ANALYTICAL APPROACHES TO BOUNDARY VALUE PROBLEMS FOR FRACTIONAL DIFFERENTIAL EQUATIONS
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Date
2026
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Abstract
This thesis presents an analytical study of nonlinear fractional boundary value prob
lems involving Caputo and Riemann–Liouville fractional derivatives. The primary
objective is to establish existence and uniqueness results for solutions of such problems
defined on finite intervals under various boundary conditions. The analysis is devel
oped within a rigorous mathematical framework that combines tools from fractional
calculus, functional analysis, and fixed point theory. Two main classes of problems are
investigated. The first class concerns Caputo fractional boundary value problems, in
cluding a nonlocal problem and a two-point problem with a convection term, for which
existence and uniqueness results are obtained using the Banach contraction princi
ple and the Schauder fixed point theorem, respectively. The second class addresses
a Riemann–Liouville fractional boundary value problem with mixed boundary condi
tions, where the problem is reformulated as an equivalent integral equation via the
associated Green’s function. A new sufficient condition for the existence of a unique
solution is then derived by applying Rus’s fixed point theorem within a two-metric
framework. The obtained criterion is shown to be less restrictive than existing results
in the literature, thereby extending the applicability of the theory to a wider class of
problems.