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Positive solutions for a class of higher order boundary value problems with fractional \(q\)-derivatives. (English) Zbl 1254.34010

Summary: The authors study the boundary value problem with fractional \(q\)-derivatives \[ -(D^v_q u)(t)= f(t,u),\quad t\in (0,1), \]
\[ (D^i_q u)(0)= 0,\quad i=0,\dots, n-2,\quad (D_q u)(1)= \sum^m_{j=1} a_j(D_q u)(t_j)+ \lambda, \] where \(q\in (0,1)\), \(m\geq 1\) and \(n\geq 2\) are integers, \(n-1< v\leq n\), \(\lambda\geq 0\) is a parameter, \(f:[0,1]\times\mathbb{R}\to [0,\infty)\) is continuous, \(a_i\geq 0\) and \(t_i\in (0,1)\) for \(i= 1,\dots, m\), and \(D^v_q\) is the \(q\)-derivative of Riemann-Liouville type of order \(v\). The uniqueness, existence, and nonexistence of positive solutions are investigated in terms of different ranges of \(\lambda\).

MSC:

34A08 Fractional ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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