Existence and nonexistence of positive solutions of a fractional thermostat model with a parameter (Q2830323)

In this paper the authors study the boundary value problem NEWLINE\[NEWLINE \left\{ \begin{aligned} {}^C\!D^{\alpha}_{0+} u(t)+\lambda f(t, u(t))=0, \;t\in(0, 1), \\ u'(0)=0,\;\beta\,{}^C\!D^{\alpha-1}u(1)+u(\eta)=0, \end{aligned} \right. NEWLINE\]NEWLINE where \({}^C\!D^{\alpha}\) denotes the Caputo fractional derivative of order \(\alpha\), \(1<\alpha\leq 2\), \(0\leq \eta\leq 1\), \(\beta>0\), \(f\) is continuous and \(\lambda\) is a parameter. The authors discuss the existence and non-existence of positive solutions when the parameter \(\lambda\) varies. The existence result relies on the classical Krasnosel'skiĭ-Guo fixed point theorem of cone-compression/expansion.