However, although many approaches have been shown to work in principle, there exists till now no general consensus regarding the best algorithm to use.
Although treatment with statins has been shown to reduce major cardiovascular events, there exists a significant unmet medical need for the treatment of cardiovascular disease.
To fulfill this, we use the Implicit Function Theorem to show that there exists a δ > 0 such that there are functions a 0 , a 0 ~ , a 1 and a 1 ~ which are continuous for 0 ≤ α - 1 < δ and a 0 ( 1 ) = a 0 ~ ( 1 ) = 0, a 1 ( 1 ) = a 1 ~ ( 1 ) = a.
Then, it follows from Lemma 3.1 that is a solution of (3.4), that is, (4.2). and suppose that this solution satisfies the condition This means that and to prove that is the recessive solution of (1.1), we need to show that there exists such that, that is, according to Lemma 3.1,.
To apply the Leray-Schauder degree to h σ, we need only to show that there exists a ball B R in C l d ( [ 0, T ] T ), whose radius R will be fixed later, such that θ ∉ h σ ( ∂ B R.
From our assumption, we have Let be the solution of (1.3) determined by Then there are eight cases to show that there exists an integer such that and.
If there exist, and such that Y k and Z k have the same sign, then will satisfy (23) if Y k = Z k = 0; or (23) will hold by taking if Y k ≠ 0. So it is sufficient to show there exist, and such that Y k and Z k have the same sign.
To verify (B.2), it is sufficient to show that there exists some neighborhood U and a non-negative (C^{2} -function V suC^{2} -function (xin E_{l}backslash U), LV isuchgathat.
end{aligned}In view of Theorem 3.1, and (3.11), to prove the proposition it suffices to show that there exists a constant (C>0) independent of (epsilon >0) such that for each (j=1,ldots,n) one has (|u_j|le Cr^{d_epsilon (i_{epsilon j})}).
Now, we need to show that there exists (M>0) such that frac{M}{Psi M Vert Omega Vertt Phi}>1, and such (M>0) exists if 1- Vert Omega Vert Phi>0.
It remains to show that there exists a minimizer x ∈ M of Φ | M. By Ekeland's variational principle [23], there exists a sequence { w j } ⊂ S + such that Ψ ( w j ) → c and Ψ ′ ( w j ) → 0 as j → ∞.
