C++ Initialize abstract base class protected member for all derived classes - c++11

Question

Is it possible to initialize abstract base class' protected member for all derived objects without writing the same initializer list in all derived class constructors? So that it acts like a static member for all derived objects. What I want is something like this (except it doesn't work) Read it like a pseudo code:
A.h
class A {
public:
A(string fn);
virtual ~A();
virtual void open_file() = 0;
protected:
string fileName;
};
A.cpp
A::A(string fn) : fileName {fn} {} //Initializer list is written only once here
A::~A() {}
B.h
class B : public A {
public:
B();
~B();
void open_file() const override;
};
B.cpp
B::B() {} //No initializer list for A::fileName here
void B::open_file() const {
ifstream SomeFile(fileName); //Use base class' protected member
..... //Do some stuff with open file
}
And imagine there's also a C derived class without an initializer list for A here that has a different overriden open_file function..
main.cpp
string fname = {"foo.txt"};
A* APtr = new B(fname); //This initializes A's fileName for all derived objects as "foo.txt"
Aptr->open_file(); //B opens foo.txt
fname = "bar.txt";
A* A2Ptr = new C(fname); //Now fileName that both B and C consume is changed to "bar.txt"
A2Ptr->open_file(); //C opens bar.txt
APtr->open_file(); //B now opens bar.txt

Answer 1

You deklared the constructor from B: B(); but you try to use it A* APtr = new B(fname); So the compiler can't find any matching constructor.
initialize abstract base class' protected member for all derived objects without writing the same initializer list in all derived class
Why not?
A.hpp
class A
{
public:
A( string fn = "") : fileName(fn){} // you can give an default path if prefered.
};
B.hpp
class B : public A
{
public:
B( string fn = "") : A( fn ) {} //c++11 feature: call base constructor.
}
Other possible solutions were:
global variable (dirty and unsafe! - Please don't do it.)
static variable in A. But you can only open one file the same time.
give A setter and getter for fileName. And use it that way:
main.cpp
B* b = new B();
b->setFileName("foo.txt");
b->openFile();