The space X is said to be a Banach function space if there exists a functional (Vert cdot Vert _{X} : mathcal{M} to[0,infty]) satisfying the following properties: Let (f, g, f_{j} inmathcal{M}) ((j=1, 2, ldots)), then (a) (fin X) holds if and only if (Vert f Vert _{X}
Since C 0 1 is dense in L 2 , we have that for any function f ∈ L 2 , there exist a functional sequence f n ∈ C 0 1 , such that ∥ f n − f ∥ → 0, as n → ∞.
Then the zero solution of system (3.1) is stable if and only if there exist a functional V : R × C ( M ) → R and a continuous function u ( x ) with u ( x ) > 0 for | x | > 0 and u ( 0 ) = 0 such that the following conditions are satisfied: (1) V ( t, 0 ) = 0. (2) V ( t, ϕ ) ≥ u ( D.
If there exists a functional such that, then (2.13) should be.
Hence x ⊥ B y. Thus, there exists a functional x * ∈ S X * such that x* x) = 1 and x* y) = 0 (cf. [6]).
There exists a functional (F in C^{1}:X_{R}^{ghtarrowmathbf {R}^{1} ) such that langle Au,Lvrangle=bigllangle -DF u),vbigrrangle,quad forall u,v in X. (2.6).
The opposite modulatory effects in acute and chronic pain states suggest that there exists a functional switch for the SI cortex at different stages of pain disease, which is of great significance for the biological adaptation.
end{aligned} From the Hahn-Banach theorem there exists a functional f in (X^) such that Vert y_{p}-y_{q} Vert =biglvert f y_{p}-y_{q}) bigrvert.
Throughout this paper, we assume that: (i) There exists a functional (F in C^{1}:X_{R}^{ghtarrowmathbf {R}^{1} ) such that langle Au,Lvrangle=bigllangle -DF u),vbigrrangle,quad forall u,v in X. (2.6) (ii) The functional F is coercive, i.e. F u rightarrowinftyquad Leftrightarrowquad {|u|}_{X_{2}} rightarrowinfty.
A function u ∈ L 2 is said to be a strong solution of Problem B, if there exists a functional sequence { u n } ⊂ W, such that u n and L u n converge in L 2 to u and f, respectively.
