Linear and nonlinear integral functional on the space of continuous vector functions

The present article is devoted to the study of a nonlinear integral functional of the form $F\left(u\right)={\int }_{\Omega }f(s,u(s\left)\right)ds$, where $\Omega$ is a closed bounded set in $R^n$, and the generating function $f:\Omega \times X\to R$ (where $X$ is real separable Banach space) satisfies Caratheodory conditions.

We study the action and boundedness of the functional $F$ on the space $C\left(X\right)$ of continuous vector functions $u:\Omega \to X$ and on the space $L_1\left(X\right)$ of essentially bounded vector functions (with natural norms).

The main results of the article are: 1) the equivalence of the action and boundedness of the functional $F$ on the spaces $C\left(X\right)$ and $L_1\left(X\right)$; 2) equivalence, for these spaces, of the numerical characteristic of the functional in the form of the supremum of the norm of the functional values on a closed ball; 3) expressing this numerical characteristic in terms of the function  that generates the functional.

Moreover, to extend the properties of the functional from $C\left(X\right)$ to $L_1\left(X\right),$ we essentially use the results of I.V. Shragin on the study of the Nemytskii operator and its generating function, as well as his ideas and methods based on the consistent proof of special auxiliary statements that use, in particular, continuous and measurable choice theorems.

The results thus obtained for the functional $F$ are specified for the case of a linear integral functional on spaces of Banach-valued functions (when $f(s,x)=a\left(s\right)\left[x\right]$ for some function $a:\Omega \to X^{*}$), and in particular, it is established that the norm of this functional on the spaces $C\left(X\right)$ and $L_1\left(X\right)$ is equal to ${\int }_{\Omega }\Vert a\left(s\right){\Vert }_{(}〖X{〗}^{*})ds$.