I have written (egrapsa). Not epistolary aorist, but refers to 1Jo 5:1-12 of this Epistle as in 1Jo 2:26 to the preceding verses.
That ye may know (hina eidēte). Purpose clause with hina and the second perfect active subjunctive of oida, to know with settled intuitive knowledge. He wishes them to have eternal life in Christ (Joh 20:31) and to know that they have it, but not with flippant superficiality (1Jo 2:3.).
Unto you that believe on (tois pisteuousin eis). Dative of the articular present active participle of pisteuō and eis as in 1Jo 5:10. For this use of onoma (name) with pisteuō see 1Jo 3:23; Joh 2:23.