Abstract
We introduce nonlinear fractional BVPs including a generalized proportional derivatives with nonlocal multipoint and substrip boundary conditions. The nonlinearities are defined on the Orlicz space and depend on the unknown function and its generalized derivative. Existence results for a nonlinear boundary value problem involving a proportional fractional derivative by utilizing some fixed point theorems are presented. The obtained results are new and are well illustrated with an example.
Highlights
The theory of fractional derivative first appeared in the 1690s by the correspondence between L’Hospital and Leibniz
In 2015, Caputo and Fabrizio [7] proposed a new definition of fractional derivative with a smooth kernel involving the exponential function
Other definition was introduced by Atangana and Baleanu [8] where the kernel appeared via the Mittag-Leffler function. These generalized fractional derivatives have been studied by many researchers
Summary
The theory of fractional derivative first appeared in the 1690s by the correspondence between L’Hospital and Leibniz. Many new results were obtained recently in fractional differential equations with nonlocal multipoint and with nonlocal multi-strip integral boundary conditions involving Caputo derivative; for example, see [2,3,4,5,6] and the references cited therein. Other definition was introduced by Atangana and Baleanu [8] where the kernel appeared via the Mittag-Leffler function These generalized fractional derivatives have been studied by many researchers. We study Caputo type fractional differential equations with nonlocal multipoint and substrips boundary conditions (1) involving the generalized proportional derivative and let f be a function in an Orlicz space. Definition 9 (The GPF dervative of Caputo type) For ρ ∈ (0, 1] and α ∈ C with (α) > 0, we define the left generalized proportional fractional derivative of Capotu type starting by a,. Lemma 13 For any f ∈ LF ([0, 1]), the solution of the fractional boundary problem
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