World View: A Philosophical and Theological Perspective

The world of today has emerged as a global village with diversity of culture, faith, religion, ideology and belief. The difference of point of view and intolerance are still left to be taken into account by the intellectuals of the world seriously with other multiple universal problems. In the present scenario, there is a need to rationalize the human existence on the face of Earth in terms of the sole objectives of human life. This study is an attempt to present a world view to the humanity through a philosophical and theological approach. Multiple questions have been raised and then answered with reference to Islamic religious philosophy of human life. It is an attempt to strengthen harmony among the world citizens.

Best Proximity Point Results in Fuzzy Metric Spaces

The aim of this research is to investigate some contractions and establish the existence results of single and multivalued mappings in frame work of some abstract space. Banach (1922) presented a fixed point theorem, namely Banach contraction theorem, which states that, a contraction mapping has a unique fixed point in a complete metric space. This theorem is widely applicable in proving the existence of solutions of functional equations under certain conditions. This theorem is based upon the iterative process, so it can be easily implemented on computer. This theorem was a revolution in fixed point theory. Later on , several generalizations of this theorem have been obtained. Fixed point results are widely studied in the framework of topological, metric and ordered oriented spaces. It is interesting to study that a nonlinear functional equation T(x) = x (where T is some linear or nonlinear operator defined on metric or fuzzy metric space) has either a solution or possess no solution. If the nonlinear functional equation has any solution, then it is interesting to find some algorithms which lead to an appropriate solution, but if nonlinear functional equation has no solution, then to try and find some approximate solution. The approximation, optimization and fixed point theory incorporate the main theme of nonlinear analysis. The best approximation theory is applicable in variety of problems arising in nonlinear functional analysis. Banach (1922) fixed point theorem is vastly applicable in proving the existence of solutions of functional equations under certain conditions. Many authors generalized Banach contraction principle in various directions. To address the question, if a nonlinear functional equation has no solution, it is recommended to approximate the solution which solves the optimization problem such that d(x, T x) is minimum. Ky Fan’s (Fan (1969)) best approximation result has been used when some nonlinear functional equations have no solution. In this thesis, best proximity point results in non-Archimedean fuzzy metric space are proved and some optimal best proximity point results are also obtained. These results provides the existence of optimal approximate solutions to some equations which have no solution. Further, fuzzy optimal coincidence point results for different proximal contractions in the framework of complete non-Archimedean fuzzy metric space are obtained. These results also holds in fuzzy metric spaces when some mild assumptions are added in the domain of involved mappings. In the next part of thesis, best proximity point results in a complete non-Archimedean fuzzy metric space with ordered structure have been discussed. Further, a class of multivalued mappings is introduced which satisfies Suzuki type generalized contractive condition in the framework of fuzzy metric spaces and some fixed point results are obtained for such kind of mappings. In next part of this thesis, Suzuki type contraction conditions are further generalized as Suzuki type F−contraction fuzzy mapping in ordered metric spaces. As an application, a common fixed point result for hybrid pair of single and multivalued mappings, the existence and uniqueness of common bounded solution of functional equations arising in dynamic programming are obtained. The obtained results generalize and extend various results in the existing literature. In each chapter, some examples are provided, which shows the validity of the results along with couple of remarks about the comparison of obtained results with the existing ones in the literature.