Here's a brief from 3 Greek lexicons providing English meanings:
Bauer-Danker, Greek-English Lexicon of the NT (BDAG)
[BDAG] ἀπειθέω
• ἀπειθέω (opp. πείθομαι; s. ἀπείθεια) impf. ἠπείθουν; 1 aor. ἠπείθησα (for ἀπιθέω [Hom.] since Aeschyl., Pla.+) disobey, be disobedient (cp. PYadin 24a, 10 [restored] of refusal); in our lit. disobedience is always toward God, God’s ordinances, or revelation (like Eur., Or. 31; Pla., Leg. 741d; Lucian, Dial. Deor. 8, 1; SIG 736, 40 [92 BC] τὸν δὲ ἀπειθοῦντα ἢ ἀπρεπῶς ἀναστρεφόμενον εἰς τὸ θεῖον μαστιγούντω οἱ ἱεροί; Dt 1:26; 9:23; Josh 5:6; Is 36:5; 63:10; Bar 1:18f). W. dat. of pers. (Num 14:43 κυρίω) τῷ θεῷ (Diod. S. 5, 74, 4 ἀ. τοῖς θεοῖς; Hierocles 24, 473 τῷ θεῷ; Jos., Ant. 9, 249) Ro 11:30, cp. Pol 2:1. τῷ υἱῷ J 3:36.—W. dat. of thing (Diod. S. 5, 71, 5 τοῖς νόμοις) τῇ ἀληθείᾳ Ro 2:8 (Theoph. Ant. 1, 14 p. 92, 5). τῷ εὐαγγελίῳ 1 Pt 4:17. τῷ λόγῳ 2:8; 3:1. τοῖς εἰρημένοις 1 Cl 59:1. τοῖς ἐμοῖς ἐλέγχοις 57:4 (Pr 1:25).—Abs. (Dicaearchus fgm. 23 [Athen. 13 p. 603b] ἀπειθήσας=disobedient) of members of a synagogue at Corinth Ac 19:9. Of a part of Israel Ro 11:31. Of people of Judea 15:31. οἱ ἀπειθήσαντες Ἰουδαῖοι the disobedient (but see below, end) Judeans Ac 14:2. λαὸς ἀπειθῶν Ro 10:21; B 12:4 (both Is 65:2). οἱ ἀπειθοῦντες IMg 8:2; 1 Cl 58:1. Of gentiles οἱ ἀπειθήσαντες Hb 11:31.—Gener. 3:18; 1 Pt 3:20. In a number of pass. NRSV and REB, among others, with less probability render ἀ. ‘disbelieve’ or an equivalent.—DELG s.v. πείθομαι. M-M. TW.
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Liddell-Scott, Greek Lexicon (Abridged)
[LS] ἀπειθέω
ἀπειθέω, f. ήσω, to be disobedient, refuse compliance, Aesch.; c. dat. to disobey, Eur., Plat.
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Louw-Nida, Greek-English Lexicon of the NT
[LN] ἀπειθέω (a) disobey 36.23 (b) reject belief 31.107
NAS John 3:36 "He who believes (pisteuo) in the Son has eternal life; but he who does not obey (apeitheo) the Son shall not see life, but the wrath of God abides on him."
- Translators always seem to get stuck on this one
- It seems in John 3:36 they see the need to translate "apeitheo" as not believe due to the contrast with believes
- The first letter "a" in apeitheo is negating a word that means (I'll put some in a passive sense): to be convinced, persuaded, believe, obey, follow. So, we can see how this word can mean "not believe" or "not obey." So, we end up with translation choices, but as you can see from the above lexicons, the negated form of the word heavily trends to meaning disobedience.
- One of the issues in translating apeitheo as not believe is that there is another Greek word that clearly means "not believe": apistia - if you look at the word pisteuo above in J3:36 and note again what happens when there is an "a" placed at the beginning of the word, we now have a negated believe, or does not believe. So, why use apeitheo when apistia can be used to clearly state not believe, assuming not believe is what is being conveyed?
- I think one of the best sections of Scripture to deal with this word is the following. I'll try to clarify why:
NKJ Hebrews 3:7-19 Therefore, as the Holy Spirit says: "Today, if you will hear His voice, 8 Do not harden your hearts as in the rebellion, In the day of trial in the wilderness, 9 Where your fathers tested Me, tried Me, And saw My works forty years. 10 Therefore I was angry with that generation, And said,`They always go astray in their heart, And they have not known My ways. 11 So I swore in My wrath,`They shall not enter My rest.'" 12 Beware, brethren, lest there be in any of you an evil heart of unbelief (apistia) in departing from the living God; 13 but exhort one another daily, while it is called "Today," lest any of you be hardened through the deceitfulness of sin. 14 For we have become partakers of Christ if we hold the beginning of our confidence steadfast to the end, 15 while it is said: "Today, if you will hear His voice, Do not harden your hearts as in the rebellion." 16 For who, having heard, rebelled? Indeed, was it not all who came out of Egypt, led by Moses? 17 Now with whom was He angry forty years? Was it not with those who sinned, whose corpses fell in the wilderness? 18 And to whom did He swear that they would not enter His rest, but to those who did not obey (apeitheo)? 19 So we see that they could not enter in because of unbelief (apistia).
- You can see the context of rebellion & sin
- Sin is lawlessness - disobedience to law
- Note the use of apistia (unbelief) in 3:12
- Most importantly, note the use of apistia (unbelief) & apeitheo (disobey) in 3:19
- It's clear why these 2 different words are being translated as they are here - to translate them both as not believe (2 different words being translated the same) would not make a lot of sense
- Clearly the author seems to be telling us something - unbelief & disobedience are being paralleled
- In my view this brings up another point to substantiate this observation: When belief is studied comprehensively, we find that it is used at times in parallel with obedience:
NKJ Romans 10:16 But they have not all obeyed the gospel. For Isaiah says, "Lord, who has believed our report?"
At the end of all this, I have come to see the Greek lexical tools as being correct in this matter.
- I primarily use "not obey" when translating apeitheo.
- I have noted these parallels of faith/belief & obedience in several places in our Text.
- Thus it makes sense to also see the parallel in unbelief & disobedience.
- John 3:36 makes perfect sense to me to contrast belief & disobedience
- Belief in Christ is ultimately Obedience to Christ since He is The Christ
- Disobedience to Christ is ultimately not believing He is the Christ
You likely didn't ask for all this, but there you have it...