Revisiting Darbo’s Fixed Point Theory with Application to a Class of Fractional Integral Equations

This chapter is devoted to define a generalized proportional \((k,\rho )\) -fractional integral operator containing the nonsingular kernel. It is a more generalized form of existing fractional integral operators like the Riemann-Liouville fractional integral operator, the Hadamard fractional integral operator, the Katugampola fractional integral operator, etc. First, we prove the existence of the generalized fractional integral operator, which follows the commutative property, and then generalize Darbo’s fixed point theory. Thereafter, by utilizing Darbo’s fixed point theorem, we demonstrate the existence of a solution to the generalized proportional \((k,\rho )\) -fractional integral equation. At the end, a suitable example is constructed, which verifies the obtained results.