I met this problem today, and fix it with bellowed steps.
1, Check the log file /var/log/mysqld.log
tail -f /var/log/mysqld.log
2017-03-14T07:06:53.374603Z 0 [ERROR] /usr/sbin/mysqld: Can't create/write to file '/var/run/mysqld/mysqld.pid' (Errcode: 2 - No such file or directory)
2017-03-14T07:06:53.374614Z 0 [ERROR] Can't start server: can't create PID file: No such file or directory
The log says that there isn't a file or directory /var/run/mysqld/mysqld.pid
2, Create the directory /var/run/mysqld
mkdir -p /var/run/mysqld/
3, Start the mysqld again service mysqld start
, but still fail, check the log again /var/log/mysqld.log
2017-03-14T07:14:22.967667Z 0 [ERROR] /usr/sbin/mysqld: Can't create/write to file '/var/run/mysqld/mysqld.pid' (Errcode: 13 - Permission denied)
2017-03-14T07:14:22.967678Z 0 [ERROR] Can't start server: can't create PID file: Permission denied
It saids permission denied.
4, Grant the permission to mysql
chown mysql.mysql /var/run/mysqld/
5, Restart the mysqld
# service mysqld restart
Restarting mysqld (via systemctl): [ OK ]