Existence and uniqueness results to positive solutions of integral boundary value problem for fractional q-derivatives

Abstract

In this paper,we are interested in the existence and uniqueness of positive solutions for integral boundary value problem with fractional q-derivative: \t\t\tDqαu(t)+f(t,u(t),u(t))+g(t,u(t))=0,0<t<1,u(0)=Dqu(0)=0,u(1)=μ∫01u(s)dqs,\\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} $$\\begin{aligned} &D_{q}^{\\alpha}u(t)+f\\bigl(t,u(t),u(t)\\bigr)+g\\bigl(t,u(t) \\bigr)=0, \\quad 0< t< 1, \\\\ & u(0)=D_{q}u(0)=0, \\qquad u(1)=\\mu \\int_{0}^{1}u(s)\\,d_{q}s, \\end{aligned}$$ \\end{document} where D_{q}^{alpha} is the fractional q-derivative of Riemann–Liouville type, 0< q<1, 2<alphaleq3 , and μ is a parameter with 0<mu<[alpha]_{q}. By virtue of fixed point theorems for mixed monotone operators, we obtain some results on the existence and uniqueness of positive solutions.