Résumé:
The objective of this thesis is to investigate fractional-order boundary value problems in
non-regular domains by examining the existence and uniqueness of solutions for various
types of abstract differential equations involving fractional operators. The study begins with
an analysis of three-dimensional fourth-order differential equations incorporating fractional
powers of the negative Laplace operator under Cauchy-Dirichlet boundary conditions in
cuspidal domains. The investigation techniques are based on transforming the main problem,
through a natural change of variables, into a complete abstract fourth-order differential
equation involving fractional powers of linear operators, which allows us to provide
results on well-posedness. Furthermore, we explore periodic-type solutions for fractional
neutral evolution equations involving Caputo and -Hilfer derivatives, utilizing classical
fixed point theorems as a preliminary step toward further investigation of fractional-order
boundary value problems in non-smooth domains.
