Existence of Solution for a Katugampola Fractional Differential Equation Using Coincidence Degree Theory

Abstract

In this paper, the authors study the existence of positive solutions to the fractional boundary value problem at resonance -(Da+α,ρx)(t)=f(t,x(t),Da+α-1,ρx(t)),t∈(a,b),x(a)=0,x(b)=∫abx(t)dA(t),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -(D^{\\alpha ,\\rho }_{a+}x)(t)= & {} f(t,x(t),D^{\\alpha -1, \\rho }_{a+}x(t)), \\ \\ t\\in (a,b), \\\\ x(a)= & {} 0, \\ \\ x(b)=\\int _{a}^{b} x(t){\ ext {d}}A(t), \\end{aligned}$$\\end{document}where 1<α≤2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<\\alpha \\le 2$$\\end{document}, and Da+α,ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^{\\alpha ,\\rho }_{a+}$$\\end{document} is a Katugampola fractional derivative, which generalizes the Riemann–Liouville and Hadamard fractional derivatives, and ∫abx(t)dA(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\int _{a}^{b} x(t){\ ext {d}}A(t)$$\\end{document} denotes a Riemann–Stieltjes integral of x with respect to A, where A is a function of bounded variation. Coincidence degree theory is applied to obtain existence results. This appears to be the first work in the literature to deal with a resonant fractional differential equation with a Katugampola fractional derivative. Examples are given to illustrate the application of their results.