Ulam stability and existence results for fractional differential equations with hybrid proportional-Caputo derivatives

Abstract

This paper is concerned with the Ulam-Hyers and the generalized Ulam-Hyers-Rassias stability results for linear fractional differential equations with hybrid proportional-Caputo derivatives in framwork of the Laplace transform method. The existence and uniqueness of solutions for nonlinear fractional differential equations with hybrid proportional-Caputo derivatives are established by means of Schaefer’s fixed point theorem, Banach’s fixed point theorem, and generalized Gronwall’s inequality. Two examples are proposed to illustrate the main results.

Subject Classification:

• 26A33
• 34K20
• 47H10

Keywords:

• Ulam stability
• Hybrid proportional-Caputo derivatives
• Laplace transform method
• Schaefer’s fixed point theorem