Abstract
In this manuscript, by using the Caputo and Riemann–Liouville type fractional q-derivatives, we consider two fractional q-integro-differential equations of the forms {}^{c}mathcal{D}_{q}^{alpha }[x](t) + w_{1} (t, x(t), varphi (x(t)) )=0 and Dqαc[x](t)=w2(t,x(t),∫0tx(r)dr,cDqα[x](t))\\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}$$ {}^{c}\\mathcal{D}_{q}^{\\alpha }[x](t) = w_{2} \\biggl( t, x(t), \\int _{0}^{t} x(r) \\,\\mathrm{d}r, {}^{c} \\mathcal{D}_{q}^{\\alpha }[x](t) \\biggr) $$\\end{document} for t in [0,l] under sum and integral boundary value conditions on a time scale mathbb{T}_{t_{0}}= { t: t =t_{0}q^{n}}cup {0} for nin mathbb{N} where t_{0} in mathbb{R} and q in (0,1). By employing the Banach contraction principle, sufficient conditions are established to ensure the existence of solutions for the addressed equations. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.
Highlights
It has been recognized that fractional calculus provides a meaningful generalization for the classical integration and differentiation to any order
2015, Zhang et al through the spectral analysis and fixed point index theorem obtained the existence of positive solutions of the singular nonlinear fractional differential equation –Dtαu(t) = w(t, u(t), Dtβ u(t)) for 0 < t < 1, with integral boundary value conditions
Benefiting from the main ideas of the above said papers, we investigate the following two nonlinear fractional q-integro-differential equations in the spaces A = C(J × R2, R)
Summary
It has been recognized that fractional calculus provides a meaningful generalization for the classical integration and differentiation to any order. In. 2015, Zhang et al through the spectral analysis and fixed point index theorem obtained the existence of positive solutions of the singular nonlinear fractional differential equation –Dtαu(t) = w(t, u(t), Dtβ u(t)) for 0 < t < 1, with integral boundary value conditions. The existence of solutions for the multi-term nonlinear fractional q-integro-differential cDαq [u](t) equations in two modes and inclusions of order α ∈
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