Existence and uniqueness results for a nonlocal \(q\)-fractional integral boundary value problem of sequential orders (Q2794941)

The authors consider the fractional \(q\)-difference integral boundary value problem NEWLINE\[NEWLINE ^C\!D_q^\beta({}^C\!D_q^\gamma+\lambda)x(t)=pf(t,x(t))+kI^\xi_qg(t,x(t)), 0\leq t\leq 1,\,0<q<1, NEWLINE\]NEWLINE NEWLINE\[NEWLINEx(0)=aI_q^{\alpha-1}x(\eta),\;x(1)=bI_q^{\alpha-1}x(\sigma),\;D_qx(1)=0, NEWLINE\]NEWLINE \(0<\eta\), \(\sigma<1\), \(0<\beta\leq 1\), \(1<\gamma\leq 2\), \(0<\xi<1\), where \(I^\xi_q\) denotes the Riemann-Liouville-type integral and \({}^C\!D_q^\beta\) denotes the fractional \(q\)-derivative of Caputo type, \(\lambda, p, k\) are constants. Several existence results for this problem are established. The main tools in the proofs are the Banach contraction principle, the Krasnoselskii fixed point theorem, and the Leray-Schauder nonlinear alternative.