Abstract
This Dissertation investigates the convergence behavior of several iterative approximation
methods in the context of fixed point theory and variational inequality problems. We extend
the viscosity approximation method by incorporating a general φ-contractive mapping
in place of the classical contraction, and establish strong convergence results for this generalized
scheme. These results ensure convergence to a unique solution of certain variational
inequality problems and are supported by illustrative examples and corollaries as special
cases.
Further, we explore split feasibility problems and split common fixed point problems, and
present a generalized viscosity approximation algorithm based on the work of Zhao and He
[68], proving strong convergence theorems under φ-contractive mappings. An application to
the split null point problem is provided to demonstrate the utility of the proposed approach.
Additionally, we study and generalize the perturbed and projection theta methods by
proposing the perturbed generalized theta method and projection generalized theta method.
We establish some weak and convergence results for perturbed generalized theta method in
uniformly convex Banach space and investigate the convergence of projection generalized
theta method in Hilbert space. Applications to the problem of finding zeros of monotone
operators are also discussed.
The results presented in this Dissertation contribute to the advancement of iterative methods
in nonlinear functional analysis and open an scope for applications in areas such as
optimization, signal processing, and computational mathematics.