are (1) Coherent Interpolative Cyclic F contraction theory (2) Generalization to partially defined metrics (3) Theorems of convergence rate depending on parameters and (4) Simpler proof techniques. Taken together, these developments provide a powerful toolbox for both theoretical investigation and applications to concrete problems in nonlinear analysis.

Keywords: Fixed point theory; Wardowski Fcontractions; cyclic mappings; interpolative contractions; partial metric spaces; Picard iteration; convergence analysis.