The authors show the existence of three positive solutions of the nonlinear boundary value problem \[ |p(t)\phi(x^\Delta(t))|^\nabla+f(t,x(t),x^\Delta)=0, \;\;t \in \mathbb {T}_\kappa, \]
\[ x(0)-\int^\infty_\sigma(0)\alpha(t)x(t)\Delta t=0 \lim_{t\to \infty}\phi^{-1}(p(t))x^\Delta(t)-\int^\infty_0\beta(t)x^\Delta(t)\Delta t=0, \] where \(\mathbb {T}\) is a times scale which may be unbounded, \(\phi(x)=|x|^{p-2}x\) with \(p>1\), \(\alpha, \beta:\mathbb {T}\to [0, \infty)\) are re-continuous with \(\alpha, \beta\in L^1[0, \infty)\), \(p\) is continuous, \(f:\mathbb {T}_\kappa\times [0, \infty)^2\to [0, \infty)\) is a Carathéodory function. The main tool is the five functionals fixed point theorem.