There is definitely no lack of heroes in Greek mythology, these people stand above mere mortals in strength or wit; however Greek heroes do not exactly match our idea of a hero. Some of their characteristics include causing more destruction than benefits to a community and being immoral beyond any social standards. It isn't easy to earn such a title, so does a mythological figure like Bellerophon have what it takes to be considered a Greek hero? I will analyze and discuss Bellerophon using Raglan and Dumezil's theories as the hero criteria for the myth.
According to Lord Raglan, Greek heroes tend to follow a pattern from when they were born to their deaths. Starting with one of the earliest written works, Homer's Iliad starts the myth by mentioning Bellerophon's grandfather Sisyphus, and his father Glaucus, starting the pattern by skipping his childhood. The story continues with him being sent to Lycia by Proitos because of an accusation by Proitos's wife that Bellerophon tried to sleep with her. At Lycia, he is given quests as his punishment and at the same time for the good of the community, most notably the killing of the Chimera, the Solymi, the Amazons and overcoming the Lycian ambush. After that, he is rewarded with a princess's hand in marriage and given half the kingdom; however he did not do this alone as Homer states him "obeying the signs of the gods" (Homer, Iliad). At the end, he loses favour with the gods for unexplained reason and meets a mysterious end. Overall, Bellerophon fits relatively well with Raglan's theory in the Iliad. He is your run on the mill Greek hero whom with the help of the Gods kills a monster, fights the Amazons and best many other warriors in combat, hence making him stand out. Following the theory, he is rewarded with marriage and half the kingdom where his reign ends tragically like most heroes when trying to settle down due to his heroic impulses. These troubles include losing favour from the gods and none of his children succeeding his kingdom.
The next oldest retelling of Bellerophon is Pindar's Olympian Ode that fills up some of the holes in Homer's Iliad, where it starts off with Bellerophon sleeping in Athena's temple. In a dream, Pallas Athena mentions that Bellerophon is related to Poseidon and therefore has the ability to tame Pegasus; she gives him a charmed bridle for Pegasus and gives him sacrificial ritual instructions for Poseidon. Like the Iliad, he also kills the Chimera, Solymi and Amazons. In this ode, Bellerophon is said to be a descendent of a God and performs his great deeds. It doesn't go into detail about his childhood and ends the ode with a vague death with "I shall pass over his death in silence" (Pindar, Olympian Ode). Pindar's Isthimian Ode however, continues the story and implies that his death happened because of his high ambition to fly to Olympus, while he was flying to Olympus, Pegasus threw him off to his doom. It is highly probable that this ending and Pegasus was added to fix the vague deaths and instructions from the Gods in Homer's Illiad. The ode does add one more heroic deed to Bellerophon's accomplishments by making him the mortal who rides Pegasus, and although his death is not mysterious like the Iliad version, it does highlight his heroic impulse more leading him to his hubris and loss of favour from the Gods.
Apollodorus also writes his take on the Bellerophon myth in The Library of Greek Mythology with not many significant changes but the reason for him staying with Proitos at the beginning of the story. Apparently, Bellerophon unintentionally killed his brother and thus, went to Proitos to be purified of his spiritual pollution. Again, similar to Homer's version Proitos's wife ends up falling in love with him and framing him when he rejects her, which leads to him performing the same quests mentioned in Homer's Iliad for her father, Iobates. Bellerophon's death is never mentioned in Apollodorus's retelling as it ends with him becoming king. Apollodorus describes his rewards given by Iobates, "he gave him his daughter Philonoe and when he died bequeathed to him the kingdom" (Apollodorus, The Library of Greek Mythology). This does give Bellerophon a happier ending, but this is not necessarily a good thing as it does not mention Bellerophon's heroic impulse and implies a smooth reincorporation into the domestic sphere; therefore compared to the Iliad, Apollodorus does not portray Bellerophon with as much Greek heroism.
On the other hand, the accidental killing of his own brother does imply about Bellerophon having an uncontrollable temper or maybe he was careless as usually Greek heroes are not know to be too bright. An example of Greek heroic foolishness would be Patroclus when he borrows Achilles's armour to fight and failed to control his rage, which lead to his death at Hector's hands (Morford, Lenardon and Sham 498). Interestingly, Apollodorus has Proitos write a letter for Bellerophon to take to Iobates while Homer has Bellerophon's message as a tablet with wicked symbols instead. Another addition to point out is that Apollodorus gives the king of Lycia and his daughter (Anteia's or Stheneboea's sister) the name Iobates and Philonoe respectively. This does point out the time difference between when the two myths were written and how new authors try to fill up the vague holes of older ones whether it's to help the myth make more sense or relating it to some sort of history.
The Astronomica by Pseudo-Hyginus is the latest of the four sources, it tells the tale of Bellerophon and explains how Pegasus became the constellation. Generally, the story is the same except that it implies that Bellerophon has tamed Pegasus before the events of Antia's accusation and there is no mentioning of Bellerophon's triumph over the Amazons, Solymi and Lycian ambush. The ending itself is the same except that after Pegasus shakes off Bellerophon, it becomes a constellation instead of staying with the Olympians. It is possible that Pseudo-Hyginus wanted to merely summarize Bellerophon's story, which could explain why only his conquest of the Chimera is written and after that he already tries to fly to Olympus. As mentioned earlier, Astronimica is a book about the origins of the constellations, so it is safe to say Pseudo-Hyginus did not feel that going into detail about Bellerophon was important and merely explained how Pegasus became a constellation.
Bellerophon does not only seem to fit relatively well for Raglan's stereotyping, but also his tragic end is explained by Dumezil's theory. By attempting to fly up to Olympus on Pegasus, Bellerophon goes against the ruler / priest society; the gods who rule among mortals are being disrespected by Bellerophon's hubris and foolish attempt to reach them without permission even though the gods helped him in his quests. To an extent, Bellerophon has also disrespected the warrior and producer class because the gods rule over all in Greece and by trying to go to Olympus, the home of the gods without any divine consent is disrespecting them all. Hence, Bellerophon went against all three societies and was punished for it.
On the other hand, Bellerophon can also be argued that he is not a greek hero, one of the biggest arguments would be he does not actually go on an otherworldly journey. Though it is true, there are other heroes that do not go on these journeys such as Clytemnestra and Cleomedes. The reason why they are still considered heroes is because of the unspeakable and / or stupid acts they commit (or attempt to) that make them stand out among mortals. In fact, an otherworldly journey is actually under the category of unspeakable deeds they are able to perform. The Greek hero's journey if any is not something any normal mortal can go through and survive, which is why these journeys themselves are another form of larger than life acts Greek heroes are capable of.
So with these four sources of Bellerophon myths, how does Bellerophon fit into being a Greek hero? His childhood is never told, he accidentally kills his own brother which is pretty extreme, he has divine allies, he kills the Chimera, Amazons, and no mortal men seems to be able to stand against him, he marries a princess, he rules over the kingdom, has trouble settling down which leads him to his hubris and to abuse his relationship with the Gods and attempts a clearly stupid feat, and meets a tragic death and / or some mysterious death. Bellerophon clearly does follow most of the steps of any Greek hero with the exception that he lacks going through an otherworldly journey. Although that does make him less of a hero compared to Odysseus and Heracles, Bellerophon's good deeds of conquering the Chimera, the Solymi and the Amazons for the benefit of Lycia, his tragic or mysterious end, and his heroic impulse denying him the leisure of settling down into the domestic sphere make up for it putting him into the category of Greek heroes.