Fixed points for mixed increasing operators in ordered Banach spaces with an application. (Q1850641)

Let \(M=[x_0,y_0]\) be an order interval in an ordered Banach space \(X\), and let \(f:M\times M\to X\) be mixed increasing, i.e., \(f(\cdot,y)\) is monotonically increasing and \(f(x,\cdot)\) is monotonically decreasing. If \(x_0\leq f(x_0,y_0)\), \(y_0\geq f(y_0,x_0)\), and \(f\) satisfies a contraction type hypothesis (either with respect to the norm or with respect to the order in an operator-bounded sense), the existence of a unique fixed point of \(f\) (i.e., a solution of \(x^*=f(x^*,x^*)\)) is obtained by an iteration technique. The result is applied to a Volterra equation of the form \[ u(t)=x(t)+\int_0^tk(t,s)f(s,u(s),u(s))\,ds. \]