PDA that accepts if there is an "a" in the second half of the string
The PDA has three phases. In the first phase, it adds $A$ to the stack per character. It switches to the second phase nondeterministically. In the second phase, it removes $A$ per character. It switches to the third phase nondeterministically, during which it empties the stack. The PDA accepts if it encountered an $a$ during the second phase.