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

Abstract

We introduce a notion of nonlinear cyclic orbital (xi -mathscr{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,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{L}(\\mathcal{D})\ heta (\\varsigma )=\\sigma \\bigl(\\varsigma , \ heta ( \\varsigma ) \\bigr), \\quad \\varsigma \\in \\mathscr{J}=[0,\\mathscr{A}], \\mathscr{A}>0, $$\\end{document} 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;\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} &\\mathcal{L}(\\mathcal{D})=\\gamma _{w} \\,{}^{c} \\mathcal{D}^{\\delta _{w}}+ \\gamma _{w-1} \\,{}^{c} \\mathcal{D}^{\\delta _{w-1}}+\\cdots+\\gamma _{1} \\,{}^{c} \\mathcal{D}^{\\delta _{1}}+\\gamma _{0} \\,{}^{c} \\mathcal{D}^{\\delta _{0}},\\\\ &\\gamma _{\\flat}\\in \\mathbb{R}\\quad (\\flat =0,1,2,3,\\ldots,w), \\qquad \\gamma _{w} \ eq 0, \\\\ &0\\leq \\delta _{0}< \\delta _{1}< \\delta _{2}< \\cdots< \\delta _{w-1}< \\delta _{w}< 1; \\end{aligned}$$ \\end{document}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],\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} &\\mathcal{L}(\\mathcal{D})\ heta (\\varsigma ) =\\sigma \\bigl(\\varsigma , \ heta ( \\varsigma ),\ heta (\\varsigma -\ au ) \\bigr), \\quad \\varsigma \\in \\mathscr{J}=[0, \\mathscr{A}], \\mathscr{A}>0; \\\\ &\ heta (\\varsigma ) =\\bar{\\sigma}(\\varsigma ),\\quad \\varsigma \\in [-\ au ,0], \\end{aligned}$$ \\end{document} 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;\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} &\\mathcal{L}(\\mathcal{D})=\\gamma _{w} \\,{}^{c} \\mathcal{D}^{\\delta _{w}}+ \\gamma _{w-1} \\,{}^{c} \\mathcal{D}^{\\delta _{w-1}}+\\cdots+\\gamma _{1} \\,{}^{c} \\mathcal{D}^{\\delta _{1}}+\\gamma _{0} \\,{}^{c} \\mathcal{D}^{\\delta _{0}},\\\\ &\\gamma _{\\flat}\\in \\mathbb{R}\\quad (\\flat =0,1,2,3,\\ldots,w), \\qquad \\gamma _{w} \ eq 0, \\\\ &0\\leq \\delta _{0}< \\delta _{1}< \\delta _{2}< \\cdots< \\delta _{w-1}< \\delta _{w}< 1; \\end{aligned}$$ \\end{document} moreover, here {}^{c}mathcal{D}^{delta} is predominantly called Caputo fractional derivative of order δ.