Working on Android Push Notification application by following this tutorials
Here is the problems I'm facing the following erros as shown in images .
You will need at least version 22.1 of the Support library. Given that ActionBarActivity is working for you, but AppCompatActivity is not, you probably have an older version of the support library that doesn't have AppCompatActivity.
From here:
The major revision for v22.1 is the deprecation of ActionBarActivity. It initially served as a convenient way to bring the ActionBar to the Android 2.1 and above, and would act as a compatibility layer for any version of Android that already included it (starting with Android 3.0 Honeycomb). Taking its place will be AppCompatActivity. On the surface, the new class will offer all of the same functionality, but it's built on top of a new AppCompatDelegate that can be used to enable most of the same features without requiring developers to inherit from AppCompatActivity. This re-engineering effort is also serving as the base for a new AlertDialog class, so every part of the user interface can benefit from the extensive list of theming features that were demonstrated for Lollipop.
Go to your SDK manager to see what version of the support library you have installed:
If you have less than 22.1, download the latest version, and then replace the jar files in your project with the new ones.