Integro-differential equations of mixed type with discontinuous terms in Banach spaces (Q1881617)

Let \(E\) be an ordered Banach space (with order generated by a cone). Moreover, let \(J= [t_0, t_0+ a]\), \(x_0\in E\) and let \(T\) and \(S\) be Volterra and Hammerstein integral operators defined by the formulas \[ \begin{aligned} (Tu)(t)&= \int^t_{t_0} k(t, s)u(s)\,ds,\\ (Su)(t)&= \int^{t_0+ a}_{t_0} h(t,s)g(s, u(s))\,ds.\end{aligned} \] The authors study the following integro-differential equation \[ u'(t)= f(t, u(t), (Tu)(t),(Su)(t)),\quad u(t_0)= x_0.\tag{1} \] Here \(f: J\times E\times E\times E\to E\) is not assumed to be continuous. Under some assumptions (rather very strong) it is shown that problem (1) has extremal solutions.