On a Second-Order Differential Problem with Fractional Derivatives of Order Greater than One

Abstract

A second-order abstract problem with derivatives of non-integer order is investigated. The nonlinearity involves fractional derivatives between 1 and 2. Existence and uniqueness of mild and classical solutions are established in appropriate spaces. This work extends similar works with or without a derivative of first order and also a work of the present author, where the order of the derivatives were between 0 and 1.

Keywords:

• Cauchy problem
• cosine family
• fractional derivative
• mild solutions
• second-order abstract problem

AMS Subject Classification:

• 26A33
• 34K40
• 35L90
• 35L70
• 35L15
• 35L07