I live in Williamson County Texas. My legal property line is backed by a city owned "greenbelt" area that supports rainwater run off through our neighborhood. The city does a very poor job of maintaining this property. I have had to take care of a section of this property in order to keep the pest and varmit population at bay from my property. I have openly mowed and maintained this section of land for 4 years now. I am preparing to plant a few trees in as well. Last year I contacted the city and complained about the lack of area maintenance and then attempted to purchase the land from the city. I was simply blown off because "it is not in the city's best interest". Since the city refuses to deal with the situation, is there a law that allows me to take ownership of this section of property by means of continual maintenace and use? I am also considering extending my personal property fence to encompass this section. I know that the city won't even notice if I do. What are my rights here? Thank you for your time.

is there a law that allows me to take ownership of this section of property by means of continual maintenace and use?

Unless there is a very unusual law in Texas, no. You can't adversely possess land against the government.