POSITIVE SOLUTIONS OF SCALAR INTEGRAL EQUATIONS

Under general conditions solutions are shared by the integral equations x(t) = p(t) -integral(t)(0) A(t - s) f (s, x(s))ds and x(t) = p(t) - integral(t)(0) R(t - s)p(s)ds + integral(t)(0) R(t - s)[x(s) - f(s,x(s))/J]ds where J is an arbitrary positive constant and R(t) is the resolvent of J A(t). A classical problem is to prove that there is a positive solution when A(t) satisfies certain conditions including A(t) > 0 and when f (t, x) > 0 for x > 0. This requires p(t) > 0 and often p(t) - integral(t)(0) R(t - s)p(s)ds > 0. We offer a constructive method of manufacturing an infinite collection of functions, p, for which this holds. Any linear combination of any of these with positive coefficients will yield a function, p, also satisfying this property. We show that the property also holds for all functions near that set. We then give a brief treatment of the non-convolution case and offer a theorem stating that if there is a solution then it is positive.