Existence Result for a Superlinear Fractional Navier Boundary Value Problems

Abstract

In this paper, we study the following fractional Navier boundary value problem Dβ(Dαu)(x)=u(x)g(u(x)),x∈(0,1),limx⟶0x1-βDαu(x)=-a,u(1)=b,\\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}$$\\begin{aligned} \\left\\{ \\begin{array}{lllc} D^{\\beta }(D^{\\alpha }u)(x)=u(x)g(u(x)),\\quad x\\in (0,1), \\\\ \\displaystyle \\lim _{x\\longrightarrow 0}x^{1-\\beta }D^{\\alpha }u(x)=-a,\\quad \\,\\,u(1)=b, \\end{array} \\right. \\end{aligned}$$\\end{document}where alpha ,beta in (0,1] such that alpha +beta >1, D^{beta } and D^{alpha } stand for the standard Riemann–Liouville fractional derivatives and a, b are nonnegative constants such that a+b>0. The function g is a nonnegative continuous function in [0,infty ) that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, we prove the existence of a unique positive continuous solution, which behaves like the unique solution of the homogeneous problem.