Sentence examples for functional belongs from inspiring English sources

The new model is driven by a nonlocal elliptic equation and the cost functional belongs to a broad class of nonlocal functional integrals.

Let X be a separable and reflexive real Banach space; let ϕ : X → R be a coercive, sequentially weakly lower semicontinuous C 1 functional, belonging to ω X, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X ∗ ; let J : X → R be a C 1 functional with compact derivative.

Let E be a separable and reflexive real Banach space; let Φ : E → R be a coercive, sequentially weakly lower semicontinuous C 1 functional, belonging to W X, bounded on each bounded subset of E and whose derivative admits a continuous inverse on E ∗ ; J : E → R a C 1 functional with compact derivative.

Let X be a separable and reflexive real Banach space; let (Phi :Xrightarrowmathbb{R}) be a coercive, sequentially weakly lower semicontinuous (C^{1}) functional, belonging to (Gamma_{X}), bounded on each bounded subset of X and whose derivative admits a continuous inverse on (X^{ast}); (J Xrightarrowmathbb{R}) a (C^{1}) functional with compact derivative.

In particular, we obtain an estimate for the Fekete-Szegö functional for functions belonging to the class, distortion, growth estimates and covering theorems.

In the present investigation, we obtain a sharp estimate for the Fekete-Szegö functional for functions belonging to the class K s.

Let (X, H, μ) be the Wiener space; we define "twisted Sobolev spaces" associated to certain compact operators on H. Wiener functionals belonging to one of these spaces are termed "hypersmooth"; we prove a measure-theoretic analog of Sard′s theorem for them.

Let X be a separable and reflexive real Banach space; let (Phi :Xrightarrow mathbb{R}) be a coercive, sequentially weakly lower semicontinuous (mathrm{C}^{1} -functional, belonging to (mathcal{W} _{X}), bounded on each bounded subset of X and whose derivative admits a continuous inverse on (X^); (J:Xrightarrow mathbb{R}) be a (mathrm{C}^{1} -functional with compact derivative.

Whether the functional (J=varphi+psi) belongs to (C^{1} X, mathbb{R})) is an interesting problem.

First of all, let us claim that the functional Φ belongs to W X. It follows from the same argument as in the proof of Theorem 3.1 in [17].

By the condition (H1) and Sobolev inequalities, it is easy to see that the functional I belongs to C 1 ( X, R ) and maps bounded sets to bounded sets.