On the solutions to a generalized fractional Cauchy problem

Fractional calculus studies problems with derivatives and integrals of real or complex order. As a purely mathematical field, the theory of fractional calculus was brought up for the first time in the XVIIth century and since then many renowned scientists worked on this topic, among them Euler, Laplace, Fourier, Abel, Liouville and Riemann [13]. When it comes to the practical applications, however, the notable development can only be observed during the last decades. Fractional operators have non–local character and consequently can be successfully applied in the study of non–local or time–dependent processes. The fields of applied sciences, where fractional calculus is found to be useful, include chaotic dynamics [19], material sciences [14], mechanics of fractal and complex media [4, 18], quantum mechanics [6], physical kinetics [20], and many others (see, e.g., [5, 16, 17, 21, 22]). In the literature many different types of fractional operators have been proposed, with the choice of a relevant definition being linked to the considered system [10, 11, 15]. Here, in order to simplify the theory, we show that two important fractional problems with different notions of derivatives– the fractional Cauchy problem with Hadamard derivatives and the fractional Cauchy problem with Riemann– Liouville derivatives– can be joined together by considering Cauchy problem with derivative defined recently by Katugampola. The Katugampola operators depend