Certain Novel Fixed-Point Theorems Applied to Fractional Differential Equations

In this paper, we introduce a new class of contractions in normed spaces, referred to as generalized enriched Kannan contractions. These contractions expand the familiar enriched Kannan contractions to three-point versions, broadening the scope of Kannan contractions. These mappings are typically discontinuous, except at the fixed points, where they exhibit continuity, similar to enriched Kannan mappings. However, through suitable examples, we demonstrate that these two classes of mappings are distinct from one another. We present new results for generalized enriched Kannan contractions. Additionally, by incorporating conditions of continuity and asymptotic regularity, we extend the class of operators to which fixed-point methods can be applied. Additionally, we derive two more results for generalized enriched Kannan contractions in normed spaces, without the requirement that they be Banach spaces. Finally, we use our main result to demonstrate the existence of solutions for a boundary value problem involving a fractional differential equation.