Question

Is it possible that a user using android application can like that application page created in Facebook from application itself?

Answer 1

Yes.

Just create a Layout with a WebView

Like

<?xml version="1.0" encoding="utf-8"?>
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
android:layout_width="fill_parent"
android:layout_height="fill_parent"
android:orientation="vertical">

<WebView 
    android:id="@+id/webview"
    android:layout_width="fill_parent"
    android:layout_height="fill_parent"
/>

</LinearLayout>

Create a activity

Like.class

WebView webview;
Then add the following at the end of the onCreate() method:
webview = (WebView) findViewById(R.id.webview);
webview.getSettings().setJavaScriptEnabled(true);
webview.loadUrl(YOUR_FB_LIKE_URL);