Fixed Point Theorems for Generalized Contractions in Complete Metric Spaces

Abstract

Present literature depicts many ways to generalize the fixed point theory, the structure, mappings, contractions and metrics are generalized to extend the results. Ordered structures are very important not only in theoretical aspects but also in application point of view. Using the notion of graphs the ordered structures are generalized and some fixed point and coincidence point theorems for single and set valued mappings are presented. By using the weak contractions, namely CG-contraction and CG-weak contractions the contractive conditions are generalized. F-contraction is also used to extend the contractive conditions for set valued mappings. Mappings are generalized by using the relations and L-fuzzy mappings. Fixed points and common fixed point theorems are presented for mappings, set valued mappings, relations and fuzzy mappings. A unique type of common fixed point theorem for two set valued mappings is presented using the idea of Picard trajectories. A generalized Hausdorff distance is presented using the notion of initial segments from set theory, as a generalization of metric. Some applications of fixed points, coincidence points and common fixed points are presented. Results for existence of solutions of ordinary and fractional BVPs are established. It has been shown that coincidence point theorem can be used to prove implicit function theorem. It is also proved that a function satisfying certain conditions involving Homotopy mapping has a fixed point at parameter value equals to zero if and only if it has a fixed point at parameter’s value one. A generalization of Kelisky-Rivlin theorem for existence of solution of a system of Bernstein’s theorem is also proved.