New generalizations for Gronwall type inequalities involving a ψ-fractional operator and their applications

In this paper, we provide new generalizations for the Gronwall's inequality in terms of a $ \psi $-fractional operator. The new forms of Gronwall's inequality are obtained within a general platform that includes several existing results as particular cases. To apply our results and examine their validity, we prove the existence and uniqueness of solutions for $ \psi $-fractional initial value problem. Further, the Ulam-Hyers stability of solutions for $ \psi $-fractional differential equations is discussed. For the sake of illustrating the proposed results, we give some particular examples.