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 ) =...

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Veröffentlicht in:Boundary value problems 2023-09, Vol.2023 (1), p.91-35, Article 91
Hauptverfasser: Panda, Sumati Kumari, Vijayakumar, Velusamy, Nisar, Kottakkaran Sooppy
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Sprache:eng
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Zusammenfassung: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 ) = 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 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 ) = 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 D δ c is predominantly called Caputo fractional derivative of order δ .
ISSN:1687-2770
1687-2762
1687-2770
DOI:10.1186/s13661-023-01778-3