Fixed point results for cyclic α-ψφ-contractions with application to integral equations

How can α-ψφ conditions be extended to cyclic mappings so that existence and uniqueness results can be established, and then connected to a concrete integral-equation problem?

The paper introduces α-ψφ-contractive and cyclic α-ψφ-contractive mappings and develops fixed-point existence and uniqueness results in metric-like spaces.

• Define the α-ψφ and cyclic structures
• Prove fixed-point existence and uniqueness results
• Develop related graph-based contractive results
• Apply the theorems to an integral equation

The fixed-point framework is used as the theoretical tool for establishing existence of a solution to the integral equation.

The paper establishes existence and uniqueness results for the introduced mappings, relates them to earlier results, and demonstrates an application to integral equations.

This research illustrates a move from an abstract mathematical structure to a concrete problem: define precise conditions, establish a theorem, and then apply the result to an integral model. That pattern also matters in problem-driven modeling and computational work.