Best proximity point theorems for nonlinear mappings in partial metric spaces

Fixed point theory is the powerful tool for solving many real-world problems since many problems can be transformed to the fixed point problem. A fundamental theorem in fixed point theory is the Banach contraction mapping principle. This principle has many applications in several branches and so it was extended in many directions. However, almost all such results dilate upon the existence and uniqueness of a fixed point for self-mappings on some appropriate space such as a metric space, a norm space, an inner product space, and etc. In the case of nonself-mappings, the fixed point problem might have no solution and hence the concept of a best proximity point is introduced for approximating the best solution. This concept is also an important tool for investigating the global optimization problems. In this thesis, we introduce several various new types of generalized contraction mappings covering many types in the literature and give the idea of several tools for proving the best proximity point results. Based on the new tools, we establish the best proximity point results for the purposed generalized contraction mappings in partial metric spaces by using two methods including the fixed point method and the direct method. Our results improve the main results of Su and Yao [Su Y. and Yao, J. C. (2015). Further generalized contraction mapping principle and best proximity theorem on metric spaces. Fixed Point Theory Appl., 2015:120.], Azizi et al. [Azizi, A., Moosaei, M., and Zarei, G. (2016). Fixed point theorems for almost generalized C-contractive mappings in ordered complete metric spaces. Fixed Point Theory and Appl., 2016:80.], and Nashine et al. [Nashine, H. K., Kadelburg, Z., Radenovi ́c, S., and Kim, J. K. (2012). Fixed point theorems under Hardy-Rogers contractive conditions on 0- complete ordered partial metric spaces. Fixed Point Theory and Appl., 2012:180.] and many results in the literature. Moreover, we will give some example for supporting our results while many results in the literature can not be applied in such example. This guarantees the proper real generalization of our results.