Oscillation of differential equations in the frame of nonlocal fractional derivatives generated by conformable derivatives

Abstract

Recently, Jarad et al. in (Adv. Differ. Equ. 2017:247, 2017) defined a new class of nonlocal generalized fractional derivatives, called conformable fractional derivatives (CFDs), based on conformable derivatives. In this paper, sufficient conditions are established for the oscillation of solutions of generalized fractional differential equations of the form \t\t\t{Dα,ρax(t)+f1(t,x)=r(t)+f2(t,x),t>a,limt→a+aIj−α,ρx(t)=bj(j=1,2,…,m),\\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}$$ \\textstyle\\begin{cases} {}_{a}\\mathfrak{D}^{\\alpha,\\rho}x(t)+f_{1}(t,x)=r(t)+f_{2}(t,x),\\quad t>a, \\\\ \\lim_{t \\to a^{+}}{ {}_{a}\\mathfrak{I}^{j-\\alpha,\\rho}x(t)}=b_{j} \\quad (j=1,2,\\ldots,m), \\end{cases} $$\\end{document} where m=lceilalpharceil, {}_{a}mathfrak{D}^{alpha,rho} is the left-fractional conformable derivative of order alphainmathbb{C}, operatorname{Re}(alpha)geq0 in the Riemann–Liouville setting and {}_{a}mathfrak {I}^{alpha,rho} is the left-fractional conformable integral operator. The results are also obtained for CFDs in the Caputo setting. Examples are provided to demonstrate the effectiveness of the main result.