The existence of a solution for nonlinear fractional differential equations where nonlinear term depends on the fractional and first order derivative of an unknown function


Suzana Aleksić, Alberto Cabada, Slađana Dimitrijević, Tatjana V Tomović Mladenović




In this paper, we consider the existence of solutions of the nonlinear fractional differential equation boundary-value problem Dα∗ u(t) = f (t,u(t),u ′(t), CDβu(t)), 0 < t < 1, 1 < α < 2, 0 < β ⩽ 1, u(0) = A, u(1) = Bu(η), where 0 < η < 1, A ⩾ 0, Bη > 1, Dα∗ is the modified Caputo fractional derivative of order α, CDβ is the Caputo fractional derivative of order β, and f is a function in C([0, 1] ×R ×R ×R). Existence results for a solution are obtained. Two examples are presented to illustrate the results.