Preheat oven to 350 degrees.
Grease a 9×9 cake pan and set aside.
Combine Splenda and sugar in a large mixing bowl. Beat in butter with mixer until crumbly, about 2 minutes.
Add eggs, one at a time, beating well after each.
Beat in mashed bananas, applesauce and vanilla. Set aside.
In a second large mixing bowl, combine flour, baking soda, and salt.
Stir into butter mixture with large spoon until just blended.
Pour batter into greased pan and bake for 20 to 25 minutes or until golden brown on top and toothpick inserted comes out dry.
Set pan on wire rack for 10 minutes before removing cake.
To remove from pan, lay a wire cooling rack over the top of the pan and then turn upside down. Put 2nd wire rack on bottom of cake and then turn over so that the top of cake is now facing upwards. Let it sit on this rack until completely cool and then it is ready for the icing.
Spread icing (see below) and chill cake until it is ready to cut. Store in the fridge.