The October Storyline
In the fall of 1989, Jim Davis produced a series of Halloween-based comic strips about Garfield waking up to an abandoned house. When Garfield finally accepted that he was alone, he's suddenly brought back to his owner Jon's house.
Naturally, this storyline had several readers question the meaning behind these series of strips. Because of the strip's surreality regarding the Halloween story arc, it is commonly believed that Garfield had either died, or that he's slowly starving to death, and that the entire strip is his dying dream.
This theory does make sense in some places. Throughout the strip, Garfield would often have dreams regarding food, and he would constantly beg Jon for his trademark dish, lasagna.
When the creator of the strip himself, Jim Davis, heard about these rumors, he was reported as to have laughed for quite a long time. According to him, he had created those series of Halloween strips, because the most common fear among several of his readers was being alone. He also confirmed that Garfield was very much alive, and that the Halloween strips weren't canonical to the main strip.
Lyman was Jon's friend as well as the former owner of Jon's pet dog, Odie. Lyman was originally a very active character in the comic strip, but he largely disappeared in the year of 1983, and there wasn't an official explanation for his disappearance.
Many theories were proposed for Lyman's disapperance. Many fans speculated that he moved from the neighborhood and simply forgot to forward his address, while other fans believed that Jon killed him and stole Odie from him. Jim Davis has also thrown in his two cents regarding his character's disappearance. He has jokingly said that Lyman was most likely in Jon's basment, or that he joined the Peace Corps and was never heard from again. However, he did state that the reason as to why Lyman left the strip was because Garfield was starting to fill the role as the snarker whenever Jon talked about his problems to him.
The Garfield Show had an hour long episode in which Garfield and friends tried to look for Lyman. The episode revealed that Lyman became a nature photographer and that he disappeared while looking for a mythological creature. this may be noncanonical, however.