Positive periodic solution of second-order difference equation with a repulsive singularity (Q6633878)

The paper consider a second-order difference equation \N\[\N\Delta^2 u(m-1)=f(m,u(m)), \tag{1}\N\]\Nwith periodic boundary value conditions \N\[\Nu(0)=u(\omega), \qquad \Delta u(0)=\Delta u(\omega), \tag{2}\N\]\Nwhere \( \omega \) is a positive integer, \N\[\Nm\in \mathbb{Z}[ 1, +\infty), \qquad f(m,u): \mathbb{Z}[1,+\infty)\times (0,+\infty )\rightarrow \mathbb{R} \N\]\Nis a continuous \(\omega\)-periodic function, and \( f(m,u) \) may be singular at \( u=0 \). \N\NThe novel contributions with respect to the provided references are: \N\N(1) Exception of one specific condition that restricts the lower bound of \( f\); the remaining conditions are related only to the value of \( f \) at \( u=0 \) and \( u=+\infty \); \N\N(2) The lower bound of \( f \) is not a fixed constant; \N\N(3) \( f \) allows both superlinearity and semilinearity; \N\N(4) The monotonicity of \( f \) is no longer required. \N\NThe authors determine existence conditions of positive periodic solutions in the context of a repulsive singularity, a weak singularity and a strong singularity.