We obtain the maximum principles for the first-order neutral functional differential equation where, and are linear continuous operators, and are positive operators, is the space of continuous functions, and is the space of essentially bounded functions defined on.
It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are a sufficient number of continuous linear functionals defined on every normed vector space to make the study of the dual space.
In the first approaches to the issue, the variational methods are applied to boundary value problems on bounded discrete intervals, which leads to the study of an energy functional defined on a finite-dimensional Banach space (see [4 9]).
Based on this fact, Bownik [31] (Theorem 2) constructed a surprising example of a linear functional defined on a dense subspace of H 1 ( R n ), which maps all ( 1, ∞, 0 ) -atoms into bounded scalars, but yet cannot extend to a bounded linear functional on the whole H 1 ( R n ).
Let be a real Banach space,, that is, is a continuously Fréchet differentiable functional defined on, and is said to satisfy the Palais-Smale condition (P-S condition), if any sequence for which is bounded and as possesses a convergent subsequence in.
Let be a convex and lower semicontinuous functional defined on.
For each, consider the functional defined on by.
Then the functional, defined on, satisfies the Palais-Smale condition.
Let X be a Banach space, f a functional defined on X.
Moreover, we establish a chain rule for (possibly nonsmooth) convex functionals defined on variable exponent spaces.
