Sentence examples for be a continuous functional from inspiring English sources

Let (f:Xrightarrowmathbb{R}) be a continuous functional and (uin X).

Let X be a metric space and (f:Xrightarrow mathbb{R} ) be a continuous functional.

Definition 2.1 Let ( X, d ) be a metric space, let I : X → R be a continuous functional and u ∈ X.

Definition 2.2 Let ( X, d ) be a metric space, let I : X → R be a continuous functional and c ∈ R. We say that I satisfies ( P − S ) c, i.e., the Palais-Smale condition at level c, if every sequence { u n } in X with | d I | ( u n ) → 0 and I ( u n ) → c admits a strongly convergent subsequence.

Lemma 2 Let ( X, d ) be a metric space, let η : [ 0, + ∞ ) → [ 0, + ∞ ) be a continuous, nondecreasing, and subadditive function with η − 1 ( { 0 } ) = { 0 }, let φ : X → be a continuous functional, and let ⪯ be the partial order introduced by (1). Then for each x ∈ X, [ x, + ∞ ) and ( − ∞, x ] are closed.

Let (V t,phi):Itimes C_{H}rightarrow mathbb{R} ) be a continuous functional satisfying a local Lipschitz condition, (V t,0)=0 ), and wedges (W_{i}) such that (i) (W_{1}( Vert phi Vert )leq V t,phi leq W_{2}( Vert phi Vert )).   (ii) (V_{text{(3)}}^{prime} t,phi leq -W_{3}( Vert phi Vert )).

In this paper, we discuss the oscillations of the fractional order differential equation D a α x ( t ) + q ( t ) f ( x ( t ) ) = 0, t ∈ [ a, + ∞ ), a > 0, where q is a positive real-valued function and f is a continuous functional; D a α denotes the Riemann-Liouville differential operator of order α, 0 < α ≤ 1.

Next we prove that J is a continuous functional of (v_{cdot}in mathcal{U}).

(t u)) is a continuous functional with respect to u in (H^{1}(mathrm{R}^{3})). (t u rightarrow+infty) as (|u|rightarrow0).

Then the measure μ defines a stochastic process on ([0,T]), (s(t)), (0leq tleq T), and (f_{i}(s)) is a continuous functional of trajectories of the process.

(3) (t u)) is a continuous functional with respect to u in (H^{1}(mathrm{R}^{3})).   (4) (t u rightarrow+infty) as (|u|rightarrow0).  .