In this thesis, we discuss several existence and uniqueness results of generalized self-similar solutions for some nonlinear partial differential equations of fractional order of Katugampola type, with boundary value, initial value, or with integral conditions in Banach space, we use the Banach contraction principle, Schauder and Guo-Krasnosel'skii fixed point theorems, and the technique of the nonlinear alternative of Leray-Schauder type.