On Volterra type generalization of monotonization method for nonlinear functional operator equations

Let $n,m,\ell,s\in\mathbb N$ be given numbers, $\Pi\subset\mathbb R^n$ be a set measurable by Lebesgue and $\mathcal{X,Z}$ be some Banach ideal spaces of functions measurable on $\Pi$. We consider a nonlinear operator equation of the form as follows:
\begin{equation} x=\theta+AF[x],\quad x\in\mathcal X^\ell, \tag{1} \end{equation}
where $A\colon\mathcal Z^m\to\mathcal X^\ell$ is bounded linear operator, $F\colon\mathcal X^\ell\to\mathcal Z^m$ is some operator. Equation (1) is a natural form of lumped and distributed parameter systems from a wide enough class. Formerly, by V. P. Polityukov it was suggested monotonization method for justification of solvability of equation (1) and obtaining pointwise estimations for solutions. The matter of this method consisted in that solvability of equation (1) was proved (besides other conditions) under following: I) operator $F$ allows some correction of the form $G=\lambda I$ to monotone operator $\mathcal F[x]=F[\theta+x]+G[x]$ such that II) $(I+A G)^{-1}A\geq0$ ($\lambda>0$, $I$ is identity operator). As our examples show, conditions I) and II) may be contradictory to each other, that narrows a sphere of application of the method. The main result of the paper is that for the case of operator $A$, possessing the Volterra property, which is natural for evolutionary equations, the requirement I) of ability to be monotonized can be replaced by the requirement of some upper and lower estimates for operator $F$ on some cone segment through linear operator $G$ and additional fixed element. We prove that for global solvability of a boundary value problem associated with a semilinear evolutionary equation it is sufficient that analogous boundary value problem associated with linear equation, derived from the original equation by estimating of a right-hand side on some cone segment, have a positive solution. The application of results obtained is illustrated by Goursat–Darboux system, Cauchy problem associated with wave equation and first boundary value problem associated with diffusion equation.