Let be a complete metric space, , and be selfmaps of such that is injective, continuous, and subsequentially convergent. Suppose that there exists such that for each in the orbit of , where is a Lebesgue integrable mapping which is summable, nonnegative, and such that for each . Then the authors show that
(ii) , and
(iii) is a fixed point of if and only if is -orbitally lower semicontinuous at .