One-Signed Periodic Solutions of First-Order Functional Difference Equations with Parameter
Received:September 05, 2017  Revised:May 17, 2018
Key Word: one-signed periodic solutions   existence   functional difference equations   bifurcation from infinity
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11626188; 11671322; 11501451), the Natural Science Foundation of Gansu Province (Grant No.1606RJYA232) and the Young Teachers' Scientific Research Capability Upgrading Project of Northwest Normal University (Grant No.NWNU-LKQN-15-16).
 Author Name Affiliation Yanqiong LU College of Mathematics and Statistics, Northwest Normal University, Gansu 730070, P. R. China Ruyun MA College of Mathematics and Statistics, Northwest Normal University, Gansu 730070, P. R. China Bo LU College of Mathematics and Computer Science, Northwest Minzu University, Gansu 730030, P. R. China
Hits: 317
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 $s\in(-\infty,s_2)\cup(s_2,0)$.