Sentence examples for functional on with from inspiring English sources

Then it is easy to check that is a nonnegative continuous concave functional on with for and is completely continuous.

Suppose that is a completely continuous operator and is a nonnegative continuous concave functional on with for all.

If and are increasing, nonnegative, continuous functionals on, let be a nonnegative continuous functional on with such that for some positive constants and, (2.33).

Let and be increasing, nonnegative continuous functionals on, and let be a nonnegative continuous functional on with such that, for some and, (3.1).

If and are increasing nonnegative continuous functional on, and let be a nonnegative continuous functional on with such that, for some and, (31).

It is easy to verify that α is a nonnegative continuous concave functional on K with α(x) < ||x|| for all x ∈ K.

Let X be a Banach space with a direct sum decomposition (X=X_{1}oplus X_{2}) with (k=operatorname{dim} X_{2} < infty) and let φ be a (C^{1}) functional on X with (varphi(mathbf{0})=0), which satisfies (PS) condition.

Let K be a cone in a real Banach space E, (A bar{K}_{c}rightarrowbar{K}_{c}) be completely continuous and β be a nonnegative continuous concave functional on K with (beta(x leq|x|) ((forall xinbar{K}_{c})).

Let A : P ¯ r → P ¯ r be a completely continuous operator and let Ψ be a nonnegative continuous concave functional on P with Ψ ( u ) ≤ ∥ u ∥ for all u ∈ P ¯ r.

Moreover, the standard argument yields that I is (C^{1}) functional on X with begin{aligned} I' u triangleq&-operatorname{div}bigl(| nabla u|^{p-2}nabla ubI' u triangleq&-operatorname{div}bigl

Let ψ and φ be increasing non-negative, continuous functionals on P, and ω be a non-negative continuous functional on P with (omega(0)=0), such that, for some (r_{3}>0) and (M>0), (varphi u leqomega u leqpsi (u)), and (|u|leq Mvarphi u)), for all (uinoverline{P varphi,r_{3})}), where (P varphi,r_{3})={uin P:varphi u)< r_{3}}).