Two-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives.
@article{bwmeta1.element.doi-10_2478_s11533-012-0112-9,
author = {Ma\l gorzata Klimek and Marek B\l asik},
title = {Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {1981-1994},
zbl = {1260.26009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0112-9}
}
Małgorzata Klimek; Marek Błasik. Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order. Open Mathematics, Tome 10 (2012) pp. 1981-1994. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0112-9/
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