Original Contributions to the Theory of Fractional Calculus Operators on Variable Exponent Lebesgue Spaces

This dissertation makes original contributions to the theory of fractional calculus operators, specifically the Riemann-Liouville and Caputo-Fabrizio fractional integral and differentiation operators, within the framework of variable exponent Lebesgue spaces, a functional analytic setting that has attracted significant research activity over the past decade due to its applications to non-standard growth problems in partial differential equations and image processing. Classical fractional calculus has been extensively studied in standard Lebesgue Lp spaces, but the variable exponent setting, where the integrability exponent is itself a function of spatial position, introduces substantial technical complications arising from the non-homogeneity of the associated function spaces and the failure of classical interpolation arguments. The dissertation establishes novel boundedness theorems for the Riemann-Liouville fractional integral operator of variable order acting between variable exponent Lebesgue spaces under minimal regularity conditions on the exponent function, extending and sharpening recent results in the literature through the construction of refined Hedberg-type inequality arguments. New extrapolation theorems for weighted variable exponent spaces are proved, enabling the transfer of boundedness results from power weights to the Muckenhoupt weight class. Applications to the well-posedness of initial value problems for fractional partial differential equations arising in anomalous diffusion processes in heterogeneous porous media, relevant to contaminant transport in Niger Delta soils, are developed. Keywords: fractional calculus, variable exponent Lebesgue spaces, Riemann-Liouville operator, Muckenhoupt weights, anomalous diffusion