Applying periodic and anti-periodic boundary conditions in existence results of fractional differential equations via nonlinear contractive mappings

We introduce a notion of nonlinear cyclic orbital (. - F)-contraction and prove related results. With these results, we address the existence and uniqueness results with periodic/anti-periodic boundary conditions for: 1. The nonlinear multi-order fractional differential equation L(D).(.) = s(.,.(.)),.. J = [0, A], A > 0, where L(D) =.w cDdw +.w- 1 cDdw- 1 + center dot center dot center dot +.1 cDd1 +.0 cDd0,. . R ( = 0, 1, 2, 3,..., w),.w = 0, 0 = d0 < d1 < d2 < center dot center dot center dot < dw- 1 < dw < 1; 2. The nonlinear multi-term fractional delay differential equation L(D).(.) = s(.,.(.),.(. - t)),.. J = [0, A], A > 0;.(.) = <overline> s (.),.. [-t, 0], where L(D) =.w cDdw +.w- 1 cDdw- 1 + center dot center dot center dot +.1 cDd1 +.0 cDd0,. . R ( = 0, 1, 2, 3,..., w),.w = 0, 0 = d0 < d1 < d2 < center dot center dot center dot < dw- 1 < dw < 1;