I just installed a new set of wipers and but it leaves the edge of the windshield unwiped when it goes back down. What can be the cause of this? Is there a way to fix this?
It's not uncommon for wiper blades that have seen a bit of use (or are approaching the end of their life) to grow less supple and to have difficulty flexing the rubber to "flip" directions - this can lead to the sort of poor clearance around the area of the change of direction that you are seeing.
With that in mind my first port of call would be to put fresh, decent quality blades on and see if the problem is resolved.
Another possibility is that the nature of the airflow over the windscreen results in the blades not being pressed particularly firmly at the vertical position - again resulting in the poor clearing performance at that point. If this is the cause then swapping to fresh blades won't cure it - unfortunately if this is the case there's very little you can do to affect it.
I have seen that wiper blades will have the wiping edge trailing the direction of motion. When the wiper arm gets to the end of range and changes direction the wiping edge will flip. This flip can make it look like arm has traversed a part of the wiper arc when in reality the wiping edge has not fully traveled over that section.
New wipers are much more likely to exhibit this behavior because the blade material is supple, flexible and wants to "stick to the window". Old blade material that has become hardened is much more likely to be so stiff that the blade will show next to none of the flipping behavior.
The reason that you may see a smeared look to the area at the end of the wiper arc where the flip occurs is that some blades are so flexible that they will lay over on their side (some may even have secondary wipe ridges). The side wipe is not a complete wipe as the wiping edge starts to flip resulting in the side lifting off the window.