Fixed Point Theorems for Non-self Mappings Using Disconnected Graphs

Abstract

Objectives: To prove the fixed point theorems for non-self mappings using disconnected graphs. Method: Graph theoretical approach is adopted to prove the fixed point theorems for non-self mappings. In all the previous works, connected graphs were used for establishing the results, but it is demonstrated in this work that disconnected graphs are best suited, and this new approach simplifies the proofs to a greater extent. Findings: The fixed point theorems by Banach, Kannan, Chatterjea, and Bianchini are proved using the new methodology. Novelty: An important part of the results concerning fixed point theorems is proving the iterated sequence to be a Cauchy sequence, and this is amalgamated with the edge sequence of the disconnected graph. Subject Classification: 54H25, 47H10 Keywords: Non-self mapping, Iterated sequence, Disconnected graph, Edge sequence, Fixed point