The Cauchy problem for fractional differential equations with worsening right-hand sides

Abstract

n the present paper, we consider existence and uniqueness conditions for solutions of the Cauchy problem for fractional differential equations with worsening operators in Banach spaces. A method for analyzing the solvability of the Cauchy problem by analyzing the convergence of the successive approximation method in scales of Banach spaces (continuously embedded in each other) was suggested in [1, 2]. Later, this method was extended to general higher-order differential equations in [3], where the results contain the classical Nagumo and Ovsyannikov theorems (e.g., see [4–8]) for integer-order equations with worsening operators. It is a natural idea to extend the method to fractional differential equations. The results of the present paper contain existence and uniqueness theorems for the Cauchy problem for equations with Caputo fractional derivatives of order α.