EXISTENCE AND STABILITY OF SOLUTIONS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS INVOLVING THE GRÖNWALL-FREDHOLM-TYPE INEQUALITY

In this work, we have explored a fractional Newton’s Second Law of motion involving the ψ-Caputo operator of order α∈(1,2]. We proved the existence and uniqueness of solutions for different classes of force functions f acting on a specific object in mention. We have used some well-known fixed point theorems and fractional calculus methods. We have proved Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability, and generalized Ulam-Hyers-Rassias stability for the main problem. Mainly, this manuscript can be divided into four major parts. In the first part, we derive the necessary conditions to assure the existence and the uniqueness of a solution. In the second part, we develop some new Grönwall-Fredholm inequalities type, and in the third part, we prove that our problem is stable with respect to all Ulam’s stability types, and we have found a simpler condition under which the UHR and HUHR stability types persist, making our findings more accessible and applicable in a large class of fractional problems. Finally, we give two examples to establish our theoretical results. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.