more like Vertcoin .

POS might be a good solution, but not with the stake based on amount of coins,  but based on a fixed amount of coins for keeping wallet synched. This way people get rewarded a certain amount of coins regardless of their wallet balance.
We call it Proof of Wallet Stake (POWS)

We're basically looking for a way to have the network continue to support transactions once all coins are mined. POS is a way to do this, but it stimulates people to not spend their coins.

Proof of Take (POT)
I personally was thinking of doing the opposite of POS, where coins deteriorate over time if they are stored and not used. Part of the stored unused coins will be redistributed over the miners in this case. When a wallet has been offline for a long time and comes online again it can be punished with a certain deterioration of funds (1% / week up to maximum of 10% or something). In this scenario, people can also prevent deterioration by synching the wallet at least once a week.

This has the following benefits:
- Transactions can be free: This would make it possible to replace the fee on transactions with a fee on keeping the coins offline.
- Decrease only for not-contributing part: Active people are in the clear, but stored funds will decrease over time.
- No inflation: The coins get redistributed, but the total amount does not have to increase.

I would be more motivated to be part of the network in such a case!

Probably a lot of things to say against it, but idea is still in early stage.