Sentence examples for be a functional defined from inspiring English sources

Let and be fixed, and let be a functional defined by (2.19).

Let E be a functional defined on ({mathcal{S}}) and taking values in R such that the following axioms hold.

Let (G_{2}:Vtimes V rightarrowmathbb{R^cup{0}) be a functional defined as follows: begin{aligned} G_{2} u,eta)= Vert u Vert _{V}^{2}-2 langle Jeta,urangle+ Vert eta Vert _{ V}^{2}, quadforall u, etain V. end{aligned} (2.2). see [29].

end{aligned} (3.4) For each (uin V), (w^{2}in A_{0}(t u)), let (tilde{F}:Vrightarrowmathbb{R}cup{+infty}) be a functional defined by begin{aligned} tilde{F} u)=tilde{F}_{0} u)+F_{1} u -langle f,u rangle, qquad tilde{F}_{0}(u)= int^{1} u -langlengle h(u)w^{2},ubigrrangle,dt,quad fin V^{ast}.

Let (G_{2}:Vtimes V rightarrowmathbb{R^cup{0}) be a functional defined as follows: begin{aligned} G_{2} u,eta)= Vert u Vert _{V}^{2}-2 langle Jeta,urangle+ Vert eta Vert _{ V}^{2}, quadforall u, etain V. end{aligned} (2.2)  .

Let (G_{1}:Mtimes V^{ast} rightarrowmathbb{R}cup{+infty}) be a functional defined as follows: begin{aligned} G_{1} u,Jeta)= Vert u Vert _{V}^{2}-2 langle Jeta,urangle+ Vert Jeta Vert _{ { V^{ast}}}^{2}+2F_{1}(u), end{aligned} (2.1) where (uin M), (etain V), (Jetain V^{ast}).

where is a functional defined in (3.14) and.

has a unique solution, and satisfies and (3.4), where is a functional defined in (3.17).

Definition 2.3 Suppose that X is a Banach space and I : X → R is a functional defined on X.

If and satisfy (3.3), problem (3.16) has a unique solution, and satisfies and (3.4), where is a functional defined in (3.17).

By virtue of Theorem 2.1, there exists a unique such that and where is a functional defined implicity in (2.24) and. is just a unique solution of (2.12) and is exactly a unique solution of (2.11).