The past decade has seen a decline in prices of smartphones. At the same time, the use of paid smartphone applications as a new entertainment tool...

This question is about determinants of supply and demand and about the way in which changes in supply and demand change the equilibrium price and quantity in a market.

Because the price of smartphones has dropped, the demand for apps will have gone up.  This is because smartphones and apps are complementary goods.  That is, the two of them are used together.  If the price of phones drops, more people will buy phones and will therefore want apps.  The demand for apps will rise.  (We can also see a rise in demand from the simple fact that, according to your question, the use of apps has grown rapidly.)

The other factor that affects the market here is the fact that more app developers are entering the industry.  This means that the supply of apps will go up.  One thing that can change supply is the number of producers.  When more developers enter the market, there are more people producing apps.  When more people produce apps, supply increases.

So what does this do to the market for apps?  The answer is that we are not sure.  We know that the quantity of apps will increase, but we do not know about the price.  When demand increases, the quantity bought and sold goes up.  When supply increases, the same thing happens.  Thus, the changes in supply and demand will both push the quantity up, so the result is clear.  However, when demand increases, price increases but when supply increases, the price drops.  That means that our two factors are pulling in opposite directions.  We cannot know what will happen to the price of apps because we do not know if the increase in supply will outweigh the increase in demand or vice versa.