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After updating to Lollipop I can not connect to the Internet via my local wifi. I find the network and have a good signal. Receiving the IP address via a DHCP server, but can then not connect to Gateway. Other devices can connect to the same network
Welcome to the community.
Does this only happen with your home WiFi? Are you able to connect to any other networks with your phone?
Can you please try removing the network (Tap and hold on the network, then choose remove) and reconnect to it?
I am having the same issue on my z2 tablet, good connection to the WiFi but no Internet access despite my Xperia z5 premium running the sky go app flawlessly and my wife's ipad and iPhone both using the Internet during the time my z2 can't.
Can you try the process above too? If that fails try the phone in safe mode and see if the same issue is occurring. If it is, run a software repair.