In the theory of indefinite sums, anti-differences and finite calculus,  ‎the following ‎difference ‎functional ‎equation ‎and ‎its ‎solutions ‎are ‎very ‎important:

‎$$‎\bigtriangleup ‎F(x):=F(x+1)-F(x)=f(x) ‎\quad‎\quad‎(1),‎$$

where‎ ‎$‎\bigtriangleup‎$ ‎is ‎the ‎forward ‎difference ‎operator when $f$ is given and $F$ is unknown. ‎ Also‎, if ‎$‎‎D_f=\mathbb{R}$‎, then there exists a special solution ‎$‎‎F_0(x)$ for equation (1) and ‎ ‎we ‎have ‎the ‎general ‎solutions of ‎it ‎as ‎follows‎: ‎‎‎$‎‎F=F_0+‎\lambda$,‎‎ ‎which ‎‎$‎‎‎\lambda$ ‎is a‎ ‎one-periodic ‎function.‎

Now‎, in my research I deal to the following functional equation
‎
$$F(x+1)+F(x)=f(x)‎‎$$

‎but I don't have any knowledge about it and its solution. ‎ What is the name of this equation and where can I learn more about it?