Look at the below code I have just changed the print statement with 'single quote' to stop the variable interpolation to use the variable as it is. In your code perl looks for scalar variable $FILE and $STRING.

use strict;
use warnings;

sub create_db_install_val(){
my $File_name=$ENV{'HOME'}."/DB_INSTALL_VALIDATION.sh";
print "\n Shell name to validate DB Installation Completion is $File_n
+ame\n";
open (FILE,">$File_name") or die "cannot open $File_name file";
print FILE "#!/bin/sh\n";
print FILE "FILE=\"/scratch/pratranj/DB_11.2.0.1.0/config/setup.log\"\
+n";
print FILE "STRING=\"Completed creating database instance\"\n";
print FILE "EXIT=1\n";
print FILE "while : ;do\n";
print FILE '    [[ -f "$FILE" ]] && grep -q "$STRING" "$FILE" && echo 
+"DB Installation is completed" && break',"\n";
print FILE "sleep 5\n";
print FILE "done\n";
close(FILE);
}
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