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We have proved that Ω i satisfies all the conditions in Lemma 3.1.

It is easy to check that ∑ i = 1 N a i F i satisfies all the conditions of Theorem 3.1 and EP ( ∑ i = 1 N a i F i ) = ⋂ i = 1 N EP ( F i ) = { 0 }.

It is easy to check that Ψ i satisfies all the conditions of Theorem 3.1 and EP ( ∑ i = 1 N a i Ψ i ) = ⋂ i = 1 N EP ( Ψ i ) = { 0 }.

The proof is to verify I satisfies all the conditions of Theorem 3.4.

We will verify that I satisfies all the conditions of Theorem B.

Thus the functional I satisfies all the conditions of Lemma 2.7, and then I has a critical point, and (1.1) has at least one solution.

Proof For any Γ ∈ Ω 2 and any i ∈ I, define the corresponding F i Γ : X i ⇉ X by F i Γ ( y i ) = { x ∈ X ∣ f i ( y i, x − i ) ≤ f i ( x ) }. Clearly, F Γ = ( F i Γ ) i ∈ I satisfies all the conditions of Theorem 2.1.

In this regard, any sequence, say { X k } in ⨂ i = 1 N Λ i satisfies all the conditions of the Arzela-Ascoli theorem on [0,T].

MAVEN will visualize all SNPs that satisfy all the conditions.

Assume l0 with 1 ≤ L < l0< H ≤ N is the point at which the inner loop finishes, i.e., μ ~ ( L, l 0 ) satisfies all the conditions from L to l0.

Since ∑ i = 1 ∞ ( 1 - b i ) λ < ∞ is satisfied, all the conditions of previous theorem are fulfilled.