Positive solutions for a new class of Hadamard fractional differential equations on infinite intervals

Abstract

In the present article, the following nonlinear problem of new Hadamard fractional differential equations on an infinite interval \t\t\t{DνHx(t)+b(t)f(t,x(t))+c(t)=0,1<ν<2,t∈(1,∞),x(1)=0,HDν−1x(∞)=∑i=1mγiHIβix(η),\\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}$$ \\textstyle\\begin{cases} {}^{H}\\!D^{\\nu }x(t)+ b(t)f(t,x(t))+c(t)=0,\\quad 1< \\nu < 2, t\\in (1,\\infty), \\\\ x(1)=0,\\qquad {}^{H}\\!D^{\\nu -1}x(\\infty )=\\sum_{i=1}^{m}\\gamma _{i} {}^{H}\\!I^{\\beta _{i}}x(\\eta ), \\end{cases} $$\\end{document} is studied, where {}^{H}!D^{nu } denotes the Hadamard fractional derivative of order ν, {}^{H}!I(cdot ) is the Hadamard fractional integral, beta _{i}, gamma _{i}geq 0 (i=1,2,ldots , m), eta in (1,infty ) are constants and \t\t\tΓ(ν)>∑i=1mγiΓ(ν)Γ(ν+βi)(logη)ν+βi−1.\\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}$$ \\varGamma (\\nu )>\\sum_{i=1}^{m} \\frac{\\gamma _{i}\\varGamma (\\nu )}{ \\varGamma (\\nu +\\beta _{i})}(\\log \\eta )^{\\nu +\\beta _{i}-1}. $$\\end{document} By making use of a fixed point theorem for generalized concave operators, the existence and uniqueness of positive solutions is established. Moreover, an application of the established results is also presented via an interesting example.