One-Signed Periodic Solutions of First-Order Functional Difference Equations with Parameter

Abstract:

      In this paper, the authors obtain the existence of one-signed periodic solutions of the first-order functional difference equation $$\Delta u(n)=a(n)u(n)-\lambda b(n) f(u(n-\tau(n))),~~n\in\mathbb{Z}$$ by using global bifurcation techniques, where $a,b:\mathbb{Z}\rightarrow[0,\infty)$ are $T$-periodic functions with $\sum_{n=1}^{T}a(n)>0$, $\sum_{n=1}^{T}b(n)>0$; $\tau:\mathbb{Z}\to\mathbb{Z}$ is $T$-periodic function, $\lambda>0$ is a parameter; $f\in C(\mathbb{R},\mathbb{R})$ and there exist two constants $s_2<00$ for $s\in(0,s_1)\cup(s_1,\infty)$, and $f(s)<0$ for $s\in(-\infty,s_2)\cup(s_2,0)$.