A THESIS SUBMITTED TO MADDA WALABU UNIVERSTY DEPARTMENT OF MATHEMATICS IN PARTIAL FULFILMENT OF THE REQUIRMENT FOR THE DEGREE OF MASTERS OF SCIENCE IN MATHMATICS.

This thesis extends the work of Rossa and Kari (2022) on best proximity point theorems for α- proximal θ- ɸ- mappings. We achieve this by generalizing the key inequality used in their proof leading to a broader class of contractive mappings. Specially, we conceder pairs of non -empty closed subsets in a complete metric space and define proximal non self- mappings. Our main novelity lies in introducing a more general in equality: . This inequality encompasses various terms not captured by the original formulation. Under suitable conditions, we prove the existence and uniqueness of a best proximity point for this generalized class of mappings. This means there exist a unique point such that minimizing the distance to the image of the mapping .Our results generalize existing theorems and demonstrate the applicability of the proposed framework. We showcase this through illustrative examples and discuss potential future directions for research. 3