Progressive contractions, product contractions, quadratic integro-differential equations

Fixed point theory has been used very successfully to obtain properties of solutions of integral and integro-differential equations of the form $ x(t) = g(t) +\int^t_0 A(t-s) v(t, s, x(s))ds $ because under general conditions the integral term may map bounded sets of continuous functions into equi-continuous sets. But quadratic integral equations have a coefficient of the integral terms of the form $ f(t, x(t))\int^t_0 A(t-s)v(t, s, x(s))ds $ which destroys the compactness of the map. Investigators have resorted to deep solutions often involving measures of non-compactness and Darbo's fixed point theorem. In an effort to obtain some elementary approaches, in this paper we develop an apparently new technique by showing that by using progressive contractions we can show conditions under which the product of two contractions is a contraction. We focus on integro-differential equations and use direct fixed point mappings which convert Lipschitz conditions into progressive contraction conditions.