A uniqueness method to a new Hadamard fractional differential system with four-point boundary conditions

Abstract

In this article, we discuss a new Hadamard fractional differential system with four-point boundary conditions \t\t\t{DαHu(t)+f(t,v(t))=lf,t∈(1,e),DβHv(t)+g(t,u(t))=lg,t∈(1,e),u(j)(1)=v(j)(1)=0,0≤j≤n−2,u(e)=av(ξ),v(e)=bu(η),ξ,η∈(1,e),\\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}$$\\textstyle\\begin{cases} {}^{H} D^{\\alpha}u(t)+f(t,v(t))=l_{f},\\quad t\\in(1,e),\\\\ {}^{H} D^{\\beta}v(t)+g(t,u(t))=l_{g},\\quad t\\in(1,e),\\\\ u^{(j)}(1)=v^{(j)}(1)=0, \\quad 0\\leq j\\leq n-2,\\\\ u(e)=av(\\xi),\\qquad v(e)=bu(\\eta),\\quad \\xi, \\eta\\in(1,e), \\end{cases} $$\\end{document} where a,b are two parameters with 0< ab(logeta)^{alpha-1}(logxi )^{beta-1}<1, alpha, betain(n-1,n] are two real numbers and ngeq3, f,gin C([1,e]times(-infty,+infty),(-infty,+infty)), l_{f}, l_{g}>0 are constants, and {}^{H} D^{alpha}, {}^{H} D^{beta} are the Hadamard fractional derivatives of fractional order. Based upon a fixed point theorem of increasing φ-(h,r)-concave operators, we establish the existence and uniqueness of solutions for the problem dependent on two constants l_{f}, l_{g}.