Existence of periodic solutions for a class of fourth-order difference equation (Q2171662)

Summary: We apply the continuation theorem of \textit{R. E. Gaines} and \textit{J. L. Mawhin} [Coincidence degree and nonlinear differential equations. Lecture Notes in Mathematics, 568, Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0339.47031)] to ensure that a fourth-order nonlinear difference equation of the form \(\Delta^4u\left( k - 2\right)-a\left( k\right) u^\alpha\left( k\right)+b\left( k\right) u^\beta\left( k\right)=0\) with periodic boundary conditions possesses at least one nontrivial positive solution, where \(\Delta u\left( k\right)=u\left( k + 1\right)-u\left( k\right)\) is the forward difference operator, \( \alpha>0,\beta>0\) and \(\alpha\neq\beta, a\left( k\right),b\left( k\right)\) are \(T\)-periodic functions and \(a\left( k\right)b\left( k\right)>0\). As applications, we give some examples to illustrate the application of these theorems.