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

Abstract

Abstract We introduce a notion of nonlinear cyclic orbital $(\xi -\mathscr{F})$ ( 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 $$ \mathcal{L}(\mathcal{D})\theta (\varsigma )=\sigma \bigl(\varsigma , \theta ( \varsigma ) \bigr), \quad \varsigma \in \mathscr{J}=[0,\mathscr{A}], \mathscr{A}>0, $$ L ( D ) θ ( ς ) = σ ( ς , θ ( ς ) ) , ς ∈ J = [ 0 , A ] , A > 0 , where $$\begin{aligned} &\mathcal{L}(\mathcal{D})=\gamma _{w} \,{}^{c} \mathcal{D}^{\delta _{w}}+ \gamma _{w-1} \,{}^{c} \mathcal{D}^{\delta _{w-1}}+\cdots+\gamma _{1} \,{}^{c} \mathcal{D}^{\delta _{1}}+\gamma _{0} \,{}^{c} \mathcal{D}^{\delta _{0}},\\ &\gamma _{\flat}\in \mathbb{R}\quad (\flat =0,1,2,3,\ldots,w), \qquad \gamma _{w} \neq 0, \\ &0\leq \delta _{0}< \delta _{1}< \delta _{2}< \cdots< \delta _{w-1}< \delta _{w}< 1; \end{aligned}$$ L ( D ) = γ w c D δ w + γ w − 1 c D δ w − 1 + ⋯ + γ 1 c D δ 1 + γ 0 c D δ 0 , γ ♭ ∈ R ( ♭ = 0 , 1 , 2 , 3 , … , w ) , γ w ≠ 0 , 0 ≤ δ 0 < δ 1 < δ 2 < ⋯ < δ w − 1 < δ w < 1 ; 2. The nonlinear multi-term fractional delay differential equation $$\begin{aligned} &\mathcal{L}(\mathcal{D})\theta (\varsigma ) =\sigma \bigl(\varsigma , \theta ( \varsigma ),\theta (\varsigma -\tau ) \bigr), \quad \varsigma \in \mathscr{J}=[0, \mathscr{A}], \mathscr{A}>0; \\ &\theta (\varsigma ) =\bar{\sigma}(\varsigma ),\quad \varsigma \in [-\tau ,0], \end{aligned}$$ 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 $$\begin{aligned} &\mathcal{L}(\mathcal{D})=\gamma _{w} \,{}^{c} \mathcal{D}^{\delta _{w}}+ \gamma _{w-1} \,{}^{c} \mathcal{D}^{\delta _{w-1}}+\cdots+\gamma _{1} \,{}^{c} \mathcal{D}^{\delta _{1}}+\gamma _{0} \,{}^{c} \mathcal{D}^{\delta _{0}},\\ &\gamma _{\flat}\in \mathbb{R}\quad (\flat =0,1,2,3,\ldots,w), \qquad \gamma _{w} \neq 0, \\ &0\leq \delta _{0}< \delta _{1}< \delta _{2}< \cdots< \delta _{w-1}< \delta _{w}< 1; \end{aligned}$$ L ( D ) = γ w c D δ w + γ w − 1 c D δ w − 1 + ⋯ + γ 1 c D δ 1 + γ 0 c D δ 0 , γ ♭ ∈ R ( ♭ = 0 , 1 , 2 , 3 , … , w ) , γ w ≠ 0 , 0 ≤ δ 0 < δ 1 < δ 2 < ⋯ < δ w − 1 < δ w < 1 ; moreover, here ${}^{c}\mathcal{D}^{\delta}$ D δ c is predominantly called Caputo fractional derivative of order δ .