Solvability of Urysohn integral equation. (Q1412462)

The authors extend the main result of \textit{J. Banaś} [J. Aust. Math. Soc., Ser. A, 46, 61--68 (1989; Zbl 0666.45008)] on the existence of integrable and monotonic solutions of the Urysohn integral equation to the nonlinear Urysohn integral equation with deviating argument of the form: \[ x(t)=g(t)+\int_0^1u\bigl(t,s,x(\phi(s))\bigr)\,ds, \quad t\in(0,1). \] It is assumed that: (i) \(g\in L^1\) is a.e. nondecreasing; (ii) \(u(t,s,x)\) satisfies the Carathéodory conditions; (iii) \(u(t,s,x)\) is a.e. nondecreasing in \(t\) for all \(x\) and almost all \(s\); (iv) the Urysohn operator \((Ux)(t)=\int_0^1u(t,s,x(s))ds\) maps \(L^1\) continuously into itself; (v) \(\phi\colon(0,1)\to(0,1)\) is increasing and absolutely continuous with \(\phi'(t)\geq M>0\) a.e.; (vi) \(| u(t,s,x)| \leq k(t,s)(a(t)+b| x| )\), where \(a\in L^1\), \(b\geq0\) and \(k\colon(0,1)\times(0,1)\to[0,+\infty)\) is measurable and such that the operator \((Ky)(t)=\int_0^1k(t,s)y(s)ds\) maps \(L^1\) continuously into itself with \(\| K\| <\frac{M}{b}\). With these assumptions it is proved that the given equation has an a.e. nondecreasing, integrable solution. The proof of the theorem is based on the classical Darbo fixed point theorem. Solutions of this equation in the space \(C_p\) have been investigated [e.g. in \textit{J. Banaś}, Rend. Circ. Mat. Palermo, II. Ser. 35, 82--89 (1986; Zbl 0625.45013)].