Fredholm and Volterra integral equations with integrable singularities
The authors present a number of results on the existence of continuous nonnegative solutions \(y\) of the Fredholm equation \[ y(t) = \int_J k(t,s) f(s,y(s))\,ds, \quad t \in J, \tag{F} \] where either \(J = [0,1]\) or \(J = [0,\infty)\), and for the Volterra equation \[ y(t) = \int_0^t k(t,s) f(s,y(s))\,ds, \quad t \in [0,T]. \tag{V} \] Here all the functions take nonnegative (real) values. The function \(f\) is allowed to be nonlinear and even singular in both its arguments. A model example of an equation of type (F) is \[ y(t) = \int_0^1 k(t,s) {1 \over [y(s)]^\alpha}\,ds, \quad t \in [0,1], \] with \(\alpha > 0\). The proofs are based on Schauder's fixed point theorem. The uniqueness of solutions is not discussed.
scientific article; zbMATH DE number 2158047