Results on Discontinuity at Fixed Point for a New Class of $F$-Contractive Mappings

Abstract

The search for contractive definitions which do not compel the mapping to be continuous at fixed points remained an open problem for a long time. Several solutions to this open problem have been obtained in last two decades. The current paper, we aim to provide another new solution direction for the discontinuity study at fixed points using $F$-contractive mappings in a complete metric space. Several consequences of those new results are also provided. This manuscript consists of three main parts. In the first part, the notion of $F$-contractive mappings has been described. In the second part, discontinuity at the fixed point assuming continuity of the composition has been investigated, whereas in the third part, discontinuity at a fixed point without assuming continuity of the composition has been illustrated.

Keywords

• discontinuity
• fixed point
• $f$-contraction
• complete metric space