At the end of Beowulf, the unknown author does not leave any direct information about what the state of the kingdom will be--given Beowulf's passing. In the final chapter of the text, the text states
the men of Geatland, his hearth-companions, mourned their hero's passing, and said that of all the kings of the earth, he was the mildest and most belovéd of his men; kindest to his kin, and the most eager for praise.
While this fails to define the future of the kingdom, it does allow readers to assume that no other king will be as great as Beowulf.
That said, Beowulf did pass the throne onto Wiglaf, prior to his death. Once again, readers can assume that Beowulf believed that Wiglaf was the best suited to possess the throne. Over the course of the text, Wiglaf proved himself to be as good a man as Beowulf. While not defined as great an epic hero as Beowulf, Wiglaf knew the importance of honor, keeping promises (battle boasts), and taking on the challenges other men would not. Given that Wiglaf had a perfect mentor, one could easily assume that Beowulf's kingdom will stand far beyond his passing.