Proverbs 1:1 The proverbs of Solomon the son of David, king of Israel;
This verse provides both the title of the book and the author of the first and main section of the book. The author is King Solomon, himself the son of David. Both were the king of Israel. It may be grammatically an open issue of whether "king of Israel" is meant to refer to Solomon or David. I believe that the latter sense is the probable, but the ambiguity is of no consequence.
Some people have attempted to deny that Solomon was the author of the book. Their rationale is that Solomon turned from God. For it is written,
1 Kings 11:6-10
6And Solomon did evil in the sight of the LORD, and went not fully after the LORD, as did David his father. 7Then did Solomon build an high place for Chemosh, the abomination of Moab, in the hill that is before Jerusalem, and for Molech, the abomination of the children of Ammon. 8And likewise did he for all his strange wives, which burnt incense and sacrificed unto their gods. 9And the LORD was angry with Solomon, because his heart was turned from the LORD God of Israel, which had appeared unto him twice, 10And had commanded him concerning this thing, that he should not go after other gods: but he kept not that which the LORD commanded.
The reasoning continues that if Solomon turned away from God, then he was not saved, and that an unsaved person could not be an author of the Holy Scripture. This reasoning is weak in two regards:
1) Scripture indicates the punishment laid upon Solomon for his disobedience. That punishment was (a) taking the ten tribes away from Solomon's son's rule and (b) stirring up adversaries against Solomon.
2) Scripture concludes the account of Solomon in the book of Kings by stating:
1 Kings 11:41-42
41And the rest of the acts of Solomon, and all that he did, and his wisdom, are they not written in the book of the acts of Solomon? 42And the time that Solomon reigned in Jerusalem over all Israel was forty years.
And Scripture concludes the account of Solomon in the Chronicles of the Kings of Israel by stating:
2 Chronicals 9:29-31
29Now the rest of the acts of Solomon, first and last, are they not written in the book of Nathan the prophet, and in the prophecy of Ahijah the Shilonite, and in the visions of Iddo the seer against Jeroboam the son of Nebat? 30And Solomon reigned in Jerusalem over all Israel forty years. 31And Solomon slept with his fathers, and he was buried in the city of David his father: and Rehoboam his son reigned in his stead.
Neither of these accounts suggest that Solomon died in ignominy, though of course that is not conclusive.
3) We are not told whether Solomon repented of his sin or not. The book of Proverbs itself speaks of repentance (see verse 23 of the first chapter, for example). This supports the idea that Solomon himself repented, though it is not conclusive.
4) There is no rule that Scripture must be written by saved men. The word of God has come by the mouth of a donkey and by mouth of King Saul, the suicide. Since we believe in verbal, plenary inspiration, it is irrelevant whether the prophet is himself a righteous or wicked man.
משׁלי שׁלמה בן־דוד מלך ישׂראל׃
Παροιμίαι Σαλωμῶντος υἱοῦ Δαυιδ, ὃς ἐβασίλευσεν ἐν Ισραηλ,
parabolae Salomonis filii David regis Israhel