Thanks for the explanation. But can we accept this language with DPDA without e-transitions? I can see that in first case we can accept this input because stack becomes empty but in other case stack is not empty.
Please suggest if there is any way to accept the language L without making a e-transition to final state or making e-transition to make the stack empty.
Also, What is general idea to identify that a DPDA exists(possibly real time DPDA (without e-transitions)) by only seeing the grammar(Since this problem is undecidable). I am asking this from GATE perspective.