Could the experience of inflatable structures in boats be applied to large aircraft design? Inflatable structures in boats have proved extremely successful and versatile since they began to find widespread use in the 1930s.

I'm aware that experiments have been made with inflatable wings - very light, delicate structures mainly on human-powered and unmanned aerial vehicles.

On boats, they're used differently: in some cases they form the main structure of the craft itself, in some, the craft has a rigid hull and the rest of the structure is inflatable, but either way, the inflatable sections are very far from delicate.

Experience gained with multiple air cells in inflatable engineering means that it's now possible to build effectively rigid structures, that are extremely tough, reliable and resilient. Their behaviour in failure modes is well-known.

Many large aircraft have built-in air compressors - the engines - that could keep the inflatable structures correctly pressured at all stages of a flight. 

Materials science has produced enormous improvements in the longevity and resistance to chemicals/sunlight/temperature ranges of the rubber skins of inflatable structures.

What opportunities are there for designing parts of an aircraft as inflatable structures (perhaps winglets, or undercarriage doors)?

Is there any research or experimentation into more heavy-duty use of inflatable engineering in aircraft?
We now have experimental use of inflatables in space as well as for boats.

If an aircraft's engines are responsible for the inflation of any aerodynamic or structural part of the aircraft, however, one would have to answer what happens if the engines stop. A conventional aircraft whose engines fail at a sufficiently high altitude can glide for some time, giving the crew a chance to restart the engines; it may even be landed without engine power if necessary. This is a very bad time for the aircraft to lose aerodynamic performance or structural strength--for example, for the winglets to deflate just when the crew need the best possible glide slope.

Unlike boats or spacecraft, moreover, most conventional aircraft are intended to descend from altitudes of low pressure to altitudes of high pressure. If the amount of gas in an inflatable structure is held constant, one has to balance structural rigidity at low altitudes against the stress on the fabric from the much greater pressure difference at high altitudes. If the amount is not constant, it has to be replenished during descent. The engines cannot be relied on to do this, since the reason for descent may be engine failure. Gas could be supplied from high-pressure cylinders, but those can be heavy.

For an aircraft not intended to climb very high, these concerns are less severe.