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

Abstract

By using the complete discrimination system for polynomials, we study the number of positive solutions in {\small $C[0,1]$} to the integral equation {\small $ϕ(x)=\int_0^1k(x,y)ϕ^n(y)dy$}, where {\small $k(x,y)=ϕ_1(x)ϕ_1(y)+ϕ_2(x)ϕ_2(y), ϕ_i(x)>0, ϕ_i(y)>0, 0<x,y<1, i=1,2,$} are continuous functions on {\small $[0,1]$}, {\small $n$} is a positive integer. We prove the following results: when {\small $n= 1$}, either there does not exist, or there exist infinitely many positive solutions in {\small $C[0,1]$}; when {\small $n\geq 2$}, there exist at least {\small 1}, at most {\small $n+1$} positive solutions in {\small $C[0,1]$}. Necessary and sufficient conditions are derived for the cases: 1) {\small $n= 1$}, there exist positive solutions; 2) {\small $n\geq 2$}, there exist exactly {\small $m(m\in \{1,2,...,n+1\})$} positive solutions. Our results generalize the existing results in the literature, and their usefulness is shown by examples presented in this paper.