Theoretical investigation on fractal–fractional nonlinear ordinary differential equations

Abstract

In this study, we examine the existence and uniqueness conditions of the solutions of the nonlinear fractal-fractional differential equations. Particular emphasis is placed on four cases: exponential decay, power law, generalized Mittag-Leffler kernels and the Delta-Dirac function. Our first contribution is the formulation of some basic inequalities inspired from Gronwall inequality setting up a solid foundation for our analysis to follow. We subsequently carefully obtain the maximal and minimal solutions in each scenario, providing a complete picture of their structure. Finally we show convergence of four different successive approximation schemes, validating their applicability in the various contexts. This is an important finding that adds to the growing literature on the use of fractional calculus in complex dynamical systems.