Existence of solutions for integral boundary value problems of singular Hadamard-type fractional differential equations on infinite interval

Abstract

We consider the existence of solutions for the following Hadamard-type fractional differential equations: {DαHu(t)+q(t)f(t,u(t),HDβ1u(t),HDβ2u(t))=0,1<t<+∞,u(1)=0,Dα−2Hu(1)=∫1+∞g1(s)u(s)dss,Dα−1Hu(+∞)=∫1+∞g2(s)u(s)dss,\\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}$$ \\textstyle\\begin{cases} {}^{H}D^{\\alpha }u(t)+q(t)f(t,u(t), {}^{H}D^{\\beta _{1}}u(t),{}^{H}D^{ \\beta _{2}}u(t))=0,\\quad 1< t< +\\infty , \\\\ u(1)=0, \\\\ {}^{H}D^{\\alpha -2}u(1)=\\int ^{+\\infty }_{1}g_{1}(s)u(s)\\frac{ds}{s}, \\\\ {}^{H}D^{\\alpha -1}u(+\\infty )=\\int ^{+\\infty }_{1}g_{2}(s)u(s) \\frac{ds}{s}, \\end{cases} $$\\end{document} where 2<alpha leq 3, 0<beta _{1}leq alpha -2<beta _{2}leq alpha -1, f:J times mathbb{R}^{3}rightarrow mathbb{R} satisfies the q-Carathéodory condition, q,g_{1},g_{2}:Jrightarrow mathbb{R}^{+} are nonnegative, where J=[1,+infty ). Nonlinear term f is dependent on the fractional derivative of lower order beta _{1}, beta _{2}, which creates additional complexity to verify the existence of solutions. The singularity occurring in our problem is associated with {}^{H}D^{beta _{2}}uin C(1,+infty ) at the left endpoint t=1 (if beta _{2}<alpha -1).