According to that video (posted December of 2013):
"the blockchain will easily reach fifty gigabytes within a year and two hundred fifty gigabytes within two years"
Actual blockchain size in December 2014? Twenty-four gigabytes. (less than half of the fifty gigabytes that the video said it would "easily reach")
Actual blockchain size six months later in June 2015? Thirty-four gigabytes.
It has grown 10 gigabytes in the past 6 months, do you really think it will "easily reach two hundred fifty gigabytes" in the next 6 months?
Now that we've established that the video is making predictions that are complete nonsense, lets move on to your concern that "in the near future only big companies will be mining it".
You ask "How to fix this problem in the future?"
What problem? What's wrong with only big companies mining it? That's exactly what it was intentionally designed for. From the very beginning it was specifically intended that eventually only big companies would be mining it. This isn't a "problem" that needs to be fixed. This is a "design feature" and the "intended behavior" of the system. If you didn't realize that, then the only thing that needs to be "fixed" is your understanding bitcoin.
Here is what Satoshi had to say about it himself (bold emphasis added by me):
- snip -
If the network becomes very large, like over 100,000 nodes
- snip -
most users should start running client-only software and only the specialist server farms keep running full network nodes, kind of like how the usenet network has consolidated.
- snip -
- snip -
I anticipate there will never be more than 100K nodes, probably less. It will reach an equilibrium where it's not worth it for more nodes to join in. The rest will be lightweight clients, which could be millions.
At equilibrium size, many nodes will be server farms with one or two network nodes that feed the rest of the farm over a LAN.
The current system where every user is a network node is not the intended configuration for large scale. That would be like every Usenet user runs their own NNTP server. The design supports letting users just be users. The more burden it is to run a node, the fewer nodes there will be. Those few nodes will be big server farms. The rest will be client nodes that only do transactions and don't generate.
- snip -