We present a rich class of exact solutions which contains radiation-dominated and matter-dominated models for the early and late universe.They include a variable cosmological “constant” which is derived from a higher dimension and manifests itself in spacetime as an energy density for the vacuum.This is in agreement with observational data and is compatible with extensions of general relativity to string and membrane theory.Our solutions are also typified by a non-singular “big bounce” (as opposed to a singular big bang),where matter is created as in inflationary cosmology.