A functional equation

Existence theorems for and the determination of continuous solutions, defined on the real axis $R$, of the functional equation $f(t)=A[t,f(at-b),f(at-c)]$, where $a$, $b$ and $c$ are real parameters, $A: R\times E\times E\to E$ is a continuous operator, and $E$ is a Banach space.