The online installers are signed using a sha256 certificate which is only supported on Windows XP with Service Pack 3. If you don't have Windows XP SP3 you can try using the offline installer available under 'Still having problems' section on Installation problems | Flash Player | Windows 7 and earlier page.
We recommend you to move to a higher OS version. If you are not on XP service pack 3 atleast install the same.
Hi mingl30342950 ,
While you have been successful in installing Flash Player using the offline installer, I'd like to investigate why you were getting the 'Application Initialization Error' using the online installer. Can you please upload the Adobe_ADM.log and Adobe_GDE.log files, saved at C:\Documents and Settings\<YourUserID>\Local Settings\Temp\ to cloud.acrobat.com/send using the instructions at How to share a file using Adobe Document Cloud? LocalSettings is a hidden directory. If you do no see it, please refer to Microsoft Windows XP - Display hidden files and folders to display hidden directories.
I located all hidden Local Settings\Temp\ directory. However, I can't find both files you mentioned.