Sentence examples for be a differentiable functional from inspiring English sources
Let be a differentiable functional on a convex set, which is called: (1) -convex [18] if (1.12).
Let X be a Hilbert space, (L Xrightarrow X) be a linear operator, and (Phi Xrightarrow R) be a differentiable functional.
Let be a differentiable functional with Fréchet derivative at satisfying the following: is sequentially continuous from the weak topology to the strong topology; is Lipschitz continuous with Lipschitz constant.
If I : E → R is a differentiable functional, u 0 is a critical point of I.
If (f:Hrightarrowmathbb{R}) is a differentiable functional, then we denote by ∇f the gradient of f.
If (f:Hrightarrowmathbb{R}) is a differentiable functional, then the gradient of f is denoted by ∇f.
Let H 1 and H 2 real Hilbert spaces and the letter I the identity mapping on H 1 or H 2. If f : H 1 → R is a differentiable (subdifferentiable) functional, then we denote by ∇f (∂f) the gradient (subdifferential) of f.
Let F be a Fréchet differentiable functional in a Banach space E and ∇F be the gradient of F, denote ( ∇ F ) − 1 0 = { x ∈ E : F ( x ) = min y ∈ E F ( y ) }. Baillon and Haddad [34] proved the following lemma.
It is clear that Ψ is a Gâteaux differentiable functional, sequentially weakly upper semi-continuous, whose Gâteaux derivative at the point (uin X) is the functional (Psi' u in X^), given by Psi' u v= int_{0}^{T}fbigl t,u(t bigr)v(t),mathrm{d}t- frac{overline{mu}}{overline{lambda}} sum_{j=1}^{n}I_{j} bigl u(t_{j} bigr)v(t_{j}) for every (vin X), and (Psi':X to X^) is a compact operator (see [33]).
It is well known that Ψ is a Gâteaux differentiable functional and sequentially weakly lower semicontinuous whose Gâteaux derivative at the point u ∈ X is the functional Ψ' u) ∈ X*, given by Ψ ′ ( u ) ( v ) = ∫ Ω ∑ i = 1 n F u i ( x, u 1 ( x ), …, u n ( x ) ) v i ( x ) d x. for every v = (v1,..., v n ) ∈ X, and Ψ': X → X* is a compact operator.
