Some existence and uniqueness theorems for nonlinearfractional differential equations of higher-order

This study aims to present global existence and uniqueness theorems for an initial value problem involving a nonlinear multiterm higher-order Riemann-Liouville fractional derivative equation. The singular nonlinearity of the equation in the problem to be considered can be shown as one of the reasons that makes this study original. One of the most used methods to examine the existence and uniqueness of solutions to this type of problem is to reduce the problem to an integral equation and examine the existence and uniqueness of the solutions of the relevant integral equation. Here, it has been proven that the problem and the weak singularity integral equation are equivalent under certain conditions. The Riemann-Liouville integral and derivative operator and their properties are given in the third part of the thesis to understand and demonstrate this proof. To search for the local solutions of this integral equation in the space of continuous functions or in "better" spaces, an operator corresponding to this equation is defined and functional analysis methods and techniques such as the compactness of the operator and the Arzela-Ascoli theorem are used for the existence of the fixed points of the operator. Since this study will deal with global solutions, the Arzela-Ascoli theorem will not be sufficient and the definition of compactness for the space in which the solution is investigated has been tried to be expanded with functional analysis techniques by following the articles of Su and Zhang (2011), Jiang (2014) and Hao et. al (2018). Finally, the theorem for the uniqueness of the solution is presented by giving the Nagumo-type uniqueness theorem for the given problem.