Investigation of properties of some derivatives and integrals of fractional order

The first aim of this study is to reveal some properties of Caputo and Riemann-Liouville fractional derivative operators, which are the two most used fractional derivative operators in fractional calculus theory, and Riemann-Liouville integral and compare the properties of derivative operators. The second aim is to investigate the local existence and uniqueness of the solutions of an initial-value problem involving a nonlinear 1<σ<2 order Riemann-Liouville fractional derivative equation. The nonlinear function on the right side of the equation in this problem has singularity with respect to its first variable. It has been shown that, under the condition of continuity on this function, the solutions of this problem are equivalent to the solutions of the Volterra-type equation with double singularity. This method provides the opportunity to investigate the existence and uniqueness of the solutions of this integral equation instead of the problem. On the other hand, the investigation of the existence and uniqueness of solutions of some integral equations with single singularity requires more effort and requires more mathematical tools than the investigation of the existence and uniqueness of solutions of their counterparts. In this study, especially while showing the uniqueness of the solutions, basic tools such as Hölder's inequality for Lebesgue spaces is used and the existing proof techniques are generalized. In this way, the Nagumo-type, Krasnoselski-Krein-type and Osgood-type uniqueness theorems, which are the most common in the theory of both classical differential equations and fractional differential equations will be given for the problem.