Monotonic positive solution of a nonlinear quadratic functional integral equation (Q984269)

Let \(a:[0,1]\to \mathbb R_+\) be an integrable and nondecreasing function, and let \(f,g: [0,1]\times \mathbb R_+\to\mathbb R_+\) be Carathéodory functions and there exist two functions \(a_1, a_2\in L_1\) and positive constants \(b_1, b_2\) such that \(g(t,x)\leq a_1(t)+b_1x\), \(f(t,x)\leq a_2(t)+b_2x\) for every \((t,x)\in [0,1]\times \mathbb R_+.\) The authors consider the following nonlinear integral equation: \[ x(t)=a(t)+g(t,x(\phi_1(t)))\int_0^tf(s, x(\phi_2(s)))ds,\quad t\in[0,1]. \] The existence of \(L^1\)-solutions to the equation is given under some further assumptions by using the Darbo fixed point theorem.