Using array:
#!/bin/sh
DATE=`date +%Y%m%d -d "1 days ago"`
DESTINATION="/Network/Storage/Backup"
LOG="/Network/Storage/Backup"
BACKUPTIMES=(_2230 _2250) #Declare an array
APPLICATION="DATADEV TESTREV PRETEST"
for C in $APPLICATION
do
cd $DESTINATION/$C/
success=false
for BACKUPTIME in ${BACKUPTIMES[@]}; do #Iterate over the array
test -f $C$DATE$BACKUPTIME.zip && success=true
done
if $success
then
echo "Backup SUCCESSFUL" >>$LOG/output.txt
else
echo "Backup FAILED" >>$LOG/output.txt
fi
done
Note: I have done minimal modification to your code, for easier understanding. There is still scope for optimization.