Fixed Point and Best proximity point theorems with Applications

The famous work of S. Banach in 1922 where the celebrated Banach contraction mapping principle was established is the source point of metric fixed point theory. The idea of the result was also taken up extended and implemented in various contexts of mathematics. The Banach contraction mapping is a continuous mapping. An early fixed point result in discontinuous contraction was proved by Kannan in 1968. Kannan type mappings occupy an extensive area in the fixed point studies. Fixed points and related problems have been studied in metric spaces with a partial ordering allowing us to develop a methodology for fixed point theory in such spaces. Particularly continuity on the function is replaced by the condition of the above mentioned type in a good number of papers. Turinici (1986) Ran and Reurings(2004) Nieto(2005) etc are some examples. Coupled fixed points are introduced by Guo Lakshmikantham (1987). Coincident point is another direction. The problem of proximity point is distance optimization between two sets. Essentially it is a global optimization problem considered as a problem of finding the optimal approximate solutions of some appropriate fixed point equation.