By the construction, (W_{1}) is bounded, and the maximum of (lambda _{i}(s)), (i=1, ldots, p), decreases along the flow lines of (W_{1}).
Indeed, for a function (fin operatorname{CB}(Lambda,X)), (f(Lambda)) is bounded and (H_{mathcal{S}}(f)(Lambda)=H_{mathcal{S}}(f)(bigcup_{iin I} Lambda_{i})=bigcup_{iin I}f_{i}(f(Lambda))). Taking into account that the family of functions ((f_{i})_{iin I}) is bounded, we obtain the desired result.
Namely, (sigma_{mathrm{ev}}(p, J_{m})) is bounded and (mathcal{N}(lambda J_{m}-p)) is not a vector space; (sigma_{mathrm{ev}}(p, J_{m})) is empty and (mathcal{N}(lambda J_{m}- p)) is trivial; (sigma_{mathrm{ev}}(p, J_{m})) is bounded and (mathcal{N}(lambda J_{m} -p)) is unbounded/infinite, respectively.
We say the locally Lipschitz functional h satisfies the nonsmooth (PS) condition if any sequence ({x_{n}}) in X such that ({h(x_{n})}) is bounded and (lambda(x_{n})rightarrow0) possesses a convergent subsequence.
If (Dsubset X ) is a convex bounded and closed set, the continuous mapping (Lambda :Drightarrow D) is a σ-contraction, then Λ has at least one fixed point in D.
Let (lambda :[a,b]rightarrow mathbb{R}) be continuous function or the function of bounded variation, and (lambda (a neq lambda (b)).
Since (E lambda)) is analytic in the upper plane (operatorname{Im}lambda>0) and (f 0,lambda)) is bounded as (|lambda|rightarrow infty), it follows that the set of zeros of (E lambda)) is bounded and forms at most countable set having as zero the only possible limit point.
Then, if (A_{1}), (A_{2}) are both full column rank, the sequence ({(x^{k},lambda^{k})}) is bounded and converges to a point ((x^{infty},lambda^{infty})inmathcal{W}^).
Moreover, (sigma ( T ) =mathbb{C} -rho ( T ) ) is called the spectrum of T. (R_{lambda} ( T ) :=T_{lambda}^{-1}:= ( T-lambda I ) ^{-1}) resolvent operator exists, (R_{lambda} ( T ) ) is bounded, and.
As assumed in many references, such as [45], the activation function (f_{i}(cdot)) of neural network (1) is continuous, bounded, and there exist constants (lambda^_{i}) and (lambda^_{i}) such that lambda^_{i}leqfrac{f_{i}(a -f_{i}(b)}{a -f_{i}lambda^_{i}, qquad f_{i}(0)=0,quad a, bin mathbb{R}, aneq b, i=1,2,lambda^_{i}
