You really need to define "influence". Yes I think he was influenced but not the way you are trying to perceive it.

I think Jon, being a fan of rock and roll, has had many influences in his early days who have helped define the artist he has become. I don't think there has been one artist who has solely influenced him per say.

Jon's obviously a self admitting fan of the Stones. Not only has he said it but he's covered a bunch of songs throughout his career while doing his Jagger moves. Jon has also stated that they are the only band he would open for and actually did. Finally, Jon met them early on during his early recording days before Bon Jovi.

Jon is also a fan of Elvis, Springsteen, Southside Johnny, The Animals, Leonard Cohen, Frank Sinatra, and many more. I'm sure they all played a role somehow within so if that's how you define "influence" then sure he was, just like all the others.

Jon's always been hell bent on being original and being true to who he is. If I hear a Bon Jovi song, it doesn't scream Stones song to me so I don't hear the influence in the sound. Sure there a few songs that sound like they could have been influenced from a song by the Stones but Bon Jovi has been known to do that with other artists as well too.

Listen to Jon's little speech before singing "Imagine" in '95. He talks about his definition of influences and how it works.