Existence of positive solutions for first order discrete periodic boundary value problems with delay (Q544180)

Let \(T>3\) be a positive integer, \({\mathbf{T}}=\{0,1,\dots, T-1\},\) let \(a,b:\mathbb Z\rightarrow[0,\infty)\) be two \(T\)-periodic functions, \(\tau:\mathbb Z\to\mathbb Z\) be a \(T\)-periodic function, \(f,g\in C([0,\infty),[0,\infty))\), and \(\lambda>0\). The authors study the existence of positive periodic solutions of the nonlinear discrete boundary value problem \[ \begin{cases} \Delta u(t)=a(t)g(u(t))u(t)-\lambda b(t)f(u(t-\tau(t))),\, t\in\mathbf{T}\\ u(0)=u(T).\,\, \end{cases} \tag{21} \] The main tool used is the Dancer global bifurcation theorem.