On the Number of Positive Solutions to a Class of Integral Equations

By using the complete discrimination system for polynomials, we study the number of positive solutions in [0, 1] to the integral equation , where , are continuous functions on [0, 1], is a positive integer. We prove the following results: when = 1, either there does not exist, or there exist infinitely many positive solutions in [0, 1]; when ≥ 2, there exist at least 1, at most positive solutions in [0, 1]. Necessary and sufficient conditions are derived for the cases: 1) = 1, there exist positive solutions; 2) ≥ 2, there exist exactly positive solutions. Our results generalize the existing results in the literature, and their usefulness is shown by examples presented in this paper