Fractional Sobolev Spaces With Applications To Boundary Values Problems

Abstract

This thesis aims to study fractional differential equations of both linear and nonlinear types, based on Riemann–Liouville derivatives of non-integer order. We begins by presenting the necessary mathematical background, including Lebesgue and fractional Sobolev spaces, and introduces the fundamental concepts of fractional inte gration and differentiation. These tools are then applied to analyze three main cases of the studied equation: the sublinear case 1 ≤ q < 2, the superlinear case 2 < q < 2 ∗ , and the linear case q = 2. In each case, we prove the existence of weak solutions within suitable functional spaces, taking into account the variation of conditions related to the spectral parameter λ.