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 (ξ− 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)θ(ς)=σ(ς,θ(ς)),ς∈J=[0,A],A>0, where L(D)=γwcDδw+γw−1cDδw−1+⋯+γ1cDδ1+γ0cDδ0,γ♭∈R(♭=0,1,2,3,…,w),γw≠0,0≤δ0<δ1<δ2<⋯<δw−1<δw<1; 2. The nonlinear multi-term fractional delay differential equation L(D)θ(ς)=σ(ς,θ(ς),θ(ς−τ)),ς∈J=[0,A],A>0;θ(ς)=σ¯(ς),ς∈[−τ,0], where L(D)=γwcDδw+γw−1cDδw−1+⋯+γ1cDδ1+γ0cDδ0,γ♭∈R(♭=0,1,2,3,…,w),γw≠0,0≤δ0<δ1<δ2<⋯<δw−1<δw<1; moreover, here Dδc is predominantly called Caputo fractional derivative of order δ.