Stability theory and existence of solution to a multi-point boundary value problem of fractional differential equations

Abstract

The aims and objectives of this manuscript are concerned with the investigation of some appropriate conditions to establish existence theory of solutions to a class of nonlinear four-point boundary value problem (BVP) corresponding to fractional order differential equations (FODEs) provided as cDωy(t)=Ft,y(t),cDω-1y(t),1<ω≤2,t∈J=[0,1],y(0)=ζy(α),y(1)=ξy(β),ξ,ζ,α,β∈(0,1),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{aligned} ^{c}&{\\mathscr {D}}^{\\omega }y(t)={\\mathcal {F}}\\left(t, y(t), {^{c}{\\mathscr {D}}^{\\omega -1}y(t)}\\right),1<\\omega \\le 2,\\,\\, t\\in {\\mathbf{J }}=[0, 1],\\\\&y(0)=\\zeta y(\\alpha ), \\,\\, y(1)=\\xi y(\\beta ),\\,\\xi ,\\ \\zeta ,\\ \\alpha ,\\,\\beta \\in (0,1), \\end{aligned}\\right. \\end{aligned}$$\\end{document}where {^{c}{mathscr {D}}^{omega }} is Caputo’s fractional derivative of order q and {mathcal {F}}in ({mathbf {J}}times {mathbf {R}} times {mathbf {R}} ,{mathbf {R}}) may be nonlinear. The required conditions are obtained by using classical results of functional analysis and fixed point theory. Further, we establish some adequate conditions for the Ulam–Hyers stability and generalized Ulam–Hyers stability for the solutions to the considered BVP of nonlinear FODEs. We include a proper problem to illustrate our established results.