Here I have a trigger whenever I create a new account it should create new contact with the account name.
if account name is updated of existing account then it will check if related contact exist with new name if not exist then create new contact .


Trigger:
trigger AccountContactTrig on Account (after insert,after update) {
     Set<Id> accIds= new Set<Id>();
    If(Trigger.isinsert && Trigger.isafter){
        List<Contact> cont = new List <Contact>();
        for(Account acc : trigger.new){
            Contact c = new Contact();
            c.AccountId=acc.id;
            c.LastName = acc.name;
            cont.add(c);
        }
        insert cont; 
    }
    if(Trigger.isupdate && Trigger.isafter){
        Set<String> accnames= new Set<String>();
        For(Account acc:Trigger.new){
            Account oldacc= Trigger.oldmap.get(acc.id);
            if(acc.name!= oldacc.Name){
                accnames.add(acc.name);
            }
        }
        Map<Id,Account> mapaccounts = new Map<Id,Account>([Select Name,(Select id, Lastname from contacts where Lastname in: accnames) from Account where id in:Trigger.new]);
        List<Contact> cont = new List <Contact>();
        if(mapaccounts.size()>0){
        for(Account ac: Trigger.new){
            if(Ac.Name != trigger.OldMap.get(Ac.Id).Name){               
            if(mapaccounts.containsKey(ac.id)){
                if(mapaccounts.get(ac.id).Contacts.size()==0){
                        Contact c = new Contact();
                        c.AccountId=ac.id;
                        c.LastName = ac.name;
                        cont.add(c);
                }
            }
        }
         insert cont;
    }
        }
    }
}