The following is an excerpt from Fresh Food from Small Spaces: The Square-Inch Gardener's Guide to Year-Round Growing, Fermenting, and Sprouting by R. J. Ruppenthal. It has been adapted for the Web.
A central problem for many of us who garden in the city is a lack of light. This is covered extensively in the vegetable chapters. I mention it again here only to note the problem that shade poses for growing fruit. In short, most fruit trees require nearly a full day of sunlight in order to set and ripen fruit. So what can you do if your only available growing space gets only a few hours of light per day? Here I have two pieces of advice: First, if you are gardening in the southern United States or anywhere with hot weather, then this small bit of sun might be enough, since some trees prefer partial shade in hot summers and can even get burned in full sun. So be courageous and try planting something there. Second, there are a few berries that are proven to be good shade producers, so you can try one of these even in partial shade conditions. In their natural environment, as lower-growing shrubs in woodland areas, many berry plants are used to partial shade. Blackberries, for example, can set nearly a full crop on a few hours of sunlight per day, and other, less common types of berries are solid shade producers as well.