Continous Combustion Cycle ?

Yes, but it takes more energy to do it than it generates.
So it acheives nothing, unless you have money to burn.
Del

Plants could do that for you and even produce a return on your investment.

Yes, it is possible. Splitting CO₂ will take more energy than is released when the carbon is burnt again, though. The Laws of Thermodynamics say so.

Burning hydrocarbons ( containing carbon and hydrogen) but recovering and reducing the carbon is not the usual idea of a recycle, but it is an answer to the problem of carbon dioxide pollution causing global warming, as it is functionally equivalent to burning just hydrogen. However, it will consume the currently limited hydrocarbon supply at an even faster rate than now, leaving us with more carbon, of which we already have a lot as coal. Thus it appears to be a lesser choice, though this is my first exposure to this idea. If we had too much hydrocarbon fuels, instead of too little, it might be a good choice.