Conditionally Compatible Mappings, Non-Compatible Mappings, Faintly Compatible Mappings and Some Results on Common Fixed Point in $C^*$-Algebra Valued Metric Space

This manuscript introduces and investigates various types of mappings in the context of a $C^*$-algebra valued metric space. These include conditionally compatible, non-compatible, and faintly compatible mappings. Additionally, the manuscript establishes fixed-point results for certain combinations of these mappings. Specifically, it considers pairs of self-mappings that satisfy either a $(\phi-\psi)$-Kannan-type or Chatterjea-type contractive condition and proves common fixed point theorems for faintly compatible continuous or reciprocal continuous along with non-compatible pairs of such mappings. Also, non-compatibility is relaxed by using the $E.A.$ property and faintly compatible continuous property or reciprocal continuous property for pairs of self-mappings. To demonstrate the practicality of these results, the manuscript includes examples that illustrate their usefulness.