Abstract
In this study, we discuss the existence of positive solutions for the system of m-singular sum fractional q-differential equationsDqαixi+gi(t,x1,…,xm,Dqγ1x1,…,Dqγmxm)+hi(t,x1,…,xm,Dqγ1x1,…,Dqγmxm)=0\\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}$$ \\begin{gathered} D_{q}^{\\alpha_{i}} x_{i} + g_{i} \\bigl(t, x_{1}, \\ldots, x_{m}, D_{q}^{\\gamma _{1}} x_{1}, \\ldots, D_{q}^{\\gamma_{m}} x_{m} \\bigr) \\\\ \\quad{} +h_{i} \\bigl(t, x_{1}, \\ldots, x_{m}, D_{q}^{\\gamma_{1}} x_{1}, \\ldots, D_{q}^{\\gamma_{m}} x_{m} \\bigr)=0 \\end{gathered} $$\\end{document} with boundary conditions x_{i}(0) = x_{i}' (1) = 0 and x_{i}^{(k)}(t) = 0 whenever t=0, here 2leq k leq n-1, where n= [alpha_{i}]+ 1, alpha_{i} geq2, gamma_{i} in(0,1), D_{q}^{alpha} is the Caputo fractional q-derivative of order α, here q in(0,1), function g_{i} is of Carathéodory type, h_{i} satisfy the Lipschitz condition and g_{i} (t , x_{1}, ldots, x_{2m}) is singular at t=0, for 1 leq i leq m. By means of Krasnoselskii’s fixed point theorem, the Arzelà-Ascoli theorem, Lebesgue dominated theorem and some norms, the existence of positive solutions is obtained. Also, we give an example to illustrate the primary effects.
Highlights
Fractional calculus and q-calculus belong to the significant branches in mathematical analysis
There appeared recently much work on q-differential equations by using different views and fractional derivatives; young researchers could use the main idea in their work
In 2010, the singular Dirichlet problem Dαx(t) + g(t, x(t), Dγ x(t)) = 0 under conditions x(0) = x(1) = 0 was investigated by Agarwal et al, where α, γ belong to (1, 2), (0, α – 1), respectively, the function g is of Carathéodory type on [0, 1] × (0, ∞) × R and Dα is the Riemann–Liouville fractional derivative [23]
Summary
Fractional calculus and q-calculus belong to the significant branches in mathematical analysis. In 2010, the singular Dirichlet problem Dαx(t) + g(t, x(t), Dγ x(t)) = 0 under conditions x(0) = x(1) = 0 was investigated by Agarwal et al, where α, γ belong to (1, 2), (0, α – 1), respectively, the function g is of Carathéodory type on [0, 1] × (0, ∞) × R and Dα is the Riemann–Liouville fractional derivative [23]. In 2014, the singular fractional problem cDα0+ x(t) + f (t, x(t), cDσ0+ x(t)) = 0 with boundary conditions x(0) = x (0) = 0 and x (1) = cDσ0+ x(1) investigated, where t, α, σ belong to (0, 1), (2, 3), (0, 1), respectively, f : (0, 1] × R2 → R is continuous with f (t, x, y) may be singular at t = 0 and cDα0+ is the Caputo derivative [31].
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