NOTE: This is a condensed excerpt from my forthcoming (2014) commentary on Philippians.
In conjunction with the Philippians’ prayers Paul envisions his perseverance enabled through the help of the Spirit of Jesus Christ. The rare Greek word translated help (epichorēgia) emphasizes the generous nature of the assistance provided, which is apt considering that the help provided is the Spirit of Jesus Christ. Paul uses similar language in Galatians 3:5 when he describes God as “he who supplies [epichorēgōn] the Spirit to you.” For Paul the gift of the Spirit was the preeminent blessing of salvation through Christ (see Gal 3:1–5:26). The Spirit is sent out by Jesus Christ to mediate his presence in the life of the believer (Gal 3:1-7). Although as a believer Paul already has the Spirit, he knows that persevering in faith to the end requires fresh supplies of the Spirit like those mentioned in Acts 4:31.
So what then is the relationship between the prayers of the Philippians and the help of the Spirit of Jesus Christ implied by the grammar? Together they are the instruments that God will use to complete his work of salvation in Paul so that on the day of Christ Jesus (1:6) Paul will be pure and blameless (1:9-11). Furthermore, God uses the prayers of the Philippians to provide fresh supplies of the Spirit to Paul to ensure his perseverance to the end. All three persons of the Trinity are involved—the Father answers the prayers of his people for fresh experiences of the Spirit of his Son Jesus Christ.
Paul did not allow his firm belief in the sovereignty of God to produce an apathy that assumes it does not matter how one lives. Instead his conviction that God was at work in and through him motivated him to persevere in his devotion to magnify Christ. But this devotion was not an individualistic faith that had no use for other believers. God uses the prayers of other believers as the conduit of God’s Spirit to empower his perseverance in the faith. What a motivation to pray for others! What a privilege that God chooses to use the prayers of his people as a means of supplying his people with fresh measures of the Spirit of Christ to enable our perseverance in the faith to the end!
 The Greek noun epichorēgia occurs just one other place in the NT (Eph 4:16). The cognate verb epichorēgeō is slightly more common, occurring five times in the NT (2 Cor 9:10; Gal 3:5; Col 2:19; 2 Pet 1:5, 11) and once in the LXX (Sir 25:22). In Greco-Roman literature it sometimes referred to generous public service (cf. BDAG), and has been found in the papyri to refer to a husband providing for his wife (cf. LSJM).
 Thus the genitive pneumatos (“Spirit”) here is objective – that is, it is the Spirit Himself who is the help given; cp. Fee, Philippians, 132-34; Bockmuehl, Philippians, 84; Silva, Philippians, 76; Fowl, Philippians, 46; Hansen, Philippians, 79-80. Others interpret the genitive as subjective, meaning that the Spirit helps or supplies assistance in various ways; see, e.g., John Young William Eadie, A Commentary on the Greek Text of the Epistle of Paul to the Philippians (Birmingham, Ala.: Solid Ground Christian Books, 2005), 44-45; Vincent, Philippians, 24; Beare, Philippians, 62; O’Brien, Philippians, 111-12; Hawthorne and Martin, Philippians, 50; Thurston and Ryan, Philippians, 63; Witherington, Philippians, 84-85. Lightfoot sees both, arguing that the Spirit of Jesus is “both the giver and the gift (Lightfoot, Philippians, 91). As for the genitive Iēsou Christou (“Jesus Christ”), it could have a variety of nuances here, including origin (“the Spirit that comes from Jesus Christ”; O’Brien, Philippians, 112; Fee, Philippians, 134-35; Bockmuehl, Philippians, 84; Witherington, Philippians, 84-85), possession (“the Spirit who belongs to Jesus Christ”; Eadie, Philippians, 45; Kennedy, “Philippians,” 427), or apposition/epexegetical (“the Spirit that is Jesus Christ”; Reumann, Philippians, 211-12; Hansen, Philippians, 80).
 Helpfully noted by Fee, Philippians, 138.
 For a thoughtful discussion of perseverance, see Thomas R. Schreiner and Ardel B. Caneday, The Race Set before Us: A Biblical Theology of Perseverance & Assurance (Downers Grove: IVP, 2001).